Sensitivity Analysis for Business, Technology, and Policymaking: Made Easy with Simulation Decomposition (SimDec)

Kozlova, Mariia (Ed.); Yeomans, Julian Scott (Ed.) Book Sensitivity Analysis for Business, Technology, and Policymaking: Made Easy with Simulation Decomposition (SimDec) Routledge Open Business and Economics Provided in Cooperation with: Taylor & Francis Group Suggested Citation: Kozlova, Mariia (Ed.); Yeomans, Julian Scott (Ed.) (2025) : Sensitivity Analysis for Business, Technology, and Policymaking: Made Easy with Simulation Decomposition (SimDec), Routledge Open Business and Economics, ISBN 978-1-040-12132-0, Routledge, Oxford, https://doi.org/10.4324/9781003453789 This Version is available at: https://hdl.handle.net/10419/312726 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode “This book on sensitivity analysis by Mariia Kozlova and Julian Scott Yeomans summarizes the state-of-the-art method for computational model analysis, the revolutionary Simulation Decomposition (SimDec). For readers like me working in the general areas of modelling, optimization, and machine learning, Ifind this book extremely useful because it essentially has everything about SimDec in one package, which makes it an effective tool to do global sensitivity analysis. The results are easy to understand, with colorful visualization, uncertainty quantification, and a wide spectrum of diverse applications. In addition, the open-source SimDec software with code packages in Python, R, Julia, and Matlab will revolutionize the ways for training next-generation scientists and practitioners to do the right kind of sensitivity analysis so as to figure out the most influential factors correctly and to support more informed decision-making.” Xin-She Yang, Reader at Middlesex University London, Fellow of the Institute of Mathematics and Its Applications (FIMA), UK “Simulation Decomposition is an incredibly powerful technique that allows researchers and engineers to identify key factors influencing the performance of a system and make targeted improvements. This book provides a simple and accessible guide to understand and apply the technique to practical problems. The accompanied open-source SimDec code is the key to enable the reader to quickly learn and apply the method.” Leifur Leifsson, Associate Professor, School of Aeronautics and Astronautics, Purdue University, USA “Simulation Decomposition methodology, a Monte Carlo–based computational algorithm, is quickly becoming a game changer in the world of engineering, industry, and finance. In this newly published book, Julian Scott Yeomans and Mariia Kozlova explore the importance of this method in providing an accurate and detailed holistic picture of the behavior of complex systems. The book delves into the real-life applications of Simulation Decomposition, highlighting its effectiveness in optimizing processes and improving product designs. Through detailed case studies and insights from industry experts, readers will gain a thorough understanding of this powerful methodology and its potential for transforming the way we approach complex systems.” Kambiz Vatan-Abadi, Chief Innovation Officer, CI Financial, Canada “SimDec is an ingenious way to tame the combinatorial complexity of systems’ behaviour in real-world decision-making. Its deceptively simple approach visually reveals the hidden factors that materially impact an uncertain outcome. This new method for determining sensitivity indices seems a hidden gem. Its transparency is invaluable to practitioners. Credit is also due to the authors for making an otherwise challenging topic most entertaining and accessible to practitioners. SimDec’s ease of use should make it the de facto standard for data analysis in industrial, engineering, and scientific environments.” Kalyan Moy Gupta PhD, Founder and Vice President of Research, Knexus Research, Washington, DC, USA “Black box functions with uncertain inputs are used to encode knowledge in many areas of science, engineering, and commerce. The challenge is to get that knowledge back out for the benefit of users. The customary approaches focus on subtle mathematics and expensive computations. This book presents SimDec, which produces interpretable graphical representations that support discussions and discovery.” Art Owen, Max H. Stein Professor of Statistics, Stanford University, USA “The fusion of sensitivity analysis with uncertainty analysis through SimDec, as presented in this book, marks a watershed moment for business professionals and GenAI developers alike. It’s a guidebook for those who dare to challenge the status quo, offering not just insights but a comprehensive toolkit for transformative decision-making. A testament to the power of interdisciplinary collaboration and open-source innovation in shaping the future of technology. Essential reading for leaders driving innovation in uncertain times.” Anferny Chen, CEO and Founder, Dataraction/DotsLive.Com, Canada “This book articulates the SimDec method for global sensitivity analysis by combining a novel visual uncertainty analysis approach with the discriminatory capabilities of a newly-created technique for calculating sensitivity indices. The real beauty of SimDec is that it can be straightforwardly applied to virtually any field of data analysis, irrespective of the mathematical sophistication of the user. Ihave been working with system dynamics, optimization, stochastic programming, and other analytical approaches for over three decades. One regret Inow have is that there was no SimDec procedure in existence at the time to support these activities.” Gordon Huang, Canada Research Chair and Professor of Environmental Systems Engineering, University of Regina, Canada Sensitivity Analysis for Business, Technology, and Policymaking SimDec is a revolution in decision-making support. SimDec “teases out” inherent cause-and-effect relationships and reveals the intricacy of relationships between sets of input and output variables. At its core, SimDec is an amalgamation of uncertainty and global sensitivity analysis with an innovative visualization technique. While straightforward and elegant, this novel approach significantly enhances the analytical capabilities of users by readily exposing seemingly, a priori, counterintuitive behaviours so that they can be readily understood by both technical specialists and non-technical users alike. This book is the first to articulate the ubiquitous applicability of SimDec and has been written by the leading proponents of the technique. The book provides the necessary background to fully understand the underlying approach and then demonstrates its applicability to a wide spectrum of fields, such as finance, entrepreneurship, energy, 3D manufacturing, geology, the environment, engineering, public policy, and even superconducting magnets. To facilitate as widespread adoption and penetration of SimDec as possible, all supporting computer codes are available, open-source, in Python, Julia, R, and Matlab. The innovative material will be of primary benefit to practitioners and researchers analyzing data from the social sciences, business, science, engineering, mathematics, and computing. Mariia Kozlova is an associate professor at LUT University Business School, Finland, and a visiting scholar at Stanford University, USA. Julian Scott Yeomans is a professor and the director for the MMAI and MBAN programs at the Schulich School of Business, York University, Canada. Routledge Open Business and Economics Routledge Open Business and Economics provides a platform for the open access publication of monographs and edited collections across the full breadth of these disciplines including accounting, finance, management, marketing and political economy. Reflecting our commitment to supporting open access publishing, this series provides a key repository for academic research in business and economics. Books in the series are published via the Gold Open Access model and are therefore available for free download and re-use according to the terms of Creative Commons licence. They can be accessed via the Routledge and Taylor & Francis website, as well as third party discovery sites such as the Directory of OAPEN Library, Open Access Books, PMC Bookshelf, and Google Books. Note that the other business and economics series at Routledge also all accept open access books for publication. Crowdsourcing in Management Research A New Tool for Scientific Inquiry Regina Lenart-Gansiniec Effective Financial Communication Key Concepts, Empirical Insights, and Implications for Practice Christian Pieter Hoffmann and Nadine Strauß Sensitivity Analysis for Business, Technology, and Policymaking Made Easy with Simulation Decomposition (SimDec) Edited by Mariia Kozlova and Julian Scott Yeomans Corporate Social Responsibility and the Supply Chain CSR Collaboration with Suppliers Monika Jedynak For more information about this series, please visit: Routledge Open Business and Economics - Book Series - Routledge & CRC Press Sensitivity Analysis for Business, Technology, and Policymaking Made Easy with Simulation Decomposition (SimDec) Edited by Mariia Kozlova and Julian Scott Yeomans First published 2025 by Routledge 4 Park Square, Milton Park, Abingdon, Oxon OX14 4RN and by Routledge 605 Third Avenue, New York, NY 10158 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2025 selection and editorial matter, Mariia Kozlova and Julian Scott Yeomans; individual chapters, the contributors The right of Mariia Kozlova and Julian Scott Yeomans to be identified as the authors of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections77 and 78 of the Copyright, Designs and Patents Act 1988. The Open Access version of this book, available at www.taylorfrancis.com, has been made available under a Creative Commons AttributionNonCommercial-NoDerivatives (CC-BY-NC-ND) 4.0 International license. Colour images are available in the Open Access version. Any third-party material in this book is not included in the OA Creative Commons license, unless indicated otherwise in a credit line to the material. Please direct any permissions enquiries to the original rightsholder. The project is funded by York University. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Names: Kozlova, Mariia (Professor of business), editor. | Yeomans, Julian Scott, editor. Title: Sensitivity analysis for business, technology, and policymaking : made easy with simulation decomposition (SimDec) / edited by Mariia Kozlova and Julian Scott Yeomans. Description: Abingdon, Oxon ; New York, NY : Routledge, 2025. | Series: Routledge open business and economics | Includes bibliographical references and index. Identifiers: LCCN 2024026905 (print) | LCCN 2024026906 (ebook) | ISBN 9781032592466 (hardback) | ISBN 9781032592473 (paperback) | ISBN 9781003453789 (ebook) Subjects: LCSH: Sensitivity theory (Mathematics)—Simulation methods. | Decision making—Simulation methods. | Decision support systems. Classification: LCC QA402.3 .S4526 2025 (print) | LCC QA402.3 (ebook) | DDC 511/.8—dc23/eng/20240628 LC record available at https://lccn.loc.gov/2024026905 LC ebook record available at https://lccn.loc.gov/2024026906 ISBN: 978-1-032-59246-6 (hbk) ISBN: 978-1-032-59247-3 (pbk) ISBN: 978-1-003-45378-9 (ebk) DOI: 10.4324/9781003453789 Typeset in Sabon by Apex CoVantage, LLC To our furry soulmates, who contributed to this book more than one could think. Tibo Chengis 2008–2023 2008–2021 xiv Figures 6.6 Decomposition of the Unit cost of batch size=1 by Deposition rate and Operation labor with equally-spaced state formation. 130 6.7 Decomposition of the Unit cost under current uncertainty levels by Batch size, Indirect costs, and Direct costs with equally-spaced state formation (top left) and sensitivity indices (top right). Detailed sensitivity indices are presented in Appendix 2. 132 6.8 Decomposition of the Unit cost under future uncertainty levels by Batch size, Indirect costs, and Direct costs with equally-spaced state formation (left) and sensitivity indices (right). Detailed sensitivity indices are presented in Appendix 3. 133 6.9 Decomposition of the Unit cost under merged current and future uncertainty levels by Batch size, Indirect and Direct costs with equally-spaced state formation. 134 7.1 Relationship between technology development paths and market opportunities. 146 7.2 Decomposition of Score by Market opportunity (58% sensitivity index). Market opportunities are sorted by the increasing mean Score. 154 7.3 Decomposition of Score by Asset Combination (28% sensitivity index). 156 7.4 Decomposition of Score by M21 (52%) and T12 (44%). The distinct colours represent different levels of M21 and different shades refer to different levels of T12 in each state of M21. 157 8.1 Life cycle diagram of single-use polypropylene-based medical mask and reusable PLA mask. 169 8.2 System boundaries of the LCA study. 171 8.3 Decomposition of the total global warming potential (GWP) of both masks by mask type (42%) and number of masks used (62%). Note that number of masks used is of different units for the different mask types. 176 8.4 Decomposition of the total global warming potential (GWP) of reusable masks by number of masks used (33%) and number of filters used (32%). 177 8.5 Decomposition of the total global warming potential (GWP) of single-use masks by means of transport (35%) and number of masks used (45%). 178 8.6 Decomposition of the total global warming potential (GWP) of single-use masks means of transport equal to Aircraft by number of masks used (69%) and distance (24%). Low, medium, and high states of distance are produced by dividing its total range [50, 1000] km into three equal intervals. 180 Figures xv 8.7 Decomposition of the total global warming potential (GWP) of single-use masks means of transport equal to Ship by number of masks used (73%) and mask disposal (26%). 181 8.8 Decomposition of the total global warming potential (GWP) of single-use masks means of transport equal to Lorry by number of masks used (74%) and mask disposal (24%). 183 8.9 Decomposition of the total global warming potential (GWP) of reusable masks (left) by LCA phases transport and use (top) and product elements filter and transport (bottom), and of single-use masks (right) by LCA phases transport and EoL mask (top) and product elements transport and mask (bottom). 184 9.1 POMDP problem formulation for the mineral exploration case. 195 9.2 State fidelity of the subsurface ore body shape for the mineral exploration problem. 197 9.3 Environment fidelity as grid resolution for the mineral exploration problem. 198 9.4 Planning fidelity using number of planning iterations in the tree search. 200 9.5 POMDP model-fidelity framework (PMFF) input variables and output metrics. 201 9.6 Example fidelity contour. Fidelity increases from the bottom left to the top right, for example, the white point at the centre is the mean runtime for the 30 × 30 grid with 1,000 planning iterations (over 500 seeds). 203 9.7 Contour maps of the discounted return for the mineral exploration POMDP. 206 9.8 Decomposition of return by state shape, planning iterations, and grid dimensions. Explained variance of the output by this decomposition is 0.002. 207 9.9 Contour maps of the 90th percentile of regret for the mineral exploration POMDP. 209 9.10 Decomposition of regret by state shape, planning iterations, and grid dimensions. Explained variance of the output by this decomposition is 0.014. 210 9.11 Contour maps of the runtime for the mineral exploration POMDP. 211 9.12 Decomposition of runtime by planning iterations and grid dimensions. Explained variance of the output by this decomposition is 0.451. 213 9.13 Contour maps of the bias in the final estimated massive ore quantity. 215 9.14 Decomposition of bias by grid dimensions and planning iterations. Explained variance of the output by this decomposition is 0.009. 216 xvi Figures 9.15 Contour maps of the number of actions (i.e. drills) for mineral exploration. 218 9.16 Decomposition of number of actions by (a) grid dimensions, (b) planning iterations, and (c) state shape. Explained variance of the output are 0.003, 0.052, and 0.052, respectively. 219 9.17 Contour maps of the accuracy in the final mine or abandon decision. 221 9.18 Comparison of histograms obtained with seed sampling (a1 & a2) and random sampling (b1 & b2). Results are similar for regret (a1 & b1) and different for runtime (a2 & b2). 224 10.1 An overview of the common synthetic drop-in fuel production pathways and their main products. MTG stands for Methanol-to-Gasoline; MTO, Methanol-to-Olefins; MOGD, Mobil olefins to gasoline and distillates; and RWGS, reverse water-gas-shift. 231 10.2 Overall diagram of the studied production route, divided into four sections. Some purification and processing steps are simplified for clarity. 233 10.3 Simulation Decomposition of NPV (Base scenario) by Gasoline price (55%) and Hydrogen price (16%). 241 10.4 Simulation Decomposition of NPV by Scenario (66%) and Gasoline price (7%). 243 10.5 Simulation Decomposition of NPV by Scenario (12%) and Gasoline price (27%). 246 10.6 Simulation Decomposition of NPV (MeOH Electrolysis) by Methanol price (56%) and Electricity price (25%). 247 11.1 4R model description. Input parameters: (a) residual stress, (b) material’s ultimate strength at the heat-affected zone (HAZ), (c) applied stress ratio, and (d) weld toe radius, and cyclic behaviour at notch. 260 11.2 Aseries of one-at-a-time sensitivity analyses of 4R model output to the four inputs (in columns). The rows represent different levels of residual stress res : high (upper row), negligible (middle row), and compressive (lower row). 263 11.3 Main decomposition of the structural reliability model output by the most influential three input parameters, explaining 96% of the variance of the output (sum of their sensitivity indices) and portraying three second-order effects causing heterogeneous input–output relationship. The histogram is stacked and exposes the entire simulation data without overlapping. The share of data in each scenario (or the probability of scenario) is displayed in the rightmost column of the legend. 269 Figures xvii 11.4 Decision tree constructed based on the most prominent heterogeneity in the effects of input variables on the output stress (denoted with stress concentration factor Kf,ref). 270 11.5 Contribution of residual stress to the variance of the output: 51% of the variance is explained by the input variable. 272 11.6 Contribution of stress ratio to the variance of the output: 35% of the variance is explained by the input variable. 273 11.7 Contribution of steel grade to the variance of the output: 10% of the variance is explained by the input variable. 274 11.8 Contribution of fatigue notch factor to the variance of the output: 4% of the variance is explained by the input variable. 275 11.9 Supplementary decomposition of the structural reliability model portraying the interaction effect between residual stress and stress ratio. Their combination explains 86% of the variance of the output. 276 11.10 Supplementary decomposition of the structural reliability model portraying the weak 4% interaction effect between steel grade and stress ratio. Their combination explains 43% of the variance of the output. 277 11.11 Supplementary decomposition of the structural reliability model portraying the correlation effect between steel grade and residual stress. Their combination explains 55% of the variance of the output. 278 12.1 Critical surfaces of superconducting alloys approximated by the correlation of Bottura (1999) and fitted to data points of Godeke etal. (2007) and Ferracin (2017). 285 12.2 (a) View of an Nb3Sn Rutherford cable showing the strands and fiberglass insulation. (b) Close-up of the strands. (c) Cross-section of the eRMC/RMM Rutherford tape consisting of 40 strands. (d) Detailed view of one strand cross section showing the Cu matrix and the 120 filaments in a honeycomb pattern. (e) An eRMC coil being wound with the cable; the bottom pancake layer has been completed. 286 12.3 CAD view of an eRMC (top) and RMM-type (bottom) coils. 287 12.4 CAD view of cross section of the eRMC coil pack, featuring the coils and the compressing pushers. 289 12.5 CAD view of cross section of the RMM coil pack featuring the RMM-type coil between two eRMC-type coils, with the bore in the centre, where the 16.5 T field is targeted. 290 12.6 Left: Picture of the end of the eRMC magnet showing the coil pack surrounded by the yoke and the shell held in place by some shims. Right: CAD transversal cross-section. 291 12.7 RMM magnet. Left: full assembly. Right: Cutoff view showing 3/8 of the magnet. 292 xviii Figures 12.8 2D model of the RMM magnet, depicting a simplified geometry. 294 12.9 2D model of the RMM magnet depicting two possible element mesh resolutions for a same geometry. The version on the right corresponds to the calculations performed in this study. 295 12.10 Example of a solution of a magnetic FEM simulation showing the magnetic field in the magnet. 296 12.11 BH curves of the ferritic materials utilized in the model (iron yoke and vertical pusher, and eRMC pole). 300 12.12 Simulation Decomposition of the RMM superconducting magnet model for output variable current. 306 12.13 Simulation Decomposition of the RMM superconducting magnet model for output variable horizontal interference. 311 12.14 Simulation Decomposition of the RMM superconducting magnet model for output variable coil stress at stage 1. 314 12.15 Simulation Decomposition of the RMM superconducting magnet model for output variable coil stress at stage 2. 316 12.16 Simulation Decomposition of the RMM superconducting magnet model for output variable coil stress at stage 3. 319 12.17 Simulation Decomposition of the RMM superconducting magnet model for output variable bladder pressure. 321 12.18 Scatter plot of horizontal interference to bladder pressure with colour-coded groups of shell thickness and yoke radius pairs consistent to Figure12.17. 322 12.19 Decomposition of bladder pressure on a limited dataset (yoke radius=360mm, shell thickness=40mm, and number of windings=264) by cable height that explains 86% of variation of the bladder pressure, and by another output of interest – horizontal interference. Absent due to correlation and rarely occurring scenarios are marked with grey font colour. 324 13.1 Distribution of future savings for different monthly saving amount (marked with different colours) and uncertain interest rate. 334 13.2 Future value of €200 monthly savings at 5% interest rate as an exponent of the number of years. 335 13.3 Future savings value under uncertain interest rate (0–10%), variable duration (1–50years), and monthly payment (€100, €200, or €300). The X-axis is truncated; the maximum amount of savings approaches seven million euros when all factors are favourable. 336 13.4 Accumulated assessment of country preferability. 339 13.5 Language comfort specific assessment of country preferability. 340 Figures xix 13.6 One realization of the stochastic reference interest rate and the corresponding mortgage rate with a cap. 342 13.7 Decomposition of the loan term (left) and corresponding overall interest expenses (right) by on and off interest cap and average interest rate. 343 13.8 Apower law curve for y=x0.3. 344 13.9 Language acquisition scenarios. 346 13.10 Car replacement option assessment. 348 13.11 Car replacement option decomposed by the costs. 349 13.12 Decomposition of body weight and fat percentage by mass of fat and lean tissue. 351 Colour versions of the artwork are available in the eBook version of this title (9781003453789). 2.1 SimDec open-source packages 41 2.2 Uses and arguments of the SimDec R package functions 46 2.3 Main functions of the SimDec Matlab package 49 3.1 The overview of application chapters and their themes 62 3.2 Modelling approaches and practices of case owners pre-SimDec 64 4.1 Classic cash flow model arrangement in a spreadsheet environment 81 4.2 Numeric assumptions for the stylized cash flow model 85 4.3 Variation in input parameters in the different taxation schemes ( =randomized) 86 4.4 Variation in input parameters in fluctuating demand case 88 4.5 Variation in input parameters in price hedging case 89 5.1 Uncertainty assumptions for input variables 100 5.2 NPV gained by the two parties, under the three supports, and in case of absence of supports 103 5.3 Results of sensitivity analysis on Rg, LPVR, and tariffs in PC regulation 104 5.4 Sensitivity indices (first-order effects) of input parameters to model outputs 109 6.1 Assumed indirect costs of 3DCP in construction 121 6.2 Assumed direct costs of 3DCP in construction 122 6.3 Single point unit cost estimation equation 122 6.4 Input variable assumptions for Monte Carlo (MC) simulation (-//- if unchanged) 125 7.1 Relationship between Assets and Market Opportunities 148 7.2 Decision-making elements for new tool: the Technical Market Opportunity Navigator 149 7.3 Variation in simulation inputs 151 7.4 Sensitivity indices of criteria for the aggregate Score across all Market opportunities 152 8.1 Variation in input parameters (all distributions are uniform) 173 Tables Tables xxi 8.2 Combined sensitivity indices for the three means of transport states separately 179 8.3 First-order sensitivity indices for LCA phases and product elements 182 9.1 Combined sensitivity indices for all considered outputs 204 9.2 Sensitivity indices for discounted return 205 9.3 Sensitivity indices for regret 208 9.4 Sensitivity indices for runtime 212 9.5 Sensitivity indices for bias 214 9.6 Sensitivity indices for number of actions 217 9.7 Sensitivity indices for accuracy 220 9.8 Decomposition of accuracy by state shape, planning iterations, and grid dimensions 222 9.9 Combined sensitivity indices for all outputs for the two sampling strategies 223 9.10 Sensitivity index totals 225 10.1 One-at-a-time sensitivity analysis 237 10.2 Scenario analysis 238 10.3 Variation in input parameters 239 10.4 Sensitivity indices for Base scenario 240 10.5 Sensitivity indices for all scenarios and the entire dataset 242 10.6 Sensitivity indices for the updated model with electrolysis-only scenarios 245 11.1 4R input parameter values and their ranges 264 11.2 Sensitivity indices 266 11.3 States of input variables for the main decomposition 267 11.4 States formation of input variables for the supplementary decomposition by residual stress and stress ratio 276 11.5 States formation of input variables for the supplementary decomposition by steel grade and stress ratio 277 11.6 States formation of input variables for the supplementary decomposition by steel grade and residual stress 278 12.1 Mechanical material properties 298 12.2 Input parameters, their evaluation points, and physical meaning 301 12.3 Output variables expected/viable ranges and consideration within the sensitivity study 303 12.4 Sensitivity indices for selected model outputs 304 12.5 Sensitivity indices for current 305 12.6 Sensitivity indices for conductor area 308 12.7 Resulting conductor area from different combinations of cable height and number of windings 309 12.8 Sensitivity indices for horizontal interference 310 12.9 Sensitivity indices for coil stress at stage 1 312 xxii Tables 12.10 Sensitivity indices for coil stress at stage 2 313 12.11 Sensitivity indices for coil stress at stage 3 317 12.12 Effects of the hkey position and cable height on maximum von Mises stresses in the coil 318 12.13 Sensitivity indices for bladder pressure 320 13.1 Modelling construct for the six cases considered 333 13.2 Inputs to multi-criteria decision-making problem: choosing a country based on seven criteria 337 13.3 Details of the considered mortgage 341 13.4 Numeric inputs to the car choice problem 347 13.5 Numeric inputs to the body fat percentage calculation 350 Antti Ahola Laboratory of Steel Structures, LUT School of Energy Systems LUT University Lappeenranta, Finland Abid Alam Economics Department Queen’s University Kingston, ON, Canada Luiz Brandao Information, Risk, and Operations Management, McCombs School of Business The University of Texas at Austin Austin, United States IAG Business School PUC-Rio, Rio de Janeiro, Brazil Jef Caers Mineral X Stanford University Stanford, CA, United States Anthony Corso Stanford Intelligent Systems Laboratory Stanford University Stanford, CA, United States Manuel García Pérez Susana Izquierdo Bermúdez CERN Méyrin, Switzerland Hannu Karjunen Laboratory of Thermodynamics, LUT School of Energy Systems LUT University Lappeenranta, Finland Sini-Kaisu Kinnunen Industrial Engineering and Management, LUT School of Engineering Science LUT University Lappeenranta, Finland Mariia Kozlova LUT Business School LUT University Lappeenranta, Finland Petteri Laaksonen LUT School of Energy Systems LUT University Lappeenranta, Finland Arto Laari Separation Science, LUT School of Engineering Science LUT University Lappeenranta, Finland Contributors xxx Preface In the past, sensitivity analysis has been employed to “shake the foundations of science” – where the actual definition of science should be considered broadly writ, here. So the purpose of the book is to test the new SimDec method by “shaking the foundations of shaking the foundations of science”. Namely, to explore how well SimDec works as a sensitivity analysis tool when applied to a diverse spectrum of disciplines. Surprisingly, collecting application cases for the book turned into a relatively straightforward task, as many researchers responded not only very positively, but also with remarkable enthusiasm for such a project. As a result, there are 13 chapters in the book. Chapters1–3 provide the necessary background for understanding SimDec and sensitivity analysis – these chapters should be read in the prescribed order and be considered “compulsory” for all readers. Chapter1 introduces the multifarious meanings of sensitivity analysis, its science and practice, and benefits and shortcomings. It shows how SimDec unifies sensitivity analysis and uncertainty analysis by combining the best features from both of their worlds. Chapter2 describes the SimDec algorithm, delivers instructions for using the open-source packages, and offers guidelines on how to interpret SimDec results. Chapter3 provides a convenient synopsis of the various applications, so the motivated reader can quickly acquire an overall comprehension of the collection of upcoming applications and determine how to navigate to the areas most relevant to their specific interests. Chapters4–13 provide the applications studied in the book. The topics of each chapter revolve around the specific projects within each of the various disciplines. Progression through these application chapters can proceed in any order, depending upon the levels of interest of the reader. The general topics of the application chapters (with chapter number in brackets) are corporate finance (4), public support (5), 3D manufacturing in construction (6), deep tech entrepreneurship (7), carbon footprint analysis (8), geology and model fidelity (9), P2X fuels (10), structural reliability (11), superconducting magnets (12), and personal decisions (13). It must be duly recognized that neither SimDec nor sensitivity analysis should be considered as “spectator sports” – all readers are firmly encouraged to get “stuck in” and participate. We want you to apply SimDec to whatever sphere of interest you might possess. Consequently, to promote as widespread an adoption and penetration of SimDec as possible, the book has been made entirely open-access and the downloadable electronic copy is free-of-charge for all readers. Furthermore, the SimDec application, itself, has been made completely open-source with a web dashboard1 and software packages readily available in Python, R, Julia, and Matlab,2 all supported by an ever-expanding discussion-board community on Discord.3 While these computer codes can be used for conducting a complete global sensitivity analysis as in the chapters, they can also be used separately as a visual analytics package for uncertainty analysis and for the standalone calculating of Preface xxxi sensitivity indices. Readers are encouraged to experiment with the use of SimDec on their own data and to extend it to suit their circumstances. When questions inevitably arise, the Discord community provides a great resource for advice, support, and feedback. Notes 1 https://simdec.io/. 2 https://github.com/Simulation-Decomposition. 3 https://discord.gg/8jkEyqXu2W. Mariia Kozlova As a habitual introvert, I would never have suspected how many people Iwould be grateful to, and enjoyed collaborations with, for successfully putting together this book project. Countless thanks go to Robert J. Moss (Stanford), Abid Alam (Queen’s), and Pamphile Roy (Austria), whose excitement about SimDec and whose eagerness to implement it in their preferred programming languages ultimately convinced me of SimDec’s pervasive usefulness. Your unbridled passions charged all endeavours with endless positive energy – including the completion of this very book. Special gratitude belongs to Andrea Saltelli, the “godfather of global sensitivity analysis”, whose interest in SimDec has been the biggest endorsement one could ever have hoped for. Andrea, your challenge, thrown down like a gauntlet in the Foreword, has been accepted. Asincere THANK YOU to all contributors to this book; it has been an unbelievably rewarding experience to work with all of you in many senses. To all who intended to contribute but eventually could not make it, you are forgiven , it is a pity that the timing did not work for us this time. My gratitude is extended to our commissioning editor, production team, the reviewers of the book proposal and the chapters, and to Robert J. Moss, who kindly prepared a superb-looking formatting style adopted throughout the text. Iam also grateful to all those who helped nurture SimDec in the past, including my Business Finland project team and advisors, and my PhD supervisors. My friends and family, you have continued to support me throughout the especially busy year of 2023 and even patiently sustained my occasional spells of grumpiness – hugs. Special thanks to Olga, who patiently listened to all my crazy ideas, doubts, and complaints, transforming the latter two into recharging laughter. Finally, it is JSY, without whom we would not be writing these lines. Iconsider myself a tremendously lucky person to have had such fulfilling long-term collaboration filled with so much support and humour! Soon, when this all-consuming book is off our shoulders, we will be able to get back to the normal levels of joke exchanges in our communication. Or should we do Volume II? Mariia Kozlova, LUT University Acknowledgements Acknowledgements xxxiii Julian Scott Yeomans As with any project of this magnitude, there are far too many parties to credit than may reasonably appear within the bounds of a short acknowledgement – but Iwill try. My sincere gratitude goes to all of our contributors for indulging us by testing this “newfangled” SimDec approach in their respective fields and for allowing us to edit their subsequent writings to fit into our book format (style, length, brevity, wording, etc.) without (too much) complaint. In doing so, the culpability for any errors and omissions firmly resides with the editors. It really has been a pleasure to work with (almost) every one of you and Ihope that these partnerships can be extended to further collaborations. Kudos to my family (you know who you are) for their tacit support – fortunately, my wife, teenaged daughter, and dog all know exactly where to position the true significance of this project within life’s broader context. Final thanks to MK for the now 5+ years of academic collaboration – this is not where Ithought we would be when we started (not even close!), but it has proved to be the most rewarding journey imaginable. We have clearly gone where angels fear to tread. And, given the time and energy commitments required, Ithink we can both pledge that it will remain a firm “no” to Volume II for another decade or two – possibly even three! Julian Scott Yeomans, York University *** The 2022–2024 research work for this book was supported in part by grant OGP0155871 from the Natural Sciences and Engineering Research Council of Canada; by funding from Business Finland, grant #6713/31/2021; by grant #00220167 from Jenny and Antti Wihuri Foundation; and by grants #220177 and #220178 from Finnish Foundation for Economic Foundation. Part I Introduction DOI: 10.4324/9781003453789-2 This chapter has been made available under a CC-BY-NC-ND 4.0 license. Abstract For a computational model to be useful for decision-making purposes, its foundations need to be “shaken” sufficiently so that one can ascertain what might happen if its underlying assumptions ever changed. This type of robustness assessment is performed via the techniques of sensitivity analysis and uncertainty analysis. These techniques supply many alternative pathways under which such evaluations can be conducted. In this chapter, we consider a full spectrum of sensitivity analysis tools – from setting up computer experiments, to analyzing their results, to discussing the variety of methods at each phase, up to investigating how the overall process might actually be deployed effectively in practice. Our conclusions on the current “state of the practice” are worrying. The most commonly used one-at-a-time sensitivity analysis methods can severely mislead decision-makers. Further to that, simple Monte Carlo simulations and the portraying of their output distributions are not sufficient to understand the actual drivers behind the model behaviour. Advanced global sensitivity analyses are often so focused on quantification that they missany insights into the nature of the effects. Conversely, the novel Simulation Decomposition (SimDec) method incorporates the best components from both sensitivity analysis and uncertainty analysis – which enables the production of holistic insights in a very straightforward fashion – thereby reinforcing its considerable potential for widespread adoption anywhere a global sensitivity analysis is performed. 1 Introduction The term sensitivity analysis (SA) can mean many things and take many forms. The fundamental idea behind it is about “shaking” the model to see what falls out of it. But it is the nature of how we actually shake it that creates a multitude of approaches. The majority of researchers and practitioners employ one-at-a-time (OAT) analysis, performed by changing the values Chapter 1 Methodological landscape of sensitivity analysis and the place of SimDec Mariia Kozlova, Samuele Lo Piano, and Julian Scott Yeomans 4 Mariia Kozlova, Samuele Lo Piano, and Julian Scott Yeomans of one factor at a time (Saltelli etal., 2019). Anecdotally, such analysis is perceived as a simple, rhetorical exercise to such an extent that the ensuing spider charts and tornado diagrams have completely flooded business reports and academic publications. Why must this ubiquitous, one-at-a-time principle be considered inadequate? Because as soon as non-additive operations enter the picture, OAT inherently disregards significant portions of the solution space, even in the simplest models. Imagine a simple model in which our losses are described by two factors: (1) the number of customers who reject our services, Ncancel, multiplied by (2) the cost of such rejection, costcancel. The base case scenario is that each customer we lose costs us $1,000. Performing a one-at-a-time sensitivity analysis on each of these factors, separately, if we vary only the first factor to see what would happen if we lost (say) five customers (i.e. varying the first factor while holding the second factor constant at $1,000), then our losses would be $5,000 (= 5 × $1,000). If, instead, we varied the second factor so that our costs per cancelled customer increased to $5,000 (i.e. holding the first factor constant at 1 customer while varying the second factor), then our losses would also be $5,000 (= 1 × $5,000). So performing this OAT-SA on the two factors would indicate that our major risks add up to a maximum of $5,000 losses. The perceptive reader, however, would clearly notice that, under the unfortunate situation in which both factors deteriorated simultaneously, the outcome would result in the much more deleterious loss of $25,000 (= 5 × $5,000). If one can observe such a discrepancy occurring in the one-at-a-time analysis of such a simple model, imagine how much misrepresentation could happen in significantly more complex models? More advanced practitioners employ Monte Carlo simulation to circumvent the shortcomings of OAT-SA, which can be attributed to uncertainty analysis (Janssen, 2013). In Monte Carlo analysis, all uncertain factors are allowed to change, simultaneously, so that extreme scenarios can be revealed in the resulting probability distribution of the model output. However, while the inadequacies of OAT are overcome, uncertainty analysis simply portrays the uncertainty ranges within the system but cannot direct the decision-maker to the most appropriate course-of-action. The entire plethora of approaches available for shaking a model all seem to be channelling the mindset of a model explorer, which results in more questions than answers. Scenario analysis is accomplished by changing several factors at once but for only a few instances – namely, the very scenarios, themselves (Hassani & Hassani, 2016). Why only these scenarios (Morgan & Keith, 2008)? Why these particular factors? And which of them drive(s) the difference(s) in the outcome of different scenarios? Threshold analysis is used to identify critical values or ranges of input variables where the model’s output changes significantly or assumes certain values – a typical use in the Methodological landscape of sensitivity analysis and SimDec 5 field of environmental impact assessment (Bojórquez-Tapia etal., 2005), risk assessment (Draper etal., 1999), and break-even analysis in investment valuation (Jovanović, 1999). Could there be other combinations of factors that result in the same changes in the output? Is there critical system behaviour outside of the studied range? Optimization, in a sense, can also be considered as “shaking” the model to find the best output (Schwieger, 2007). But should we not also learn how to avoid undesirable output values? Or which combinations of factors can lead us to good-enough outputs rather than the best? The Holy Grail of sensitivity analysis is the aspirational global sensitivity analysis1 – a field of science that incorporates a spectrum of mathematically advanced methods to compute sensitivity indices that ascribe a degree of influence of the individual (or groups of) inputs on the model output (Saltelli etal., 2004). The practice of this analysis is considered to be exemplary, because of its global nature, in which the entire space is studied as all factors are changed simultaneously. Unfortunately, sensitivity indices alone do not tell the full story, no matter how good or reliable they may seem. Sensitivity indices only describe the strength of an effect, not its shape. The very shape of an effect is often crucial for understanding the model behaviour and approaching the decision-making surrounding it (Kozlova, Moss, etal., 2024). Simulation Decomposition, or SimDec, is a novel approach that cleverly aggregates all of the benefits of these other methods. In essence, it is a visualization of a model output distribution, decomposed and colour-coded by the multi-variable scenarios, the variables for which are chosen based on the global sensitivity indices calculated (Kozlova, Moss, etal., 2024). Thus, SimDec inherently amalgamates uncertainty analysis with global sensitivity analysis. More importantly, it exposes the character of the effects within a model, thereby enabling deeper insights into its fundamental behaviour. While many previous studies have adopted competing perspectives on the typology of sensitivity analysis methods (Pianosi etal., 2016; Lo Piano & Benini, 2022; Saltelli etal., 2008), we have chosen to focus on the insight derivation capabilities of these different approaches. To motivate this viewpoint, we consider the actual process of the analysis as the most appropriate lens for conducting our methodological overview and extensively examine its corresponding rationale throughout each analytical phase. SimDec is subsequently grounded within this landscape, and its role is contrasted with the capabilities of these other methods. Consequently, the goal of this chapter is to depict the existing landscape of the various sensitivity analysis methods practiced throughout business, science, and academia and to firmly establish the pre-eminent need for applications of SimDec analysis within this context. 12 Mariia Kozlova, Samuele Lo Piano, and Julian Scott Yeomans based on a cumulative distribution function (Pianosi & Wagener, 2015) rather than a probability distribution function. However, several criticisms of its robustness have been raised (Puy etal., 2020a). Owen (2014) proposed the use of approaches from cooperative game theory in sensitivity analysis, such as Shapley values. Shapley values were originally proposed to define the surplus each player in a cooperation would be entitled to (i.e. the value of an n-person game). Shapley values are being increasingly used for the interpretability of machine learning models (Molnar, 2020). Their use in sensitivity analysis has increased recently due to their suitability for handling correlated inputs. One of the recent developments, the simple binning method (Kozlova etal., 2023), converts the general concept of variance-based sensitivity indices directly into mathematics for their computation. The method involves binning the data by X, computing the averages of Y in those bins, and then taking the variance of those averages. First-order, second-order, and combined (or closed) sensitivity indices can be readily computed. The method works with given data and captures correlated inputs, which show up as negative second-order effect. The sum of combined indices equals one even in the presence of interactions and correlations, if all randomness in the model has been accounted for. In all the described methods and for global sensitivity analysis, in general, the sole goal is to compute sensitivity indices that show how much uncertainty in the model output is attributed to different input variables (Saltelli etal., 2008). All previous phases were just stepping-stone preparations for this revelation. Most of these methods do not work with dependent inputs. There are numerous works that introduce modifications to existing methods to enable them to capture dependencies in the model (Da Veiga, 2015; Lamboni & Kucherenko, 2021; Mara etal., 2015; Borgonovo etal., 2024), but these all require different treatments of the data with dependent variables. The only exceptions to this additional complication are Shapley values and the simple binning method, which work with correlations without a need for supplemental adjustments. Furthermore, most of these methods require access to the model in order to conduct multiple simulations. Only a handful of methods are able to work with a given dataset, including (1) the approaches that compute first-order indices (Marzban & Lahmer, 2016; Plischke, 2012) and (2) the simple binning method (that also produces second-order indices as well as capturing interactions), which has been shown to produce reliable estimates even with a simple random sampling of 1,000 points (Kozlova etal., 2023). 2.1.5 Visualization The two main visualization types employed in one-at-a-time sensitivity analysis are tornado plots (Figure1.2A) and spider charts (Figure1.2B) (Kozlova, Methodological landscape of sensitivity analysis and SimDec 13 Moss, etal., 2024). Tornado plots are restricted to two evaluation points in addition to the base case and, thus, lack the capability to communicate nonlinearities in the underlying model. However, since the X-axis of the tornado chart depicts the actual values of the output, any assumptions made regarding input variation do not have to be symmetric and can, therefore, better reflect reality. On the other hand, spider charts place their output values on the y-axis with the corresponding per cent of factor change values on the x-axis. Hence, this structure requires that all inputs vary within symmetrical ranges, which is rarely a realistic assumption (Saltelli etal., 2019). The ability of spider charts to portray several evaluation points for each factor (e.g. ±10%, ±20%, and ±30%) enables them to communicate the shape of underlying relationships and to expose the existence of nonlinearities within the model. Regardless of the specific advantages and disadvantages of these two visualization types, the predominant failure of their approach is that only one factor is varied at a time, which means that any interactions of inputs in the model cannot be captured. Studies that employ global sensitivity analysis usually stop at one of the previous phases and do not systematically proceed into visualization. The only commonly employed visualization technique is a box plot that characterizes the convergence of index estimates under different conditions (Shi etal., 2023; Barr & Rabitz, 2023; Shang etal., 2023). Occasionally, indices themselves are depicted with a bar chart (Puy etal., 2021; Vuillod etal., 2023; Thapa & Missoum, 2022). Consequently, neither of these visualizations can uncover the inherent characteristics of the effects. Nevertheless, various types of visualizations do sporadically make an appearance in one form or another in different global sensitivity analysis studies. Scatter plots (Figure1.2C) are sometimes used in conjunction with GSA (Saltelli etal., 2008; Wainwright etal., 2014; Palar etal., 2023) and are also recommended for the initial visual inspection of the input–output relations. The scatter plots are produced from global simulations but are restricted visually to a single input variable, thereby limiting the perception of complex joint effects in a model. For example, if a scatter plot consists of a non-linear trend, it is unclear which other input(s) are responsible for it and how. Heteroskedasticity (i.e. a variation of the output range uncertainty against the values of the input value) is a clear sign of higher-order interactions with (an) other input variable(s), but does not suffice to identify which variables are responsible for it. Often, as a part of uncertainty analysis (Lo Piano & Benini, 2022), researchers display a distribution of the model output in the form of a histogram (Figure1.2D). This visualization, however, possesses limited information content on the model behaviour. It merely displays the overall uncertainty in the model output and its shape. Descriptive statistics of the output variation (such as expected mean, minimum, maximum, standard deviation, etc.) can be derived. However, the histogram alone cannot be used to determine 14 Mariia Kozlova, Samuele Lo Piano, and Julian Scott Yeomans which input variables affect the output and does not display the nature of their interaction. Heat maps (Pleil etal., 2011; Owen etal., 2019) (Figure1.2E) and response surfaces (Myers etal., 2016) are frequently used to depict the relationship between two input variables and the output. However, they both require dimensionality reductions concerning all other inputs (they are normally fixed, and the analysis happens in a two-at-a-time mode). Both visualization types require a two-dimensional matrix of output values associated with discrete values of the two inputs. Thus, such graphics are more often produced with a full-factorial design sampling strategy, because all other randomized approaches would require further data transformation. Parallel coordinate plots (Figure1.2F) are another type of graphic, where each parallel coordinate represents each input and output, and the lines connecting them provide a visual association within each simulation run (Pianosi etal., 2015). Chaotically intersecting lines between an input and the output signify no influence, while all other patterns indicate the opposite. However, simulations are often comprised of thousands of runs, which translates into thousands of lines on the parallel coordinate plots, making them unreadable for all practical purposes (Heinrich & Weiskopf, 2013). Various remedies to the visual overloading problem have been proposed (Steinparz etal., 2010; Roy etal., 2018). Nevertheless, the increasing number of input variables and the problem of the optimal ordering of coordinates prevent this visualization type from widespread adoption. Overall, although many different visualization types exist, they all possess limitations in depicting the underlying behaviour of a computational model. None of the visualizations has been systematically recognized as the universal standard for global sensitivity analysis studies. Hence, the lack of visualization is a major shortcoming in global sensitivity analysis as it prevents an understanding of the nature of the models and proves to be a major impediment to well-informed decision-making (Kozlova, Moss, etal., 2024). 2.2 The place and role of SimDec The SimDec procedure comprehensively spans two specific phases of sensitivity analysis: (4) computing sensitivity indices and (5) visualization. Because it fully encompasses phase 4, SimDec inevitably engages all the previous data generation phases. Specifically, SimDec receives a simulated dataset as input, calculates sensitivity indices using the simple binning approach, and then creates a visualization of the most influential relationships in a model (Kozlova, Roy, etal., 2024). Consequently, SimDec captures the entire quantification process of sensitivity analysis and communicates the “shape” of these effects via an intelligent visualization mechanism. This process results in a more holistic sensitivity analysis approach and provides considerable quantities of actionable insight (Kozlova, Moss, etal., 2024). Methodological landscape of sensitivity analysis and SimDec 15 SimDec projects a visualization of key multidimensional model relationships onto a two-dimensional graph. The SimDec visualization is generated by its own distinct algorithm that requires certain operations to be performed on the data, in contrast to other, more “direct” visualization approaches. The mechanism involves decomposing the dataset into scenarios based on combinations of states of the most influential input variables. The colour-coding logic is designed in accordance with formed scenarios, where the states of the most influential input are assigned distinct primary colours, and all other state subdivisions assume shaded gradations of these primary colours. The Figure 1.2 Schematic representation of different visualization types. (colour image is accessible via the link) 16 Mariia Kozlova, Samuele Lo Piano, and Julian Scott Yeomans decomposition produces a legend which depicts which colours and states of inputs are attributed to specific scenarios and indicates how each data point is assigned to a corresponding scenario. This information is used to construct the actual visualization. The prevalent SimDec visualization approach is to construct a stacked histogram in which the coloured scenarios are directly mapped onto the distribution of the model output. Potentially, the series could be overlayed as opposed to stacking, but the perceived relative simplicity attributable to overlaying possesses several notable deficiencies (Kozlova & Yeomans, 2020). Box plots could also be employed if either the distribution has a poorly readable shape or some scenarios have low probabilities and are not visible (for examples, see Chapter2, Kozlova, Roy, etal., 2024). To summarize the major new contributions of SimDec to the overall landscape of sensitivity analysis methods: (1) it relies on a more efficient and accurate quantification approach (Kozlova, Moss, etal., 2024); (2) it produces a more powerful visualization in terms of information content and overall readability (Kozlova etal., 2023); and (3) it forces a much more systematic deployment of visualization so that the procedure of sensitivity analysis does not stop at quantification but continues to explore model behaviour more holistically. Essentially, it is the only visualization technique for sensitivity analysis that readily (1) exposes interactions, (2) displays the data of more than three dimensions, and (3) preserves readability simultaneously (see Chapter2 (Kozlova, Roy, etal., 2024) for specific details on using SimDec and for guidance in its interpretation). 3 Usage of methods 3.1 The process of sensitivity analysis in reality The process of sensitivity analysis portrayed in Figure1.1 has rarely been used wholly in practice, resulting in incomplete experimental designs. The majority of studies that adopt sensitivity analysis have opted for the one-at-a-time methods (Lo Piano & Benini, 2022; Pianosi etal., 2016; Saltelli etal., 2019), focusing on only two phases out of five – (3) sampling and (5) visualization – which have then been implemented within frequently questionable experimental designs. As a result, both a proper “shaking” of the model (phase 2, variation) and quantification (phase 4, computing sensitivity indices) are missing. Even studies by the authors of this chapter can be considered guilty of such insufficiencies (Kozlova etal., 2019; Lo Piano & Mayumi, 2017). Another fairly common practice has been the adoption of uncertainty analysis in the form of Monte Carlo simulation, with the resulting output displayed in distributional form, thereby skipping the quantitative phase (phase 4, computing sensitivity indices). Coupled with the limitations of the distribution as a visualization type, such analysis remains completely deficient with respect to identifying exactly which factors contribute to the variability of the output. Methodological landscape of sensitivity analysis and SimDec 17 The expanding body of research indicates that global sensitivity analysis normally concludes at phase 4, computing sensitivity indices, rather than assessing the shape of the discovered effects by proceeding to the (5) visualization phase (Shi etal., 2023; Barr & Rabitz, 2023; Vuillod etal., 2023; Wang & Jia, 2023; Jung & Taflanidis, 2023; Shang etal., 2023; Ballester-Ripoll & Leonelli, 2022; Thapa & Missoum, 2022; Xiong etal., 2022; Yang, 2023). Rare exceptions to this convention possess no systematized logical reasoning, with the various approaches ranging from coloured scatter plots (Palar etal., 2023) to 3D Kiviat charts (Roy etal., 2018). However, the nature of this unsystematic implementation prevents the visualization phase from becoming an integral part of the sensitivity analysis process. 3.2 Uptake of sensitivity analysis in business and academia Corporate uptake of sensitivity analysis methods is not encouraging. In the areas of corporate finance and risk management, the concept of global sensitivity analysis has not been recognized at all (Ryan etal., 2002; Graham & Harvey, 2001; Hubbard, 2020). Arecent (2022) inquiry by the authors into the operations of a dozen European companies of different sizes and industries showed that more than a third did not use any sensitivity analysis at all, half employed simple one-at-a-time sensitivity analysis, and one company used Monte Carlo simulation for decision-making. In academia, the uptake of global sensitivity analysis has featured much more prominently. Since its inception in the 1980s, a community of numerical modelers has shown great interest in employing the computational approaches provided by the methodology (Iooss & Lemaître, 2015). More recently, sensitivity analysis has been recognized as possessing natural synergies in the variable selection problems of statistical learning, especially for machine learning (Da Veiga etal., 2021). Irrespective of the disciplinary field, all the stages of mathematical model building can clearly benefit from performing appropriate sensitivity analyses (Borgonovo & Plischke, 2016), as described in the “Methodological landscape” section. In general, estimates of the use of global sensitivity analysis range between 12% (Lo Piano & Benini, 2022) and 20% (Saltelli etal., 2019) cases. However, the majority of academic reviews have evaluated research that already includes some kind of sensitivity analysis. What about studies that do not? Namely, how high is the uptake of sensitivity analysis in computational modelling in general? This is a difficult question to answer, considering the magnitude of the field and the physical impossibility of reviewing all publications within it. Akeyword search in the SCOPUS database using a query with “model*” in title, abstract, and keyword fields returned more than 15million results. Constraining this output to those which included “sensitivity analysis” reduced the total to only 1% of these results for those using a one-at-a-time sensitivity analysis (OAT)2 and to only 0.03% for those using global sensitivity analysis (GSA)3 (Figure1.3). 18 Mariia Kozlova, Samuele Lo Piano, and Julian Scott Yeomans Figure 1.3 Adoption of sensitivity analysis (SA) and global sensitivity analysis (GSA). (colour image is accessible via the link) Methodological landscape of sensitivity analysis and SimDec 19 These numbers must be viewed with an element of scepticism since we cannot ensure that all 15million papers that mention “model*” actually perform a computational modelling exercise.4 Irrespectively, the breakdown of the results by the subject area magnifies the expressed concerns (Figure1.4). Psychology, veterinary, and health-related areas are less likely to refer to computational models. However, many other areas (including mathematics, computer science, and environmental science) provide a reasonable match. Clearly, there is a degree of imprecision in the estimate of the share of SA studies (first column), but far less so for the estimate of the share of GSA in SA studies – since the more specific keywords return less noisy results. Figure 1.4 Adoption of sensitivit y analysis (SA) and global sensitivity analysis (GSA) by topical area. (colour image is accessible via the link) 20 Mariia Kozlova, Samuele Lo Piano, and Julian Scott Yeomans Figure 1.5 Adoption of sensitivity analysis (SA) and global sensitivity analysis (GSA) in selected modelling fields. (colour image is accessible via the link) To increase the accuracy of the exercise, we examined a selection of more specific modelling domains and approaches (Figure1.5). The share of SA in modelling papers appears to be the highest at 7% in several small domains. These domains include life cycle assessment (LCA), cost–benefit analysis (CBA), and multi-criteria decision-making (MCDM). The remainder of the frameworks exhibit no more than 3% of papers referring to SA. The share of global studies among SA is under 10% in each category. More concerning are the numbers for the share of studies that employed GSA overall – which all fall below 1%. 4 Conclusions Sensitivity analysis has become a term that can be understood very differently by different groups of researchers and practitioners. This chapter attempts to unify these different perspectives and frame them into the lenses of sensitivity analysis as a process for investigating a model. Our main finding is the failure of the entire spectrum of methods to be sufficiently useful in understanding model behaviour and for supporting decision-making. Specifically, one-at-a-time sensitivity analysis methods prove insufficient (and even misleading) because they fail to capture interactions and do no cover the space of variability properly. On the other hand, more advanced methods (i.e. global sensitivity analysis) tend to be prematurely satisfied with a quantification of the strength of the effects but fail to consider the shape of these effects. Employing visualizations within the sensitivity analysis field is rare and not systematic. Methodological landscape of sensitivity analysis and SimDec 21 Assessing existing literature reviews and searching through appropriate scientific databases leads one to the conclusion that the adoption of sensitivity analysis (especially global sensitivity analysis) throughout all fields of computational modelling is extremely limited. Fewer than 10% of the existing computational studies adopted any sensitivity analysis at all, and virtually none of these actually performed global sensitivity analysis. Other reviewers have reported similar findings (Lo Piano & Benini, 2022; Pianosi etal., 2016), while some have indicated slightly higher adoption rates but selectively for highly-cited papers (Saltelli etal., 2019). Furthermore, in industry and business, we have not encountered the use of global sensitivity analysis at all. The only exceptional cases occurred when the employees, themselves, were trained in it and were developing software for it (Baudin etal., 2015). Otherwise, the mathematical complexity and the lack of instruction in global sensitivity analysis appear to be the overriding impediment to its dissemination and implementation (Saltelli etal., 2019). SimDec possesses the significant potential for remedying all these shortcomings. Its algorithm and focus on visualization seem to make it the inevitable tool for effectively covering the entire process of sensitivity analysis. Its sophisticated quantification of input variable influence and its accessible visualization of the shape of the underlying effects are inherently incorporated into its unifying methodology. At the same time, the simplicity of the procedure and the fact that hard-to-understand quantification is contained within a hidden instrumental phase of the method make it easier to adopt for people not expressly trained in either sensitivity analysis or design of experiments. Thus, the value of SimDec appeals to the entire spectrum of users – from novice users to highly experienced sensitivity analysis practitioners. To accelerate its adoption, SimDec open-source codes have been made freely available5 in Python, R, Julia, Matlab, and Excel/VBA – together with an interactive web-based dashboard – all supported with extensive instructions on how to use them and how to interpret their results (Kozlova, Roy, etal., 2024). Acknowledgements This research was supported in part by grant OGP0155871 from the Natural Sciences and Engineering Research Council; by funding from Business Finland, grant #6713/31/2021; and by grant #220177 from Finnish Foundation for Economic Foundation. Notes 1 As indicated by several review studies, global sensitivity analysis methods have verylimited penetration to the practice (Pianosi etal., 2016; Lo Piano & Benini, 2022a), which is often attributed to its limited appearance in the teaching programs (Saltelli etal., 2019). 28 Mariia Kozlova et al. the consideration of appropriate corrective initiatives, such as: What are the alternatives? What can be done to achieve the desired outcome range? Are there any readily identifiable alternate manoeuvres? What shielding from risk is truly possible? Simulation Decomposition (SimDec) provides an alternative, methodological pathway to redirect the analysis of computational models towards actionability. In essence, SimDec intelligently maps various multivariable scenarios onto the distribution of an output. This mapping exposes actionable insights that are critical for decision-making, including: • How to achieve the desired output range while avoiding undesirable ones (which combinations of input variable states result in desirable output ranges)? • What is the extent of control that can be exercised over the system under the uncertainty conditions present (how much overlap is there between different scenarios)? • Are there specific scenarios that reduce the levels of uncertainty more than others (what is the relative width of the scenarios)? • What is the nature of interactions within the model (does an input variable influence the output only under certain circumstances)? SimDec can facilitate the overall interactivity of many decision-making processes and can prove remarkably powerful in revealing the underlying nature of model behaviour. It also provides a useful tool for assisting in model creation and for ensuring that models function in their intended way. This chapter provides guidelines for those seeking a better understanding of the algorithm behind SimDec (Section2), wishing to learn how to interpret SimDec results (Section3), or in using any of the available open-source packages (Sections4 and 5). 2 SimDec algorithm Fundamentally, the SimDec algorithm maps multivariable scenarios onto a distribution of model output in an intelligent fashion, thereby enabling a visualization of the critical cause–effect relationships inherent within the model. As its name implies, SimDec decomposes all data (from simulation runs or measured dataset) into automatically created scenarios constructed from the combinations of ranges (states) of the influential input variables. SimDec is a fully automatic procedure that returns a visualization depicting model behaviour that explains the most important sources of variation within the output (see Figure2.1). The algorithm consists of two fundamental parts: (1) a computation of sensitivity indices (Kozlova etal., 2023) that discern which influential SimDec algorithm and guidelines for its usage, interpretation 29 Figure 2.1 Core idea behind SimDec. (colour image is accessible via the link) 30 Mariia Kozlova et al. variables should be used in the decomposition, followed by (2) a core visualization of the decomposed output distribution (Kozlova & Yeomans, 2022). 2.1 Sensitivit y indices computation The sensitivity indices computed at the beginning of the SimDec procedure are global variance–based indices. The first-order index SXi is computed as a variance of the expectation of Y conditioned on Xi weighted by the total variance of Y. Var(( YX ) Si) Xi= (1) Var () Y The second-order indices have a similar formulation, but the expectation of Y is now conditioned on a pair of input variables, and the corresponding first-order effects are deducted to capture the pure excess effect (i.e. it would equal zero in a purely additive model, indicating no interaction). Var(( YX|,X Sij )) XX ij=--SS Var () YXX ij (2) A combined (or closed) index is created to aggregate the firstand second-order effects by adding the first-order index to the sum of all second-order effects related to this input variable. The “halving” is necessary to enforce a condition that the sum of all combined effects equals to 1 when the full variability of the output is explained. Kinput S SS c XX ii =+ sXX ij (3) j=12 ij Equations (1) and (2) can be determined using various mathematical methods and are abstractions of classical Sobol’ (1993) expressions. The sensitivity indices used in SimDec are computed via an innovative binning approach (Kozlova etal., 2023) in which the conditional expectation of Y to X (s) is determined by binning the X (s) and then calculating the averages of Y in those bins. Kozlova etal. (2023) have shown that indices calculated by this binning approach possess consistently high accuracy, even on very small datasets. These sensitivity indices can be calculated without any modifications to the algorithm for either simulated data or any measured dataset (i.e. containing input and output variables). SimDec algorithm and guidelines for its usage, interpretation 31 2.2 Decomposition procedure Figure2.2 presents a step-by-step illustration of the decomposition procedure on a stylized example model. The decomposition algorithm requires six steps. Step 1 and step 2 are used to determine which input variables to select for the visualization. Steps 3–6 construct the explicit visual decomposition of the model. 1. Compute sensitivity indices. The combined (firstand second-order) effects are calculated. 2. Order and select inputs. Athreshold of cumulative importance is established. Inputs are selected (in decreasing order of their sensitivity index values) until the cumulative sum of the indices equals or exceeds the threshold. (The threshold could be thought of as establishing “how much explanation of the output variability should be captured by the selected variables”.) 3. Divide inputs into states. The automatic procedure breaks down the numeric range of each selected input into two or three states – with the same number of observations in each. Three states are chosen if there are two or fewer variables selected in step 2. Otherwise, two states are formed. If an input variable consists of five or fewer unique values, the number of states created corresponds to the number of values (i.e. each value corresponds to exactly one state). 4. Form scenarios. All combinations of all states of the selected input variables form an exhaustive set of scenarios. This association is used to construct the legend in the visualization. 5. Map simulated outputs to scenarios. Each output value from the dataset is matched to a specific scenario based on the corresponding values of its inputs and the association created in step 4. (The scenario allocations enable a subsequent visualization of the data. As with “classic simulation”, the visualization in SimDec is a histogram depicting the distribution of output values.) 6. Colour-code the output distribution. The simple output histogram is converted into a stacked histogram (which preserves the original shape), with the scenarios from step 4 as the series. The colour-coding follows a specific rule: the states of the most important variable are assigned distinct primary colours, while all other partitions assume shades of these main colours. The stacked histogram colouring logic for SimDec can be used in other types of visualizations. For example, box plots can be used when some scenarios contain very little data, the shape of the distribution is inconvenient for visualization, or the data sample is too small to enable the creation of a meaningful histogram (see Section5.3). 32 Mariia Kozlova et al. Figure 2.2 SimDec algorithm illustrated on an example model.1 (colour image is accessible via the link) Source: Kozlova etal. (2024). SimDec algorithm and guidelines for its usage, interpretation 33 3 Ho w to read SimDec The various concepts needed to construct an effective interpretation of the SimDec results include an understanding of: • What do the sensitivity indices actually mean? • How to read a histogram? • How to judge the degree of influence of one input on the output using SimDec? • How to read joint effects of several inputs on the output using SimDec? 3.1 Sensitivity indices The sensitivity indices used in SimDec are global variance–based indices. Global means that they are computed when everything is changing simultaneously (as opposed to one-at-a-time analysis), and variance-based means that the index shows how much variability/variation of the output is explained. The simple binning algorithm (Kozlova etal., 2023) used in SimDec computes three types of indices: first-order (or main) effects, second-order (or interaction) effects, and combined (or closed) indices. The combined indices aggregate the firstand second-order effects and provide the default measures used to identify the most influential input variables selected for decomposition. The first-order indices indicate how much each input variable contributes individually to the variance of the output. For example, in a situation of YX=, the first-order index of X would be 1.0 (as it explains 100% of the variability). In an additive model YX=+ 12 X, where X1 and X2 have identical distributions (or numeric ranges), both inputs would have first-order indices of 0.5 (as, for this model, each explains 50% of the variability). An input variable can have close to 0 influence, if its numeric range is small compared to other variables or if the model mechanics dictate that there is little impact from it. The second-order indices describe how much a pair of input variables contributes to the variance of the output on top of their individual influence. These indices would necessarily be zero for additive models. For example, in YX=+ 12 X, the second-order index for the pair XX 12 would be equal to 0. Second-order indices can be positive if the input variables are multiplied in the model or possess a more complex interaction (see 3.4.1, “Interactions”). Apositive second-order index means that the pair of variables affects the output synergistically (i.e. together they produce more influence than simply a sum of their individual effects). Second-order indices can assume negative values, which indicate an overlapping effect of these variables (i.e. a correlation or dependency) in the model. For example, if X1 and X2 assume the same values in every single simulation run of the model YX=+ 12 X, their second-order effect would be −1.0, denoting a full overlap of their effects. In situations 34 Mariia Kozlova et al. where both correlations and interactions are present, the second-order index takes the sign of whichever effect is more pronounced (Kozlova etal., 2023). Combined sensitivity indices are calculated for every input variable as the sum of their first-order index and a halved sum of their second-order indices with all other input variables. The halving is needed to avoid double-counting when summing up all the indices. In the previous example of YX=+ 12 X, with X1 and X2 taking identical values in every simulation run, the first-order effects of both will be 1.0 (since each input separately explains the full variability of the output), the second-order effect of this pair of inputs would be −1.0, the combined index for each input is then 0.5 [= 1.0 + (−1.0)/2], and the sum of the combined indices is 1.0 [= 0.5 + 0.5]. The final summation value of 1.0 means that, overall, the entire variance of the output is explained by these two input variables. The sum of the combined indices provides a good estimation of whether the selected input variables fully explain the variation of the model output. If the sum is lower than 1.0, it might indicate that there is unaccounted randomness occurring within the model (e.g. if some input variables are not registered for the analysis, there might be some stochasticity coming from the model mechanics or its environment). An alternate explanation might involve the existence of considerable third-order effects. However, third-order effects are a rare phenomenon. Asum of combined indices considerably higher than 1.0 indicates a significant overlapping of information content in the model/ system and is a common attribute from the analysis of different model layers (e.g. aggregates of random input variables) or empirical data. One should bear in mind that sensitivity indices represent approximate estimates and are prone to numeric noise, especially in small-sample/ high-number-of-variables situations. Under such circumstances, all indices (especially the second-order ones) can be affected by noise (e.g. many indices would hold values in the 0.01–0.02 range). The focus of analysis should be redirected onto those input variables with indices over 0.05, while all these low-value-effect variables can be safely discarded. In SimDec, the sensitivity indices prove instrumental in helping to select which inputs to choose for the decomposition, while any actual reporting of their calculated values remains optional. 3.2 Probability distribution/histogram A histogram provides a representation of the distribution of numerical data. Its horizontal axis shows the range of the variable of interest, and its vertical axis denotes the count (also called frequency) or the probability (if the count has been divided by the total number of data points). One could think of creating a histogram as the deliberate act of distributing cubes with numbers (data points) across baskets (bins) that designate the specified number range. Figure2.3 demonstrates an example of such an action on a small scale. SimDec algorithm and guidelines for its usage, interpretation 35 Figure 2.3 A histogram built for an array of Y={11, 12, 25, 28, 28, 29, 31, 35, 39, 41}. (colour image is accessible via the link) In Figure2.3, the Y-axis can be converted from count to probability if divided by the total number of data points, 10. Its labels would be converted from 1 to 0.1 and from 4 to 0.4. The 0.4 mark implies that bins which reach it contain 40% of data (since the bin 20–30 contains 4 cubes, which is 40%). One should note that changing a bin width would also affect the Y-axis of a histogram. A distribution alone can supply only limited information about the data – its minimum, maximum, shape (where most of the data occurs), together with some additional descriptive statistics. However, an explicit mapping of which input values lead into which specific regions of the output distribution (as provided by SimDec) enables a more definitive exposure of the underlying model behaviour. 3.3 Streng th of influence When decomposing a histogram by a specific single input variable, one can visually perceive its degree of influence. Figure 2.4 demonstrates various single-variable decompositions that project commonly observed scatter plot patterns onto their congruent SimDec visualizations. If an input variable has no effect on the output, then its states (e.g. low and high) would lie on top of each other in the SimDec histogram, with fully overlapping output ranges. In such a case, the border between the states would be essentially horizontal, and the corresponding sensitivity index would be equal to 0. If an input variable has a strong effect and explains most of the variance of the output, the borders between its states on the SimDec histogram would appear more vertical. Such visualizations have important decision-making implications (e.g. if the high state of X can be fixed by the decision-maker, it would guarantee a certain range of values for Y). 36 Mariia Kozlova et al. Figure 2.4 Schematic visualization of different degrees of influence of an input variable on a model output in two different visualization types. (colour image is accessible via the link) The cases in-between possessing low-to-strong effects would display a more “diagonally-appearing” border division between states. The less the states overlap each other, the larger the effect of X on Y. While horizontal displacements of sub-distributions on the SimDec histogram are key to interpreting the results, vertical positionings occur based solely on the technical plotting order of the series in the stacked histogram. 3.4 Joint effects When two or more input variables are used for decomposition, it becomes possible to examine their joint effects. There are, fundamentally, three ways that a pair of input variables can jointly affect the output: • The same as the sum of their individual effects (i.e. an absence of correlation or interaction) • A synergy or extra effect on top of the sum of their individual effects (i.e. interaction) • A redundancy or overlapping effect (i.e. correlation) SimDec algorithm and guidelines for its usage, interpretation 37 This subsection illustrates how SimDec portrays the different cases behind interactions and correlations in contrast to the no-joint-effect situation. 3.4.1 Interactions The schematic visualization in Figure2.5 depicts how different types of interactions of input variables on the output appear in SimDec visualizations. Figure2.5A shows how the various sub-distributions (the different colours) of an additive model in which both input variables are equally important would be uniformly shifted. The corresponding second-order effect of such inputs would be equal to zero. Figure 2.5B illustrates the linear interaction effect that is characteristic of multiplicative models. In the SimDec histogram, the sub-distributions become shifted more-and-more along the horizontal axis. The effect of one input on the output becomes increasingly more magnified with the increasing value of the other input. The sensitivity index computed for the second-order effect of such input variables would be non-zero. The model of an electric aircraft flying range as a function of the capacity of its batteries and the power of its electric motor provides an example of such a linear interaction effect (Kozlova etal., 2021). In another type of interaction, Figure 2.5C demonstrates how one input variable can switch the direction of influence on the output in different states of the other input variable. Such an effect might occur due to a sign change in a model. The calculated second-order effect would be non-zero. Such an interaction was observed in the carbon footprint model of Kozlova and Yeomans (2022), where, in the case of disposal via landfilling, an increased usage improved the footprint. However, for the case of disposal via incineration, the opposite footprint effect happened. Namely, an increased usage deteriorated the footprint because of accounting for negative carbon emissions. Figure2.5D demonstrates that other types of nonlinear interactions can occur in models. For example, an input variable might have no effect on the output in one state of another variable (the red-shaded sub-distributions lying on top of each other) but exhibit a strong effect otherwise (the shifted blue pattern sub-distributions). Such non-linear effects will possess non-zero second-order sensitivity indices. The crying baby model of Kozlova etal. (2024) illustrates such an interaction in which the model parameters only affect the output when a particular type of optimization is used. Figure2.5 displays one example without interaction (Figure2.5A) and three cases possessing very different types of interactions (Figures2.5B–2.5D). In Figures2.5B–2.5D, the interaction effects are detected by the calculation of non-zero second-order indices. However, it is impossible to ascertain exactly what types of interactions are present without the accompanying SimDec visualizations. Teasing out the nature of the underlying interaction effects in a computational model and understanding the behaviour, in general, is crucial for effective decision-making. 44 Mariia Kozlova et al. Figure 2.7 The output of the SimDec Python package. (colour image is accessible via the link) > auto_vis$simdec_plot > auto_vis$legend_table The variables for decomposition can also be user-defined (rather than determined automatically), and the colours and appearance of the resulting histogram can be customized manually. For example, to modify the look of the stacked histogram, one could execute the following lines of code. SimDec algorithm and guidelines for its usage, interpretation 45 Figure 2.8 Web-based SimDec dashboard simdec.io. (colour image is accessible via the link) 46 Mariia Kozlova et al. > colors <- c(‘#8c5eff’, ‘#ffe252’, ‘#0dd189’) > custom_vis <- simdec_visualization(output, inputs, SI, main_colors = colors) > custom_vis$simdec_plot > custom_vis$legend_table Table 2. 2 Uses and arguments of the SimDec R package functions Function Purpose Inputs Outputs sensitivity_ Computes – input – SI – combined sensitivity indices.R sensitivity – output indices indices – FOE – first-order effects – SOE – second-order effects simdec_ Automatically – input – scenarios – vector of visualization.R generates – output scenario indices of size SimDec’s – SI Nruns * 1 stacked – scen_legend – association histogram between inputs’ states and visualization scenarios – boundaries – numeric boundaries of the formed states – simdec_plot – SimDec stacked histogram visualization – legend_table – a legend for the plot A deliberate choice was made during the development process so that the function sensitivity_indices would not include any dependencies in order ensure that future “update”-related maintenance issues were minimized. Despite this design decision, the overall SimDec package remains computationally efficient. Conversely, the simdec_visualization function does possess several dependencies. However, these dependencies have been carefully chosen to ensure that only regularly maintained packages available through CRAN have been relied upon. Currently, simdec_visualization depends on ggplot2, dplyr, colorspace, gridExtra, and kableExtra. The mandatory arguments for each of the functions are described in detail in Table2.2, whereas the up-to-date list of optional arguments can be accessed in SimDec R package documentation in GitHub. SimDec algorithm and guidelines for its usage, interpretation 47 4.3 Julia The scientific computing language Julia is designed for numerical computation with a syntax like Matlab and Python but with the speed of C++ (Bezanson etal., 2017). The lightweight Julia package, SimulationDecomposition. jl, can be installed using the following commands: After installation, the user can import the SimulationDecomposition package into the Julia REPL. Astructure containing the data table and bins is created by first loading the data, selecting the input variables, selecting a target output variable, and number of bins, then calling. The coloured histogram and table can then be displayed for further analysis using the functions plot and table, which result in a similar visualization to that of Figure2.7. An example notebook using the Pluto.jl package can be found in the GitHub repository.5 4.4 Matlab The SimDec Matlab package employs two main functions (see Table2.3). The Matlab functions6 must be downloaded and their corresponding folder must be explicitly activated as a path in Matlab. The input data needs to be provided in the form of two variables: inputs (of the size NK runs *inputs) and output (of the size Nruns *1). julia>] # to enter package mode (@v1.9) pkg> add https://github.com/Simulation-Decomposition/SimulationDecomposition.jl julia> using SimulationDecomposition julia> data = load_data(“data_engineering.csv”) julia> inputs = [:Battery, :Motor] julia> target = :Distance julia> nbins = 50 julia> simdec = SimDec(data, target, nbins) julia> plot(simdec) julia> table(simdec) 48 Mariia Kozlova et al. After the output and the input variables have been formulated in the Matlab workspace, the entire SimDec procedure can be run via these two functions, which will produce a visualization similar to Figure2.7. % getting the data Matrix = xlsread (“example_data.xlsx”); output = Matrix(:,1); inputs = Matrix(:,2:end); % running SimDec [SI, FOE, SOE] = sensitivity_indices (output, inputs) [scenarios, scen_legend, boundaries] = simdec_visualization (output, inputs, SI); Several optional arguments are available for the simdec_visualization.m function in order to customize a decomposition (see the up-to-date list of optional arguments in the documentation of SimDec Matlab function on GitHub). The optional arguments are set as in any standard Matlab instance. For example, the following code specifies the names of the variables and changes the colour palette. % custom names and colors output_name = ‘Output’; input_names = {‘Input1’,’Input2’,’Input3’,’Input4’}; colors = {‘#3F45D0’,’#DC267F’,’26DCD1’}; [scenarios, scen_legend, boundaries] = simdec_visualization (output, inputs,. . . SI,’OutputName’,output_name,’InputNames’,input_ names,’MainColors’,colors); 4.5 Excel template The Excel template is designed to work with spreadsheet models. The template is downloadable via GitHub7 and contains an example model, a main sheet for the SimDec interface, and a VBA macro that performs the requisite SimDec functionality. The SimDec interface (see Figure2.9) consists of (1) the Monte Carlo simulation area, (2) a decomposition set-up, and (3) the SimDec algorithm and guidelines for its usage, interpretation 49 resulting output graphics, together with appropriate summary statistics and a legend. Detailed instructions for utilizing the template can be found either in a specifically dedicated video tutorial8 or in Kozlova and Yeomans (2022). The most important distinction between this Excel tool and all remaining SimDec packages is that sensitivity indices are not computed in the template. Any decision to select which variables to use in the decomposition (and their ordering) remains entirely at the discretion of the user. 5 Usage nuances Several questions come to mind when studying a model with SimDec: How many input variables should be randomized? What should the sample size be? How should the data be sampled? Which variables to choose for decomposition? How to form scenarios for the decomposition? What are the alternatives to stacked histogram visualizations, and when to use them? These questions all are addressed in this section. 5.1 Selection of input variables for decomposition By default, SimDec uses sensitivity indices to indicate exactly which variables to select for decomposition. The de facto method-of-choice is the simple binning approach of Kozlova etal. (2023) and this procedure is incorporated into all SimDec packages. However, any other method for computing sensitivity indices could be used, if preferred. Variance-based methods make more sense, since they straightforwardly translate higher values into more widely dispersed variable states in the histogram. It is also easier to work Table 2. 3 Main functions of the SimDec Matlab package Function Purpose Inputs Outputs sensitivity_ Computes – inputs – SI – combined sensitivity indices.m sensitivity – outputs indices indices – FOE – first-order effects – SOE – second-order effects simdec_ Automatically – inputs – scenarios – vector of scenario visualization.m creates Sim-– outputs indices of size Nruns * 1 Dec stacked – SI – scen_legend – association histogram between inputs’ states and visualization scenarios – boundaries – numeric boundaries of the formed states – stacked_histogram – object that returns the visualization and the legend 50 Mariia Kozlova et al. Figure 2.9 SimDec Excel template. (colour image is accessible via the link) SimDec algorithm and guidelines for its usage, interpretation 51 with methods that can operate on the given data (Plischke, 2012; Puy etal., 2024; Kozlova etal., 2023), since the same dataset is later used to build the visualization. Conversely, selection of input variables for decomposition could also be done manually – whether for exploratory data analysis purposes or to satisfy alternate decision contexts. In complex nonlinear models, exploring the nature of separate interaction effects or the shape of a single-input variable influence on the output can produce additional insights into the model behaviour (Ahola etal., 2024). Some decision problems might dictate the specific choice of variables (e.g. the public policy requirements for project performance in Kozlova etal. (2016)). 5.2 States and scenario formation One important distinction of SimDec from scenario analysis is that the set of scenarios is not arbitrarily decided upon but results from listing all state combinations in the actual decomposition. However, the choice of the number and numeric boundaries of states is more flexible. By default, SimDec creates three states if two input variables are selected and two states otherwise. An exception occurs when an input variable can assume no more than five unique values, in which case, each value instance becomes its own separate state. The number of states can always be modified in response to the specific needs of the decision context. For example, in one of the SimDec application chapters, a decomposition into nine states is created in order to depict the sub-distributions of nine distinct market opportunities (Myers etal., 2024). It is imperative to supply the corresponding number of HEX codes for the main colours to the function in order to ensure its proper functioning. For establishing the numeric boundaries between states, the default procedure is to ensure an equal apportionment of data into each state. An alternative, inbuilt option is to choose equally-sized numeric intervals for the states. The two approaches create an identical set of states if the input variables are uniformly distributed. For non-uniformly distributed variables, the “equal-interval” principle allocates different amounts of data into each state. The choice should be based on the specific decision context. For example, use the “equal-amount-of-data” option to reflect the equal probabilities of occurrence of different states for external variables not controlled by the decision-maker. However, equal-sized-interval states would be preferable if the variable is under the decision-maker’s control. Custom numeric boundaries have been prescribed to reflect certain key thresholds imposed by a decision-maker, where SimDec can then be used to see whether achieving that threshold (or not) proves beneficial (Kozlova etal., 2016). However, if the purpose of an analysis is to study the behaviour of the underlying model, then the default state formation method is advised in order to prevent possible visual distortions in the visualization. For example, 52 Mariia Kozlova et al. an increasing number of data points in states of a uniformly distributed input variable can be confused with linear interaction (see Figure1.5B), while erroneous boundary setting that results in no data in a state can be confused with correlation (see Figure1.6C). If empirical data is analyzed, algorithms that detect natural breaking points for defining the state boundaries might provide the judicious choice. Studying the application of SimDec to empirical data provides a potentially fruitful avenue for the direction of future research and development. 5.3 Sample size and sampling strategies Sample size, in conjunction with the number of randomized input variables, can affect SimDec performance in two ways: (1) accuracy of the computed sensitivity indices, and (2) smoothness of the stacked histogram visualization. For computing sensitivity indices, the larger the sample size and the fewer the number of variables, the higher the accuracy of resulting estimations. First-order indices converge sooner (Marzban & Lahmer, 2016) than second-order ones (Kozlova etal., 2023). In general, as few as 1,000 data points are sufficient to generate stable and reliable sensitivity indices for models possessing six input variables (Kozlova etal., 2023). Furthermore, quasi-random sampling can be used to improve the accuracy (Kozlova etal., 2023). Increasing noise combined with higher numbers of input variables results in much noisier second-order effects (most of the pairs of input variables show joint 0.01–0.02 effects instead of zero). In such situations, the resulting sum of indices has been observed to overshoot the expected 100%. Nevertheless, even with either a smaller sample size or a larger number of variables, first-order indices have been reliably used to judge the relative importance of input variables. Examples of this range of reliability can be observed for an application with 29 inputs and 1,000 sample (Pellegrino etal., 2024) and for a case of a partial dataset of only 152 points (Pérez etal., 2024). In a visualization, the more data points there are, the smoother the histogram itself and the more distinct the borders between the scenarios appear (see Figure2.10). One thousand data points appear to establish the basic minimum amount of data for clear readability. However, if the computational costs are bearable, then 10,000 data points are recommended, as this can produce a very smooth and crisp visualization. An increased number of uncertain variables causes more uncertainty in the model output. This uncertainty results in a larger overlap of consecutive scenarios. The choice for the number of randomized input variables for the simulation (or data analysis) should appropriately reflect the task at hand – a higher number for more realistic modelling of the system and a lower number for studying the key behaviours in the model. Iterative analysis with consequent removal of SimDec algorithm and guidelines for its usage, interpretation 53 less-influential inputs (or the adding of other variables if something important had been missed) enables in-depth exploration of a model behaviour. It can be observed that different sampling strategies do not produce significant differences (see each column in Figure2.10). Quasi-random sampling and full factorial designs result in slightly smoother visualizations at sample sizes of 10,000. However, full factorial designs involve rapid escalations in computational costs as the number of input variables increases. Thus, quasi-random sampling can be recommended as the best approach for data generation in SimDec, as this simultaneously improves both the quantitative and visual aspects of the results. 5.4 Alternative visualization types Figure2.11 shows that the same information depicted in stacked histograms can also be visualized using box plots. In the figure, each scenario indicated previously with a specific-coloured sub-distribution in the stacked histogram is now represented by a separate box in the box plot. Furthermore, a detailed tracing indicates that each box is located precisely under the correspondingly coloured sub-distribution of the histogram above. Box plots provide a useful alternative under the following circumstances. • Some of the scenarios contain very little data and are not visible on the histogram (see an example in Figure2.12). • The shape of the distribution is inconvenient for histogram visualizations (for example, the too-skewed distribution in an exponential model). • The data sample is too small, and the histogram is too dissected (e.g. Figure2.10, bottom row). The colour-coding of scatter plots according to the values of another input have occasionally appeared in the literature (see, for example, Palar etal., 2023). However, a multivariable decomposition colouring on the scatter plot did not yield any informative visualizations. Scatter plot visualizations tend to be obscured by the overlapping of the dots in the scenarios and by the need to read the effect of one input variable relative to the dots’ location while reading (an)others relative to their colours. Overlay charts have also been considered as an alternative visualization format and have been incorporated as an optional display in a number of the commercial spreadsheet add-ins. The idea is analogous to the scenario portrayal in a stacked histogram, but instead of the stacking, these “scenarios” are overlayed instead. However, this overlaying approach possesses multiple drawbacks, including problems in visualizing multiple scenarios, simultaneously, which is an essential “must-have” pre-requisite provided by SimDec (Kozlova & Yeomans, 2020). Consequently, stacked histograms (and box DOI: 10.4324/9781003453789-4 This chapter has been made available under a CC-BY-NC-ND 4.0 license. Abstract This chapter provides an overview of the collection of SimDec application cases in this book. Ageneral summary of case themes, angles of analysis, modelling choices, and post-processing practices is presented. The chapter also introduces several of the brightest “aha moments” generated from SimDec, whether the uncovering of complex effects previously hidden within the models or the veritable “unshackling” of mindsets experienced by the modelers themselves. 1 Introduction The biggest contributions of this book appear in the application chapters used to demonstrate how SimDec has been implemented in several, very different contexts. The sole goal of this chapter is to preview these applications in order to provide the reader with an overall impression of what is to come and/or to direct them to a relevant portion of the book that might captivate their interest. However, it is not only the context that the reader might be willing to consider. Independently of the context, some cases might present a relevant mathematical framework, or a curious approach to sensitivity analysis, or an intriguing effect that the specific model harbours. The chapter proceeds by outlining the various themes and rationale behind each of the cases. Then it provides an overview of the modelling approaches, with a specific focus on the model analysis practices that the contributors had been using prior to SimDec. Finally, it highlights the most “peculiar” characteristics and findings observed across the chapters. These aspects include: (1) the apparent conventional inertia behind deterministic thinking, (2) the relative ubiquity of intertwined heterogeneous effects existing within even the simplest of models, and (3) how the plethora of novel functionality accompanying SimDec leads to a liberation of sensitivity analysis from its prevailing custom of examining the effect of inputs on the outputs. The chapter concludes by summarizing how much more can be done Chapter 3 Overview of SimDec applications Mariia Kozlova and Julian Scott Yeomans Overview of SimDec applications 61 analytically with SimDec and projections of the authors’ future aspirations for the method. 2 Themes Thematically, the application chapters have been partitioned into four distinct domains: (1) Business, (2) Environment, (3) Engineering, and (4) Behavioural Science (see Table2.1). Each chapter has been assigned an abbreviated “code name” so that it can be referred to concisely within the body of the text. The Business domain encompasses cases devoted to various facets of strategic decision-making. The first chapter in the domain builds on a classic corporate finance example of investment valuation and expands it with considerations of uncertainty and design of suitable managerial actions (4_Invest). Next, the perspective shifts from an investor to a policymaker in a study examining public policy instruments for facilitating private–public partnerships (5_Public). Further, the context shifts from regular to cutting-edge: an emerging technology of construction via 3D printing is analyzed from the unit cost perspective (6_Constr). The Business domain concludes with an audacious experiment that translates a qualitative framework (Market Opportunity Navigator) into a quantitative tool (7_Deeptech). The Environmental domain consists of three very diverse applications. 8_ Carbon provides a classic study of emissions forming part of the contentious debate on whether single-use or reusable products are more sustainable. 9_ MinEx examines a sequential decision-making model built for mineral exploration which, by concentrating on the model fidelity, shifts the focus from “how good is the decision” to “how good is the model”. 10_P2X advances the conservative practices of classic techno-economic analysis using an example of power-to-X technology. The Engineering domain considers two distinctly captivating studies. 11_Reliable introduces a model to predict fatigue in the welded joints of steel structures that possess a very peculiar nonlinear behaviour that the authors had struggled to convey prior to SimDec. 12_Magnet presents a simulation-based study of a computationally intensive, three-layered model of a superconducting magnet at CERN that effectively contrasts the performance of alternative magnet design options. From the Behavioural Science domain, the applications conclude with 13_ Choice. This chapter marks a departure from the realm of formal academic and industrial applications towards more personal decision situations – ranging from pondering mortgage conditions to choosing a country of residence. Within this domain, SimDec is used to transform intuition and intertwined personal preferences into a visual representation of the different choice outcomes which, in many cases, led to new realizations through the reframing and/or redesigning of the specific decision situation. 62 Mariia Kozlova and Julian Scott Yeomans Table 3. 1 The overview of application chapters and their themes ) ) ) )v etal., 2024 ) eref ellegrino etal., 2024 W)alzer etal., 2024 )ers etal., 2024 P Moss etal., 2024) ence My Vinitskaia etal., 2024 o etal., 2024 R Karjunen etal., 2024 Ahola etal., 2024) ) Stepano enk érez etal., 2024P Sidor ( ( ( ( ( ( ( ( ( ( ent elop tunities y finding es under dif- -use and viour - - . erf tgage, y dev ellness estment b y of dif . . elopment oppor . v ’ incentiv . y y choice, mor . s . ysi able in ter estors urpose of the anal ofit v t of technolog ootprint of single tions that mat ypes anding in ec . y t ting a set of dev s f zing model fidelit Designing a pr Underst erent polic ac eusable mask estigating economic viabilit ent design choices , car choice, w y options erf ting personal decision-making situa , countr estigating dif vings f Studying the ef ment on the unit cost tions: sa P Selec to pursue. Comparing carbon f r Analy v technolog Communicating complex nonlinear beha of the model. vIn Suppor language learning In onment Engineering vioural Domain Business virEn Beha Code 4_Invest 5_Public 6_Constr 7_Deeptech 8_Carbon 9_MinEx 10_P2X 11_Reliable 12_Magnet 13_Choice turing t of Deep tech mark ootprint tion et s tunities vid mask Carbon f of Co Critical mineral e estment Public suppor ting tur valuation infrastruc ts in construc oppor exploration y ojec 3D manufac X --toer tural conduc pr Struc reliabilit Case w v o Super magnet design ersonal decisions In P P # 4 5 6 7 8 9 10 11 12 13 Overview of SimDec applications 63 3 Models and pre-SimDec practices The cases in the book capture a wide variety of modelling conventions, with the authors more accustomed to their different, discipline-specific, follow-up practices for model analysis shown in Table3.2. Most of cases had been analyzed previously using pre-existing computational models. The relatively simple models of cash flows (4_Invest, 5_Public, and 10_P2X) had all been implemented in Excel. Reflecting the nature of its field, the arithmetically straightforward life cycle assessment (LCA) model (8_Carbon) possesses heavy underlying data collection requirements. Accounting for material and joint properties, 11_Reliable builds upon a novel custom model that predicts the fatigue in welded joints far better than the more commonly used regression models. One mathematically complex model in the book involved stochastic optimization by partially observable Markov decision process (POMDP) for arranging sequential decisions (9_MinEx). The most computationally-intensive model in the book is the one for evaluating the magnetic and mechanical properties of a complicated geometric structure approximated with a finite element method (12_Magnet). Asingle evaluation of this model took 13 minutes on average to solve using a supercomputer. Three application chapters did not possess any pre-existing models, so that their entire modelling exercise was inspired solely by the capabilities of SimDec. The unit cost estimation model was built within 6_Constr when it became clear that uncertainty in technology development, the effect of economies of scale, and variations in many other parameters could be studied simultaneously. The proponents of the 7_Deeptech case were frustrated by the existing qualitative approaches to uncertainty in deep tech ventures and eagerly availed themselves of the opportunity to transform their problem into a quantitative model. The resulting multi-criteria decision-making framework contributed a harmonic solution requiring only minimal adjustments to their existing qualitative tool. 13_Choice was inspired by the power of SimDec to solve more “unconventional” applications and transcends the experiences of any single person. The underlying models include (1) two basic functions (summation/wellness and power/learning), (2) computations based on annuities in two other cases (savings and mortgage), and (3) two simple multi-criteria decision-making models (car and country choice). 13_ Choice demonstrates that modelling “regular” life choices with SimDec can encourage more astute behaviour (delayed gratification), while converting decision-making into decision-situation-making. All case contributors had used other model analysis prior to their work with SimDec, with some using a single analysis approach and others employing several methods (see Table3.2). One-at-a-time sensitivity analysis and scenario analysis were the most common methods employed by the contributors. Heat maps and Monte Carlo simulation had been considered less frequently. Nevertheless, neither of these types of analysis can create 64 Mariia Kozlova and Julian Scott Yeomans Table 3. 2 Modelling approaches and practices of case owners pre-SimDec MC§ – – – Heat maps‡ – – – s ios† ysi Scenar – – – . odel anal *AT – – – M O ) evaluation cases al Ball celx yst , EA e C , Matlab ouree to f war thr t cel cel, Cr cel cel Sof x x E E E GaBi L cel x x x E Julia E Matlab ANSYS celx ew ( E -magnetic models or a f e varied at the same time. criteria . o y analysis tr , multi- . ain inputs ar low model tions y analysis t ype unc odel t w modello criteria decision-making eral inputs at a time, but only f Unit cost estimation cle assessmenty decision-making M MultiSequential decision-making w model atigue assessmentF Mechanical and elec e all uncer lo Cash f -a-time sensitivit es changing sev -at-a-time sensitivit Cash f e c o Lif Cash f Simple f w odel olv ted -at s – – – tically a t xi M or onev e ands f terChap 4_Invest 5_Public 6_Constr 7_Deeptech 8_Carbon Reliable 9_MinEx 10_P2X 12_Magnet T stA _11 13_Choice * O † Scenario analysis in ‡ Heat maps is prac § MC is Monte Carlo simulation, wher Overview of SimDec applications 65 a holistic picture of the model behaviour (Kozlova, Lo Piano, etal., 2024). Better opportunities arise when an analysis is performed iteratively over different sets of model conditions in the model (i.e. the 9 iterations of OAT in 5_Public and the 12 iterations in 11_Reliable). However, discerning any appropriate insights from so many separate graphs is a highly complicated exercise, matched only by the complexities of communicating such insights to other stakeholders. Notably, pursuing a more concise systematic solution to this iterative exercise established the initial groundwork for creating SimDec (Kozlova etal., 2017). For each case contributor, their work with SimDec provided their first exposure to global sensitivity analysis (GSA). This widespread “ignorance” of GSA illustrates another manifestation within modern modelling culture. Saltelli etal. (2019) had indicated that the neglect of GSA occurs due to its relative analytical complexity, combined with a paucity of exposure to it in academic teaching. 4 SimDec pearls 4.1 Away from determinism Deterministic models possess several pitfalls. Firstly, a modeler’s capacity to find mistakes is limited. In general, only when the output values deviate significantly from their expected range can one determine that an error has crept into the model. Spreadsheet models (which appear in several chapters) are especially prone to these mistakes. In spreadsheets, a line of code can be represented across multiple cells, with each cell editable separately (e.g. the same equation might be spread over different time periods). Whenever a single cell in one “line” is edited, it does not become immediately apparent, because the formula is hidden and only visible when that cell is selected. Even if the cell is edited intentionally, once the model is copied and adapted to another case, such specific edits are easily forgotten and may continue to distort the results for many cases in the future. Multiple instances of such inherited mistakes have occurred in the authors’ corporate experience. When applying SimDec for first time, several authors discovered model inconsistences that had to be corrected prior to the formal analysis. Several additional nuances were uncovered. In the CERN model of 12_Magnet, SimDec uncovered a previously unnoticed relationship pattern between interference, bladder pressure, and the cable height, for which the authors have still not yet been able to establish a physical explanation. SimDec informed the model building process for 7_Deeptech, in which the first iteration contained flawed links that only became apparent after the SimDec visualization. In the 8_Carbon case, SimDec identified which input variables and process elements had negligible importance even with extreme numeric ranges, thereby enabling the investigators to adjust the direction of their burdensome data 66 Mariia Kozlova and Julian Scott Yeomans collection process. Under such circumstances, SimDec proves to be an incredibly convenient tool for assisting with model building and testing. Secondly, deterministic models can fail to capture the varying degrees of uncertainty in different model elements or computational scenarios. Two similarly profitable investment opportunities in the base case may be very different in terms of their risks and potential. Thus, they would attract a different type of investor or, as in the case of 5_Public support, also entail a different degree of budget depletion for the government. Similarly, different 7_Deeptech opportunities encompassed different level risks. In the multi-criteria choice for country of residence in 13_Choice, the different width in the scoring of country alternatives was caused by the variation of characteristics within a country and imprecision of their estimates – both adding an important slant to the decision-making situation and prompting a narrowing of the scope. Thirdly, basing decisions on deterministic outputs leads to tunnel vision and precludes strategizing. Aclassic and widely used rule in investment appraisal is to “invest if NPV>0”. Chapter4_Invest criticizes such an approach and shows how much more can be achieved if a proper uncertainty and sensitivity analysis has been allowed to take place. Arigid investment opportunity containing a lot of risks was transformed into an alluring endeavour with built-in flexibility to address the most critical source of profitability uncertainty – demand. It was suggested that the static term investment appraisal be substituted with the more actionable and uncertainty-encompassing term investment design. In LCA studies comparing single-use and reusable products, the key question is an examination of the breakeven usage for the reusable product. Beyond this point, it becomes more sustainable to utilize the reusable option rather than the single-use product. Chapter8_Carbon emphasizes the absurdity of such questions when there are so many dynamic elements involved. Abreakeven usage number is anything but deterministic, being, in fact, nonlinearly dependent on several factors in the product life cycle. Thus, designing a policy based on a single breakeven number would create an undesirable effect under all circumstances (except, of course, for the extremely rare case when all factors just happen to assume their mean values as in the solution of the deterministic model). The deeply-rooted custom of policymaking based on such crisply precise narratives has been heavily criticized by many thought-leaders (Saltelli etal., 2020; Savage & Markowitz, 2009). 4.2 Heterogeneous effects It is not simply ranges that matter (i.e. how uncertain the output values appear); it is what drives them that makes the difference in decision-making. This driver captures the essence of SimDec – to be able to display the output distribution and the most influential factors behind it on a single graph. Overview of SimDec applications 67 Figure 3.1 SimDec displaying nested heterogeneous effect, adopted from 11_Reliable. (colour image is accessible via the link) Source: Ahola etal. (2024). Colour A B C Low Low 1 2 High 1 2 Medium Low 1 - High 1 - High Low - 2 High - 2 68 Mariia Kozlova and Julian Scott Yeomans The degree of that influence is the most critical piece of information for decision-making. Kozlova, Moss, etal. (2024) show that the effects of input variables on the output can be heterogeneous: (1) one input can gradually increase its influence on the output in combination with the higher values of another input, (2) the direction of influence on the output of one input can be reversed based upon different ranges of another input, or (3) the very presence of the influence of one input on the output can be conditioned to another input. In the presence of heterogeneous system behaviours, the decision-maker must carefully account for such case-specific behaviour in devising an effective way forward. The applications illustrate various cases of heterogeneous effects. 11_Reliable showcases a beautifully intricate, nested heterogeneous effect involving three input variables. To illustrate it, let us denote the most important input variable as A, the second-most important as B, the third-most important as C, and the output variable as Y (Figure2.1). Figure3.1 displays a very complex nested what-if effect. If A is in its high or medium state, the Y distribution is narrow, B does not play much role, and C has only one state present in each state of A. Conversely, if A is in its low state, then B has a lot if influence on Y, and in this situation, C affects Y a lot if B is high and only moderately if B is low. All the variables A, B, and C are controllable in the decision context, and their ranges represent possible variation. So depending on which portion of Y the decision-maker wants to achieve, either only A should be manipulated to medium or high or A and B if both A and B are low, or all three of them if A is low and B is high. The corresponding output ranges partially intersect, so several of the scenarios can lead to the same ranges of Y, thereby creating flexibility for the decision-maker. The richness of this triple joint heterogeneous effect is so comprehensive that a decision tree was built to support its readability. Another example of complex heterogeneous effects comes from the superconducting 12_Magnet model, where one of the outputs was affected by two input variables in a nested U-shaped fashion. Again, for illustration purposes, the context is skipped and the variables have been renamed. In Figure3.2, one can observe the U-shape formed by B in the states of A. For example, in low A (blue shades), the high and low B results in medium values of Y (scenarios of the darkest and the lightest shades of blue are on top of each other), but the medium B leads to high and low values of Y (other blue scenarios are located to the sides of the previously mentioned ones). While a BY scatter plot alone fails to reveal this pattern, SimDec readily teases out a visualization of the nested pattern due to the decomposition based on two input variables. Sometimes, heterogeneity presents itself based on determining which sets of input variables affect which different portions of the output data. Such multidimensionality can manifest itself in seemingly unexplained patterns on the SimDec chart and requires additional partitionings in the data in order to Overview of SimDec applications 69 obtain a clearer picture (as was done in 8_Carbon). Decomposing the entire dataset revealed several peaks in the output distribution (Figure3.3, top). Clearly, some other input variables were causing these peaks, but their influence was not apparent when the entire dataset was analyzed. The authors fixed the most influential variable (mask type) and examined the decomposition of only the single-use mask portion of data. The peaks still contained several overlapping colours. The most important input variable in that Figure 3.2 SimDec displaying nested U-shape effect, adopted from 12_Magnet (top row), and a scatter plot of the variable that causes it B to Y, but where the U-shape effect is not visible. (colour image is accessible via the link) Source: Pérez etal. (2024). Colour A B Low Low Medium High Medium Low Medium High High Low Medium High DOI: 10.4324/9781003453789-6 This chapter has been made available under a CC-BY-NC-ND 4.0 license. Abstract The assessment of investment projects relies on plentiful assumptions and combines significant uncertainty with rapidly accumulating complexity. However, the tools commonly employed in this discipline are far from ideal and often fail to capture the required level of complexity to adequately assist the wide spectrum of disparate needs of the various decision-makers. This chapter shows how Simulation Decomposition (SimDec) can improve investment decisions by shifting the focus from static invest-or-not decision rules towards dynamic, actionable project design mindsets. After introducing cash flow modelling, a case model is analyzed using SimDec to illustrate the method’s context-specific capabilities. SimDec not only captures the uncertainties and reveals which factors are most important but also shows how different combinations of them can shape the overall profitability of the project. The chapter adopts a multistakeholder perspective, informal language, and analogies to capture the shortfalls of traditional approaches to investment appraisal. 1 Introduction Companies are perpetually faced with the inevitable challenge of identifying and implementing value-creating investment projects. It comes as no surprise that financial strife (even bankruptcy) awaits those who are inefficient or complacent in this search for best projects. On the other hand, companies capable of identifying good investment opportunities consistently find their fortunes growing. The decisions that need to be taken are often complex because of the inevitable reliance on future estimates and plentiful assumptions. At best, the discipline of finance supplies structural frameworks and decision rules that generally lead to more favourable outcomes. The computation of Net Present Value (NPV) is widely regarded as the most established investment appraisal technique by providing a clear-cut decision rule for valuing investments. The result of NPV computation is nothing less than a “to be or not to Chapter 4 Unlocking actionability in financial modelling with Simulation Decomposition Roman Stepanov, Mariia Kozlova, and Julian Scott Yeomans 78 Roman Stepanov, Mariia Kozlova, and Julian Scott Yeomans be” moment, conveniently determined by the sign of a single number (Ross etal., 2014). If the NPV yields a positive number, the project is “to be” and should be accepted. Conversely, if the NPV is negative, the project is rejected as a “not to be”. The straightforward binary nature of the decision rule contributes to its popularity due to its intrinsic clarity and simplicity. However, this textbook NPV rule is not immune from rightful criticism usually attributable to the static nature of its underlying assumptions. Namely, a single number can never capture and convey the full complexity associated with future events. Adeterministic model can never successfully incorporate all of the multiple uncertainties and conflicting stakeholder perspectives. Nor can it offer sufficient insight to intervene and redesign any given investment project. Much more actionable information is needed to extrude an efficient business plan that sufficiently integrates all of the diverse complexities that exist. While the task of building the most detailed NPV models is relevant across the board, the subsequent approaches presented below in order of usefulness and sophistication, are also equally important (Graham & Harvey, 2001; Ryan & Ryan, 2002). The most pervasive approach is to do nothing more than rely simply on the single value of the NPV calculation. The oversimplification in this case is recklessly precarious. The second approach is to repeat the initial oversimplification absurdity three times under the guise of a “scenario analysis”. The third approach amounts to an iterated “what-if” function (aptly called one-at-a-time sensitivity (OAT) analysis) that generates data for the creation of a spider chart or a tornado diagram. This represents a next-level advance in analytical sophistication because the variables can be ranked in order of their individual effect on profitability and the most impactful ones can be identified. The fourth approach is to employ Monte Carlo simulation to examine the likelihoods and probability distributions of the outcomes. Simulation can be considered an advanced approach as it exposes the dynamic links between NPV and simultaneous changes in multiple variables. The outputs from this method can be displayed as probability distributions in the form of a histogram. One criticism of the Monte Carlo approach is that it does not provide a roadmap of what exactly needs to be done so that the project remains confined to the attractive portions of its distribution while avoiding the negative risks. However, this missing actionability feature has been addressed by the recently developed Simulation Decomposition (SimDec) approach of Kozlova and Yeomans (2022). SimDec shows precisely how certain combinations of input factors need to be changed in order to steer the project into the desirable portions of its profitability distribution. In effect, SimDec prompts and answers a series of actionable questions: Can anything be done? What? What else can be done to achieve the same result? How can it be done? By providing these capabilities, SimDec raises yet another question. How is it that the entire financial world is still satisfied Unlocking actionability in financial modelling with SimDec 79 with only the first four approaches of NPV modelling? Consequently, this chapter discusses the nuances of cash flow modelling and then demonstrates how SimDec can elucidate actionable investment road maps by testing it in a number of financial modelling situations. 2 Nuances of cash flow modelling 2.1 Stakeholder perspective on cash flow modelling The cash flow perspective on financial modelling and decision-making can be viewed metaphorically as essentially nothing more than the creation of a multilayered cake. At first glance, the cake appears entirely non-threatening and straightforward to make. However, a thorough knowledge of the recipe, an understanding of the inherent interaction between the ingredients, and the underlying multiplicity of the different layers all represent challenges to the task of successful baking. Analogously, cash flow analysis involves combining numerous ingredients, such as revenues, costs, capital expenditures, taxes, etc., and then spreading their planned involvement judiciously throughout the lifetime of the given investment project as if they were individual layers of a financial cake. Just like an experienced baker, a CEO can pursue their creative instincts when combining ingredients and layers or adopt a more scientific approach to investment planning. Under the more technical approach, each component must receive due consideration in terms of balancing its overall contribution and net value creation with respect to the other ingredients. People face investment decisions all the time. While some of these decisions may be more straightforward than others, they all contain an inevitable complexity associated with future uncertainties. Irrespective of investment type (e.g. the purchase of a fixed income security, an investment in real estate, construction of a production facility, a new service, etc.), all decisions can be captured by the following generic NPV valuation formula: CF NPVI 0 n =- +i i (1) i=1 () 1+r where I0 is the amount of the investment, CFi is the after tax difference in all revenues and costs in the ith period, n is the number of time periods, r is cost of capital, and NPV is the net present value. 2.1.1 T he perspective of the corporate finance expert A person trained in finance will be instantly drawn to the discount factor 1 which for the n th year is () 1+rn (see equation1). This factor needs to be 80 Roman Stepanov, Mariia Kozlova, and Julian Scott Yeomans Figure 4.1 Metaphor for cash flow modelling. (colour image is accessible via the link) Unlocking actionability in financial modelling with SimDec 81 applied to all future cash flows because, for investment evaluation, they should be expressed in terms of today’s money. According to convention, the most convenient representation of the investment project is a spreadsheet of the following format (Table4.1). Having performed this transformation, one can obtain estimates corresponding to the terms of the equation and demonstrate that if one invests 200,000 in return for an annual cash flow of 12,000 and, in five years’ time, sells the asset for a higher price of 210,000, no value will be generated, given the 8% cost of capital (see the last entry in the “Cumulative DCF” row, which is also known as the NPV of the project). This number is not encouraging, given that at this stage no maintenance, tax, and other expenses have been factored into the evaluation. Atable not too dissimilar to the preceding one, but typically displayed in Excel, is a common representation format for equation (1) by a corporate finance expert. 2.1.2 The perspective of the accountant The main focus for an accountant is the production of three financial statements: the income statement, the cash flow statement, and the balance sheet. In combination, these statements are designed to portray the overall financial position of a company in a true and fair way. Therefore, the preceding scenario would be viewed through the lens of their corresponding accounting standards. Accountants and investment appraisers “talk money” in different terms, and it is important for them to know the translation rules (see modelling traps in Sections2.2 and 2.3). Furthermore, accountants can provide extensive advice on various taxation schemes (see Section4.1) and other nuances that might potentially improve the outlook of a project (Fazzari etal., 1988). 2.1.3 The perspective of the budget holder In principle, budget holders ought to be concerned primarily with the feasibility of initial investments, the quality of the underlying assets, and their Table 4.1 Classic cash flow model arrangement in a spreadsheet environment Year (n) 012345 Capital –200,000 210,000 expenditure CF 12,000 12,000 12,000 12,000 12,000 Net CF –200,000 12,000 12,000 12,000 12,000 222,000 Discount factor 1.00 0.93 0.86 0.79 0.74 0.68 DCF –200,000 11,111 10,288 9,526 8,820 151,089 Cumulative –200,000 –188,889 –178,601 –169,075 –160,254 –9,165 DCF 82 Roman Stepanov, Mariia Kozlova, and Julian Scott Yeomans recoverable value at the end of the project. Since investments are frequently made upfront as a precondition for generating future cash flows, there is a natural temptation to receive the benefits as early as possible and incur the costs as late as feasible over the time horizon. However, the reality of the budget holders’ position within many organizations may be different. Allocated budgets generally need to be realized within tightly constrained periods, and thus, investments need to be made at the earliest possible opportunity in the project cycle. Otherwise, the money sits idle without generating due returns, an organization is exposed to potentially higher taxes, or in cases of public institutions, the entire budget allocations may be drastically reduced or withdrawn completely. 2.1.4 The perspective of the bank manager Bank managers are naturally conservative since their primary concern is loan repayment in accordance with their pre-specified repayment schedule. Consequently, evaluations of investment projects are performed on the basis of pessimistic cash flows, longer payback periods, and higher cost of capital (reflecting more risks) (equation (1)). An additional concern is to ensure that the lending rate (which constitutes the primary source of interest income) is appropriately matched to the proposed investment project under the prevailing economic conditions. 2.1.5 The perspective of the risk controller The risk controller must identify all possible project risks and measure them in terms of severity of impact and probability of occurrence; this is a well-known approach called a risk matrix. The reports containing the risk matrix look sophisticated and certainly contribute to a “feel good” factor for most executives. However, it has been shown that the subjective nature underlying this exercise has little to do with the actual probabilities of risk occurrence or their monetary damage to the company. Adoption of more quantitative methods has been strongly advocated for this crucial corporate function (Hubbard, 2020). This, however, requires additional retraining of the associated personnel (Sidorenko, 2023). 2.1.6 The perspective of the engineer Engineers tend to focus on the practicalities of project implementation as opposed to any underlying cost and cash flow aspects. This financial disconnect often places them at loggerheads with financiers during the project design phase. The challenging job of an investment committee is to balance a budget that ensures that the engineers receive financial resources sufficient to achieve their project-related specifications. Lean manufacturing (Zhu & Lin, 2017) and Design for Manufacturing and Assembly (DFMA) (Lu etal., Unlocking actionability in financial modelling with SimDec 83 2021) are innovative examples of how frontrunning organizations have bridged this gap. 2.1.7 The perspective of the CEO The role of the CEO is to view the investment project without being overwhelmed by the multiplicity of details, perspectives, and nuances of the investment project. Their most valued skill is the holistic ability to simultaneously parse a large quantity of interdisciplinary and interdepartmental information that may be incomplete, biased, and contradictory (Mitchell etal., 2016). From a financial perspective, the pinnacle of such simplification is frequently the single NPV number, which acts as an accept/reject criterion for the given investment project. 2.2 Modelling trap #1: actual vs accounting cash flows In practice, corporations represent the structure of cash flows in a model in two different ways: (1) actual and (2) accounting. In a simplified form, actual cash flows (Figure4.2) include investment cost (often referred to as CAPEX), operating profit (calculated by deducting operational expenses (OPEX) from revenues), and taxes (computed on the basis of accounting cash flows). Free cash flow (FCF) is the summation of these terms which forms the basis for computing numerous profitability indicators. Earnings before interest and taxes (EBIT) represent the accounting value used to determine the tax base (i.e. the value that is then multiplied by the tax rate). EBIT employs accounting conventions that convert the initial investment cost into a depreciation stream that allocates the costs throughout Figure 4.2 Interlink bet ween two possible structures of a cash flow model: actual cash flows (left) versus accounting cash flows (right). (colour image is accessible via the link) 84 Roman Stepanov, Mariia Kozlova, and Julian Scott Yeomans the lifespan of the project. To calculate free cash flow from EBIT, the initial investment cost needs to be included (with a minus sign) and depreciation added back (with a plus sign). This step of adding depreciation (after previously deducting it from EBITDA to get EBIT) is often a source of confusion for modellers unaware that depreciation is not a real cash flow item. This confusion is easily avoidable if cash flows are computed using only actual flows: investment cost plus operating profit minus taxes. 2.3 Modelling trap #2: real vs nominal, and constant vs current Another confusion often arises in maintaining consistency between cash flows and interest rates (cost of capital) when adjusting the calculation to account for inflation. Figure4.3 summarizes all possible terms for the two types of cash flows and interest rates. Cash flows and interest rates need to be expressed in the same terms and either adjusted for inflation (equivalently – excluding inflation effect/constant dollars/real terms cash flows) with real interest rates or including inflation (equivalently – current dollars/nominal terms cash flows) with nominal interest rates. Interest rates are normally reported in nominal terms. Thus, the discount rate, usually computed as weighted average of nominal equity rate and Figure 4.3 The connection between constant/current dollar and real/nominal interest rate. (colour image is accessible via the link) Unlocking actionability in financial modelling with SimDec 85 nominal debt rate, is also nominal. Following the instruction of Figure4.3, in order to stay consistent, nominal discount rate should be used with cash flows that include inflation. Alternatively, the discount rate should be converted into the real rate, and then it can be used with cash flows that exclude inflation. Both ways of computation would result in the same value for NPV in the absence of taxes and other complexities (Kozlova, 2019). 3 SimDec analysis In this section, SimDec is used to visualize the impact of changes in taxation, demand, and pricing on the case model’s overall profitability. The SimDec approach extends the uncertainty analysis of classic Monte Carlo simulation (Metropolis & Ulam, 1949) and simultaneously conducts a global sensitivity analysis (Borgonovo & Peccati, 2006). The main analysis is performed using the SimDec Excel template (Simulation Decomposition, 2023a; Kozlova & Yeomans, 2022), while additional SimDec Matlab scripts have been used to compute the sensitivity indices (Simulation Decomposition, 2023b; Kozlova etal., 2024). The notation of highlighting names of variables with bold italic and their states with italic is adopted henceforth. 3.1 Computational model and numeric assumptions The classic cash flow approach (Figure4.1 and equation (1)) for modelling investment profitability. The subsequent assumptions are used for the stylized model, which are further modified for each case condition (see Section4) (Table4.2). The cash flow model is simulated in the next section under different uncertainty assumptions specific to each circumstance considered (Sections3.2–3.4). The simulation data is further used for the sensitivity analysis with SimDec. Table 4.2 Numeric assumptions for the stylized cash flow model Input variable Base case value Investment, K$ 30,000 Resale value 3,000 Price, K$/pcs 2.5 Volume, pcs/year 20,000 Fixed costs, K$/year 9,000 Variable costs, K$/pcs 1.3 Other costs, K$/year 5,000 Writing down allowance 18% Income tax 25% Revenue inflation 3% Cost inflation 4% Cost of capital (real) 5% Cost of capital (nominal) 9% 92 Roman Stepanov, Mariia Kozlova, and Julian Scott Yeomans References Borgonovo, E., & Peccati, L. (2006). Uncertainty and global sensitivity analysis in the evaluation of investment projects. International Journal of Production Economics, 104(1), 62–73. Fazzari, S., Hubbard, R. G., & Petersen, B. (1988). Investment, financing decisions, and tax policy. 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Design for manufacture and assembly (DfMA) in construction: The old and the new. Architectural Engineering and Design Management, 17(1–2), 77–91. Metropolis, N., & Ulam, S. (1949). The Monte Carlo method. Journal of the American Statistical Association, 44(247), 335–341. Mitchell, R. K., Weaver, G. R., Agle, B. R., Bailey, A. D., & Carlson, J. (2016). Stakeholder agency and social welfare: Pluralism and decision making in the multi-objective corporation. Academy of Management Review, 41(2), 252–275. Ross, S. A., Westerfield, R., & Jordan, B. D. (2014). Fundamentals of corporate finance. Irwin. Ryan, P. A., & Ryan, G. P. (2002). Capital budgeting practices of the Fortune 1000: How have things changed. Journal of Business and Management, 8(4), 355–364. Sidorenko, A. (2023). Controversial thoughts about modern day risk management in non-financial companies, training and consulting services. Risk-Academy Blog. https://riskacademy.blog/author/alexausrisk/ Simulation Decomposition. (2023a). SimDec excel spreadsheet with macro. GitHub. https://github.com/Simulation-Decomposition/simdec-excel Simulation Decomposition. (2023b). SimDec Matlab package. GitHub. https:// github.com/Simulation-Decomposition/simdec-matlab Zhu, X., & Lin, Y. (2017). Does lean manufacturing improve firm value? Journal of Manufacturing Technology Management, 28(4), 422–437. DOI: 10.4324/9781003453789-7 This chapter has been made available under a CC-BY-NC-ND 4.0 license. Chapter 5 Unpacking the role of contextual factors in public support for mitigating revenue risk in public–private partnership projects Roberta Pellegrino, Mariia Kozlova, Luiz Brandao, and Julian Scott Yeomans Abstract Revenue risk due to demand fluctuations is one of the major issues affecting public infrastructure projects (building bridges, airports, schools, etc.). It becomes much more critical when private partners have been involved in the construction, financing, and operation of the infrastructure projects, as it enters into the realm of Public–Private Partnerships (PPP). In PPP projects, revenue risk impacts the project profitability for the private investor. To attract private financing in PPP infrastructure projects, governments must frequently include supplementary public guarantees to mitigate this risk. However, the impact on PPP projects is often difficult to estimate, since it depends both on uncertainty and on the actual exercise of the guarantee. All these challenges, along with the intrinsic characteristic of PPP projects being partnerships among different actors lasting a long period, complicate the choice of which form of guarantee is more suitable for ensuring project success. Such a choice is also strongly affected by contextual factors and project characteristics. In this chapter, we employ the hybrid sensitivity–uncertainty analysis technique, Simulation Decomposition (SimDec), to investigate how contextual factors and project characteristics affect the choice of the optimal form of public subsidy for revenue risk mitigation. To this aim, we focus on the case of an Italian airport. Through this real case, we provide useful guidelines that can be used by the government in the selection of public subsidies to mitigate revenue risk in PPP projects. 1 Introduction Public–Private Partnerships (PPP) have been used increasingly to deliver public infrastructure through the involvement of private expertise and financing in public enterprise (Irina & Veronica, 2022). The major challenges preventing private participation are frequently linked to the numerous uncertainties 94 Roberta Pellegrino et al. which characterize these projects. These uncertainty shortcomings can strongly impede the long-term profitability for private entities (Osei-Kyei & Chan, 2015). To overcome these issues and to continue to attract external private financing, governments often incorporate public supports intended to share risks with the private entities (such as revenue risk), thereby guaranteeing an adequate return to the potential participants. While such risk socialization incentives are frequently used in PPP projects, assessing their worth is not an easy task (Pellegrino, 2021; Hemming, 2006). Although the value of the guarantee is not preset, the payoff is exercised whenever the uncertainty thresholds triggering the guaranteed payout conditions are met. Under a Minimum Revenue Guarantee (MRG), the government agrees to cover any shortfall of revenue up to some predefined threshold. The exact amount to be paid to the private entity cannot be determined with certainty, a priori (Carbonara etal., 2014a, 2014b; Carbonara and Pellegrino, 2018). Consequently, it is imperative that these guarantees be correctly designed so that all risks are shared mutually between both parties and the government is not overwhelmed by the fiscal liabilities. For example, an excessive overpayment occurred in the case of the Salvador–Itaparica bridge system when disproportionately generous public guarantees resulted in the elimination of all risk to the private investor (Sant’Anna etal., 2022). Conversely, private entities can incur losses when the risk guarantees are not designed properly to sufficiently protect their interests. The Least Present Value of Revenue (LPVR), which does not involve government payouts, represents another type of risk-mitigating mechanism in which the government consents to extending the concession period to ensure a minimum level of return. The exact extension period is not known, a priori, but depends on the evolution of revenues (Engel etal., 2001; Xiong & Zhang, 2014; Pellegrino etal., 2013). Hence, the issue of proper choice of public supports is crucial for ensuring PPP projects’ success, for both public and private parties. Although several studies have focused on identifying specific models or instruments for public supports (Brandao & Saraiva, 2008; Almassi etal., 2013; Carbonara etal., 2014b; Carbonara & Pellegrino, 2018), only a few have explored the actual assessment and benchmarking of the different types of guarantees (Pellegrino, 2021; Song etal., 2018; Liu etal., 2017). Pellegrino (2021) developed a model to compare guarantees and to select the support mechanism which optimizes both parties’ interests according to a win–win principle. Pellegrino (2021) concluded that the choices were based upon both (1) endogenous factors related to the specific project or design of the guarantee and (2) exogenous factors related to the context in which the projects are developed. All sensitivity analyses performed within these existing studies have examined how each individual factor, taken one-at-time, influences the overall choice. Recent research has underscored how imperative it is in decision-making to consider the joint behaviour of all factors, simultaneously, in order to Contextual factors in mitigating revenue risk in PPP projects 95 capture how the interaction of different input factors affects outcomes (Kozlova, Moss, etal , 2024) To fill this gap in the PPP context, this chapter employs the hybrid sensitivity–uncertainty analysis technique, Simulation Decomposition (SimDec), to examine how the different contextual factors and project characteristics concurrently impact the choice of public subsidy in PPP revenue risk mitigation SimDec has been successfully used previously for several energy policy contexts, including renewable energy policy design (Kozlova etal , 2017), studying interaction of investment subsidies and carbon trading (Kozlova & Yeomans, 2019), and examining the effects of biofuel support instruments (Ruponen etal , 2021) The remainder of the chapter is structured in the following way Section2 outlines the computational model for the overall PPP decision process The underlying computational model is described, including the details on the three considered support types (Section2 1); the various input assumptions are described (Section 2 2); and the earlier conducted sensitivity analysis results are replicated (Section2 3) Section3 introduces the SimDec analysis set-up and the results generated Finally, Section4 concludes with a discussion on whether the SimDec analysis has actually contributed any significant additional insight into the mitigation process of revenue risk in PPP projects 2 Computational model 2.1 Modelling public support The model to assess and benchmark different public supports to mitigate revenue risk has been designed in order to more equitably satisfy the interests of both the private and public partners (Pellegrino, 2021) For the private sector, the intent is to enable them to recover their initial investments For the public sector, the objective is to ensure that the investments are compliant with accounting specifications regarding revenue guarantees and that these guarantee do not prove to be an onerous burden on society To achieve these objectives, a two-stage economic approach has been adopted (Pellegrino, 2021) In the first stage, the net benefit for each party is calculated using the Net Present Value (NPV) NPV is the difference between the discounted value of cash inflows and the discounted value of cash outflows for project The net benefit for the private entity is computed by equation (1), while the net benefit for the public sector is determined using equation (2) NPVI CF r V r CC tt Tc t C t R C T constr C =− ++ () ++ () =+ 1 11 (1) NPVCF r CF r V rG Gt Tc t G G ttTc Ft G t R G TC =+ () ++ () −+ () − ==+ 01 11 1 (2) 96 Roberta Pellegrino et al. where: • T C is the actual concession period • rC is the discount rate of the private entity • tconstr is the construction period I • I= tconstr t Ct=0 () 1+rt is the private entity’s investment in the infrastructure, with It being equal to the investment (capital expenditures) in year t • CFtt =−ROCt is the cash flow in year t, with Rt being the revenue in year t, OCt being the operational expenditures (including cost of maintenance) in year t Cash flow is received by the private entity during the concession period The infrastructure is transferred to the public entity at the end of the concession period Consequently, cash flow is received by the public from the time of transfer until the end of the project life • VR is the residual value of the infrastructure in the amount that the public party agrees to pay when the infrastructure is returned to the government • CFG t is the cash flow received by the government during the management of the project by the private sector, such as concession fees, profit-sharing, etc • G is the amount of guarantee (determined in the next subsections) • rG is the discount rate of the government According to the NPV criterion, the private party is satisfied when NPV is positive (equation (3)); therefore, the risk that the private entity is not satisfied is the probability that NPVC<0 (equation (4)) NPVC 0 (3) PrivateEntitys’risk =<Prob () NPVC0 (4) The government is satisfied when the value of the guarantee G (i e the cost to the government) is economically sustainable and politically acceptable Namely, it should be less than a fixed amount in order to be compliant with the accounting standards specifying treatment guarantees (5) G (5) The estimation of depends upon the criterion adopted in the specific country For example, the Eurostat rule establishes that assets should be considered on-balance sheet when the net cost of guarantees covers more than the 50% of the capital investment costs In this case, according to this rule ( =0 5IC), equation (5) expresses the condition of satisfying the public sector’s fiscal management interests and allows for setting a level of G that keeps the investment off-balance sheet Contextual factors in mitigating revenue risk in PPP projects 97 The risk to the public sector may be expressed as the probability that its NPV (considering the value of released guarantee) is lower than 0 (equation (6)) Government’sriskP=<robN () PVG0 (6) Once the interests of the two parties are computed, the public supports are benchmarked and chosen in the second stage To be compliant with the win–win condition, the form of guarantee minimizing the difference between the net profits (NPV) gained by the contractual parties and the risk borne by the two parties is selected The following sections illustrate how to calculate the effect of three support instruments on PPP projects and the corresponding logic for selecting the optimal one Monte Carlo simulation is employed to account for the variability of factors affected by the contracting uncertainty Both historical data and expert opinion are used to define the uncertainty of different input distributions Monte Carlo simulations provide “more realistic” probabilistic representations of the outputs, in contrast to the more commonly used, single “deterministic” values obtained through more traditional techniques 2.1.1 Minimum r evenue guarantee (MRG) Under the MRG, the government agrees to pay the private company all possible revenue shortfalls up to some predefined level (Rg) MRGs mitigate the inherent risks and make these investment types attractive to private investors The revenue Rt received by the private company while operating the infrastructure is calculated by equation (7) Rm tt =ax () PR ,Rg (7) where PRt is the project revenue in year t without guarantee The total amount of guarantee G is determined by equation (8) maxR () −PR GTc gt ;0 = t (8) tt=constr () 1+rG where the actual concession period T C coincides with the contractual one TC, that is, TT CC = 2.1.2 L east present value of revenue (LPVR) Under an LPVR schema, the government agrees to extend the concession period when the present value of revenues equals the minimum predefined threshold (LPVR) in equation (9) 98 Roberta Pellegrino et al. Tc C R T t t LPVR with TT > tt=constr () 1+CC (9) rG where the actual revenue coincides with the project revenue, that is, R t=PRt If equation (9) is satisfied for TT CC , then the concession will end at TC as contractually established Under this schema, the guarantee is the value of the cash flows renounced by the government due to the flexible term contract, as in equation (10) CF G= Tc t t if T C > TC (10) tT=c () 1+rG 2.1.3 P rice cap regulation (PCR) The price cap regulation (PCR) provides a mechanism aimed at preventing monopolistic infrastructure firms from earning excessive returns Under such arrangements, the private company must deliver a service or product subject to a maximum price ceiling that is negotiated with the government If the firm’s costs fall below this ceiling, the firm earns a profit while society experiences a “loss” Otherwise, when the firm’s costs rise above the price ceiling, the price does not exceed the limit and the firm will be penalized for the inefficiency (Pellegrino, 2021; Pellegrino etal , 2011) According to PCR, the private company develops an investment plan that is shared with the regulatory agency The expected annual depreciation is dependent upon the programmed investments, the expected operating costs for managing the infrastructure, and the expected profits Based on these data, the regulatory agency estimates a “calculated” fee for each year, which is the fee for recouping operating costs and depreciation for new investments This approach ensures an adequate return on investments, as calculated in equation (11) VT exp C t = t () CA++RC t (11) where CC represents the calculated operating costs (calculated opext), including maintenance cost; AC is the calculated depreciation according to the investment plan; RC is the calculated remuneration as a percentage (set by the regulator a priori) of the invested capital; and Vexpt is the expected value at t of the projected traffic volume Given “real-world” uncertainties, the actual situation may be different from the expected (or calculated) one, either in terms of actual cost or in Contextual factors in mitigating revenue risk in PPP projects 99 terms of traffic volume If the actual costs () CA++Ractual t are lower than the calculated ones, the extra profit will be shared between the public and private sectors Otherwise, it represents a loss for the private firm If w is the prearranged percentage of extra profit recognized by the regulatory agency, the extra profit for the public is determined in equation (12) w Profitsharingtt = () CC C−actual t (12) 100 where CC t are the calculated operating costs, and Cactual t are the actual operating costs Additionally, patterns may deviate from initial projections, providing excess revenues for the private firm if they turn out to be higher than expected 2.2 Case bac kground and numeric assumptions To illustrate the usefulness of the PPP model developed in the previous section, we applied it to the case of an Italian airport The project is between the National Civil Aviation Authority (ENAC)1 and a private contractor managing the airport From an operations perspective, the revenues arise from both aviation and non-aviation activities Aviation revenues are directly attributable to any aeronautically-related activities carried out at the airport, including airport charges, security services, centralized infrastructures, and other related activities Non-aviation revenues include commercial activities (sub-concessions, utilities, parking, advertising), real estate, and other third-party ancillary activities The operational costs consist of service costs, personnel costs, and consumptions costs The estimation of the operating revenues and costs has been based on historical data and expert opinion The input variables are categorized into deterministic and uncertain variables The deterministic variables are: • The concession period corresponds to the time when the infrastructure is operated by the private and is fixed at 33years • The project lifetime covers the entire lifespan of the project and is set at 67years • The total investment costs over 33years is 11,470M€ • The discount rate is assumed to be equal to the risk-free rate for both parties (rG=rC=5%) 100 Roberta Pellegrino et al. Table 5.1 Uncertainty assumptions for input variables Group Input random variables Probability Source distribution1,2 Revenues Real estate subconcession, € PERT(50, 56, 60) Expert Commercial subconcession, € PERT(96, 99, 105) opinion Parking (in subconcession), € U(9.73, 11.90) Advertisement, € U(11, 15) Other revenues from aviation U(27, 34) activities, € Other revenues from nonU(16, 20) aviation activities, € Costs Cost for services, € PERT(182, 300, 600) Expert Cost for personnel, € PERT(72, 100, 150) opinion Cost for fuel and lubricant, € PERT(1.42, 3.3, 8) Consumption materials, € PERT(2, 6, 15) Traffic Traffic – passengers GBM(0.0332, Historical 0.0385) data Traffic – freight GBM(0.0.25, 0.0.80) Other Electric energy price, €/MWh MR(21.10, 0.08167, Historical 0.1774) data 1 Parameter values are in millions. 2 PERT stands for BetaPERT distribution; U, uniform; GBM, Geometric Brownian Motion; and MR, mean reverting. Source: Based on Pellegrino (2021) (https://ascelibrary.org/doi/10.1061/%28ASCE%2 9CO.1943–7862.0002098), Table1, p.6. The uncertain variables are: • Aviation revenues arise from security services, centralized infrastructures, and other related minor activities (aviation revenues) Historical data and expert opinion were used to estimate them (see Table5 1) • Non-aviation revenues include subconcessions and utilities, parking, and advertising (see Table5 1) • Traffic is modelled as a random variable following a Geometric Brownian Motion (Pichayapan etal , 2003; Garvin & Cheah, 2004; Brandao & Saraiva, 2008; Iyer & Sagheer, 2011) (equation (13)) 2 −Q + tt = Q2 Q QQ tt1e + (13) where Q is the traffic growth rate, Q is the annual volatility of the traffic, and ~N(0,1) is the standard Wiener process Table5 1 reports the parameter values used in the case The unitary price charged to the users has been calculated based on historical data (airport charges: landings and departure rights: €3 30; parking and hospitalization rights: €0 17; passenger boarding fee: €13 85; freight loading and unloading taxes: €0 04) • Operations costs, such as service costs, personnel costs, and consumptions costs, are based on historical data We model the electricity price Contextual factors in mitigating revenue risk in PPP projects 101 as a random variable according to a Mean Reverting Process (Blanco & Soronow, 2001a, 2001b; Blanco etal , 2001; Deng, 2000), according to the equation: dStt = () sS −dt +dWt where s is the long run mean (the mean reversion level), is the annual volatility of the price, is the mean reversion rate, and dWt is a Brownian motion (so dWt Nd () 0t) Table5 1 reports the assumptions used for modelling the statistical distributions of uncertain variables The input variables of groups C and D are stochastic processes, where the value at each time depends on the value in the previous year All other input variables are assumed to independently follow the distributions specified in Table5 1, each year The model is implemented in a spreadsheet, and the uncertainty handling, simulation, and sensitivity analysis are performed using the Crystal Ball software package 2.3 P revious sensitivity analysis studies Pellegrino (2021) performed an extensive uncertainty and sensitivity study of the case In particular, a Monte Carlo simulation was run, and the resulting probability distribution examined in the absence of support Subsequent simulation experiments were repeated for each support type, and the descriptive statistics are provided in Section2 3 1 In addition, one-at-a-time (OAT) sensitivity analyses were conducted for the different levels of support and for different values of the uncertain inputs (see Section2 3 2) 2.3.1 M onte Carlo simulation Evaluating the project without any form of guarantee, we obtain the net benefits (NPV) for the public and private parties, as shown in Figure5 1 As seen in the figures, the project is positive for the government and negative for the private party in the absence of government support While the risk of loss is negligible for the government, it is high for the private entity Clearly, any infrastructure project exhibiting such characteristics would not be appealing to private investors and would, therefore, require additional government support guarantees to become more attractive Table5 2 shows the net profit statistics of the two actors (private party and government) for the project when considered under each of the three forms of public support (MRG, LPVR, and PCR) It is clear that introducing government support increases the project profitability for the private entity and reduces the risk that the private interests are not satisfied From the government perspective, it can be noticed that under LPVR, the NPV remains the same as the case without supports, while MRG 108 Roberta Pellegrino et al. single annual value of the same variable, such input variables were transformed into mean and variance values. This introduces 26 parameters into the analysis from the 13 input variables specified in Table5.1. Three additional variables representing the support level for each of the three types are also recorded. Consequently, the overall dataset consisted of 1,000 iterations of 12 outputs and 29 input parameters. Furthermore, a combined dataset was created for the three main output values using an extra artificial discrete input variable to represent the support case. This combined dataset enables a direct comparison of all policy options simultaneously using a single graph by employing the extra support input to designate different support types in the decomposition. 3.2 Sensitivity indices This case poses two significant computational challenges for the effective determination of sensitivity indices. Firstly, the highly uncertain conditions are not completely reflected by the selected input parameters (i.e. the aggregation of individual annual inputs into summarized means and variances). Secondly, the number of inputs is quite high compared to the relatively low number of simulated iterations. For these reasons, only first-order indices (i.e. sensitivity values of individual input variables) are computed as the estimation of second-order effects becomes too noisy (see Table5.4). In Table5.4, the notable sensitivity indices are highlighted with green shading; all index values below 2% are greyed out, as are the actual names of the inputs that have all sensitivity index values below 2% for each output. Table5.4 indicates that the Support type is the most influential variable in the merged dataset. The corresponding support levels appear influential in corresponding support types, for Tariff in Price Cap, LPVR in LPVR, and Rg in MRG. However, the impacts of these effects are not symmetric for the different parties. The tariff affects NPV of the private investor but has only a minor influence on the profitability for the government. LPVR has modest effect on the government NPV, but none for the private investor, and exhibits a considerable difference between the two. Rg plays a significant role for NPV of private investor and the difference between the two but has only a modest effect on the government NPV. Besides the support-level variables, the mean traffic of passengers is the only other variable that contributes a noticeable first-order influence over all cases. The first-order effects of all other input variables are negligible. Afinal influence observation is that the sum of all indices is considerably below 100% in the no support case. This observation is characteristic of a highly uncertain system containing many completely random inputs. 3.3 Decomposition Figure5.3 displays the distributions for each of the three outputs of the merged dataset, NPVG – NPVC, NPVC, and NPVG, decomposed by Support type (the most influential parameter identified in Section3.2). The decomposed graphs Contextual factors in mitigating revenue risk in PPP projects 109 Table 5.4 Sensitivity indices (first-order effects) of input parameters to model outputs Inputs Price Cap LPVR MRG No support Both NPVCNPVGBoth NPVCNPVGBoth NPVCNPVGBoth NPVCNPVG Support Mean Electricity Price (normal) 1% 1% 1% 1% 1% 1% 1% 1% 1% 0% 1% 1% Variance Electricity Price (normal) 0% 1% 0% 1% 0% 1% 2% 2% 1% 1% 1% 0% Mean Traffic – freight (normal) 1% 0% 1% 0% 1% 1% 1% 1% 1% 1% 0% 1% Variance Traffic – freight (normal) 1% 1% 0% 1% 1% 1% 1% 1% 0% 1% 1% 0% Tariff 4% 29% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% Mean Advertisement 0% 2% 1% 1% 1% 1% 1% 2% 1% 1% 1% 1% Variance Advertisement 1% 1% 1% 1% 1% 1% 0% 1% 1% 0% 1% 1% Mean Commercial subconcession 0% 0% 0% 0% 0% 0% 1% 1% 1% 1% 0% 0% Variance Commercial subconcession 1% 0% 0% 1% 0% 1% 0% 1% 0% 1% 0% 0% Mean Consumption materials 0% 0% 0% 1% 0% 0% 0% 0% 0% 0% 0% 0% Variance Consumption materials 1% 1% 1% 0% 1% 1% 1% 1% 1% 1% 1% 1% Mean Cost for fuel and lubricant 2% 1% 2% 1% 1% 2% 1% 1% 1% 1% 1% 2% Variance Cost for fuel and lubricant 1% 1% 1% 0% 1% 2% 1% 1% 1% 1% 1% 1% Mean Cost for personnel 1% 1% 2% 0% 1% 1% 1% 2% 1% 1% 1% 2% Variance Cost for personnel 2% 1% 1% 1% 1% 1% % 1% 2% 2% 1% 1% Mean Costs for services 0% 1% 1% 3% 1% 0% 2% 3% 1% 2% 2% 1% Variance Costs for services 1% 1% 1% 2% 1% 1% 0% 1% 1% 1% 1% 1% LPVR 1% 1% 1% 85% 0% 14% 1% 0% 1% 2% 0% 1% Mean Other revenues from aviation 1% 1% 0% 0% 1% 0% 1% 1% 1% 1% 1% 0% Variance Other revenues from aviation 1% 1% 1% 2% 1% 1% 0% 1% 0% 1% 1% 1% Mean Other revenues from non-aviation 0% 0% 0% 1% 0% 1% 1% 1% 1% 1% 1% 0% Variance Other revenues from non-aviation 1% 0% 1% 0% 1% 1% 1% 1% 0% 1% 1% 1% Mean Parking (in subconcession) 1% 1% 1% 0% 1% 1% 0% 1% 1% 1% 1% 1% Variance Parking (in subconcession) 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% (Continued) 110 Roberta Pellegrino et al. Inputs Price Cap LPVR MRG No support Both NPVCNPVGBoth NPVCNPVGBoth NPVCNPVGBoth NPVCNPVG Mean Real estate subconcession 0% 1% 0% 1% 1% 1% 1% 1% 1% 1% 1% 0% Variance Real estate subconcession 1% 1% 1% 2% 1% 1% 1% 1% 1% 1% 1% 1% Rg 1% 0% 1% 1% 1% 1% 82% 82% 38% 1% 1% 1% Mean Traffic – passengers 44% 25% 48% 6% 45% 36% 7% 7% 27% 19% 35% 48% Variance Traffic – passengers 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% Sum of first-order indices 71% 76% 72% 117% 69% 72% 113% 116% 88% 46% 61% 71% Note: “Both” stands for NPVG – NPVC. Table 5.4 (Continued) Contextual factors in mitigating revenue risk in PPP projects 111 Figure 5.3 Simulation Decomposition of profitability distributions for the government (right), private party (middle), and their difference (left) by the policy type. (colour image is accessible via the link) 112 Roberta Pellegrino et al. visibly indicate that the profitability distribution of the government and the distribution of the difference with the private party profitability are clearly differentiated by the four policy cases. Because the profitability distribution of the private investor appears more condensed, it has been decomposed further by including the moderately influencing (7%) Mean traffic variable to provide additional explanatory clarity. For the private investor (Figure5.3, middle), Price cap appears to be the most attractive support type because it always provides guaranteed profitability (the entire red sub-distribution is in the positive NPV range). MRG is also always profitable, but with less upside compared to Price cap. LPVR has a negligible, but non-zero, probability of losses. In contrast, No support makes the project look unattractive, with an expected mean below zero. The increased passenger traffic slightly shifts profitability upwards, but does not show any major influence. This confirms its non-zero, but low, sensitivity index of 7%. The government (Figure 5.3, right) benefits the most from Price cap, whereas the distributions of the other support types generally remain grouped together. No support and LPVR occupy almost the same profitability range as LPVR, though they extend more into lower-value regions. However, neither variable produces results in the negative profitability range. MRG has a non-zero probability of loss for the government. The difference of the two NPVs (Figure5.3, left) compares the relative economic positions of the two parties by showing how much higher the profitability is for the government in comparison to the private investor. The Price cap sub-distribution is high in the positive range, even though it is also the most profitable scheme for the private investor. This occurs because, for this support type, the consumer pays the profits directly to both parties. Under LPVR, the government benefits more than the private investor most of the time, but not by as much as under Price cap. This effect can be observed in the figure because the yellow sub-distribution occupies a narrower range close to zero, though it occasionally enters into the negative zone. Conversely, under MRG, the private investor profits more than the government most of the time. Finally, the decompositions for each individual support type do not uncover any interesting patterns. All individual effects reported in Table5.4 assume a rather monotonic appearance, and thus, these graphs have been omitted from the chapter. In summary, the SimDec approach has considerably eased the overall complexity of the analysis process for the risk mitigation decision-maker. While the previously conducted sensitivity analysis approaches all required multiple simulation experiments to produce their results (Figures5.1–5.2 and Tables5.2–5.3), only a single simulation was necessary to produce all of the SimDec results. SimDec automatically incorporates an analysis of all of the variables that had to be generated individually in the prior approaches. The three graphs in Figure5.3 Contextual factors in mitigating revenue risk in PPP projects 113 combine the information content from Figure5.1 (probability distributions), Table5.2 (that offers 72 different values for analysis), and Table5.3 (that further uncovers the details with 315 more values). SimDec generates profitability distributions for all of the policy options simultaneously on a single graph. This visualization enables all the information to be directly comparable and all of the comparative insights to become readily obvious. Furthermore, while the earlier model exhibited only monotonic behaviour, the SimDec visualization has uncovered far more complex nonlinear relationships, including significant underlying heterogeneous behaviour. Consequently, SimDec should become an essential analysis support approach for any well-informed risk mitigation in PPP decision-making (Kozlova, Moss, etal., 2024). 4 Discussion and conclusions This chapter has outlined a SimDec analysis for mitigating the revenue risks for various public infrastructure support types. In general, the PPP model is challenging due to the significant levels of uncertainty inherent within the model, the combination of numerous input variables that change concurrently, and the limited number of simulation runs. Furthermore, there has been a need to integrate extensive prior uncertainty and sensitivity analyses into the decision process. Nevertheless, it has been shown that SimDec performs admirably by contributing several additional analytical benefits together with numerous supplementary insights. For the PPP mitigation case considered, SimDec provided global insights into the economic balancing of different policy options, revealed the most important factors, and directly highlighted the effects of several modelling choices. The global sensitivity analysis from SimDec revealed that, in contrast to the earlier one-at-a-time analysis (Figure5.2), only passenger traffic volume is important for the airport investment profit when all uncertain conditions are considered simultaneously (Table5.4). Moreover, in the absence of support, the variability of the output cannot be sufficiently explained by the aggregated input parameters due to the considerable randomness of the multiple input variables. In conclusion, for the general case, it can be recommended that SimDec should be broadly applied to the analysis of public infrastructure investments for investigating partnership arrangements, for examining policy options, and for conducting investment risk mitigation analysis. Acknowledgements The work is supported by grant 220178 from the Finnish Foundation for Economic Education and by grant OGP0155871 from the Natural Sciences and Engineering Research Council. 114 Roberta Pellegrino et al. Notes 1 The National Civil Aviation Authority (ENAC) is a non-economic public body with regulatory, organizational, administrative, patrimonial, accounting, and financial autonomy. 2 https://github.com/Simulation-Decomposition. References Almassi, A., McCabe, B., & Thompson, M. (2013). Real options–based approach for valuation of government guarantees in public–private partnerships. Journal of Infrastructure Systems, 19(2), 196–204. https://doi.org/10.1061/(ASCE) IS.1943-555X.0000117 Blanco, C., Choi, S., & Soronow, D. (2001). Energy price processes used for derivatives pricing & risk management. Commodities Now, 74–80. Blanco, C., & Soronow, D. (2001a). Jump diffusion processes-energy price processes used for derivatives pricing and risk management. Commodities Now, 83–87. Blanco, C., & Soronow, D. (2001b). Mean reverting processes-energy price processes used for derivatives pricing & risk management. Commodities Now, 5(2), 68–72. Brandao, L. E., & Saraiva, E. (2008). The option value of government guarantees in infrastructure projects. Construction Management and Economics, 26(11), 1171–1180. Carbonara, N., Costantino, N., & Pellegrino, R. (2014a). Revenue guarantee in public-private partnerships: Afair risk allocation model. Construction Management and Economics, 32(4), 403–415. Carbonara, N., Costantino, N., & Pellegrino, R. (2014b). Concession period for PPPs: Awin–win model for a fair risk sharing. International Journal of Project Management, 32(7), 1223–1232. Carbonara, N., & Pellegrino, R. (2018). Revenue guarantee in public–private partnerships: Awin–win model. Construction Management and Economics, 36(10), 584–598. Deng, S. (2000). Stochastic models of energy commodity prices and their applications: Mean-reversion with jumps and spikes (Working Paper). University of California Energy Institute. https://haas.berkeley.edu/wp-content/uploads/pwp073.pdf Engel, E. M., Fischer, R. D., & Galetovic, V. A. (2001). Least-present-value-of-revenue auctions and highway franchising. Journal of Political Economy, 109(5), 993–1020. https://doi.org/10.1086/322832 Garvin, M. J., & Cheah, C. Y. (2004). Valuation techniques for infrastructure investment decisions. Construction Management and Economics, 22(4), 373–383. Hemming, R. (2006). Public-private partnerships, government guarantees, and fiscal risk. International Monetary Fund. Irina, C., & Veronica, B. (2022). World practice in the evolution of public-private partnership of infrastructure projects. International Journal of Economics, Business and Management Studies, 9(1), 1–12. Iyer, K. C., & Sagheer, M. (2011). Areal options based traffic risk mitigation model for build-operate-transfer highway projects in India. Construction Management and Economics, 29(8), 771–779. Kozlova, M., Collan, M., & Luukka, P. (2017). Simulation Decomposition: New approach for better simulation analysis of multi-variable investment projects. Fuzzy Economic Review, 21(2), 3–18. Kozlova, M., Moss, R. J., Yeomans, J. S., & Caers, J. (2024). Uncovering heterogeneous effects in computational models for sustainable decision-making. Contextual factors in mitigating revenue risk in PPP projects 115 Environmental Modelling & Software, 171, 105898. https://doi.org/10.1016/j. envsoft.2023.105898 Kozlova, M., Roy, P., Alam, A., Moss, R. J., & Yeomans, J. S. (2024). SimDec algorithm and usage instructions. In M. Kozlova & J. S. Yeomans (Eds.), Sensitivity analysis for business, technology, and policymaking made easy with Simulation Decomposition. Routledge. Kozlova, M., & Yeomans, J. S. (2019). Multi-variable Simulation Decomposition in environmental planning: An application to carbon capture and storage. Journal of Environmental Informatics Letters, 1, 20–26. Liu, T., Bennon, M., Garvin, M. J., & Wang, S. (2017). Sharing the big risk: Assessment framework for revenue risk sharing mechanisms in transportation public-private partnerships. Journal of Construction Engineering and Management, 143(12), 04017086. Osei-Kyei, R., & Chan, A. P. (2015). Review of studies on the critical success factors for public–private partnership (PPP) projects from 1990 to 2013. International Journal of Project Management, 33(6), 1335–1346. Pellegrino, R. (2021). Effects of public supports for mitigating revenue risk in public– private partnership projects: Model to choose among support alternatives. Journal of Construction Engineering and Management, 147(12), 04021167. Pellegrino, R., Ranieri, L., Costantino, N., & Mummolo, G. (2011). A real options-based model to supporting risk allocation in price cap regulation approach for public utilities. Construction Management and Economics, 29(12), 1197–1207. Pellegrino, R., Vajdic, N., & Carbonara, N. (2013). Real option theory for risk mitigation in transport PPPs. Built Environment Project and Asset Management, 3(2), 199–213. Pichayapan, P., Hino, S., Kishi, K., & Satoh, K. (2003). Real option analysis in evaluation of expressway projects under uncertainties. Journal of the Eastern Asia Society for Transportation Studies, 5, 3015–3030. Ruponen, I., Kozlova, M., & Collan, M. (2021). Ex-ante study of biofuel policies– analyzing policy-induced flexibility. Sustainability, 14(1), 147. Sant’Anna, R. L., Brandão, L. E. T., Bastian-Pinto, C. D. L., & Gomes, L. L. (2022). Liability cost of government guarantees in highway concession projects: Case of the Salvador–Itaparica bridge. Journal of Infrastructure Systems, 28(2), 05022003. https://doi.org/10.1061/(ASCE)IS.1943-555X.0000690 Song, J., Zhao, Y., Jin, L., & Sun, Y. (2018). Pareto optimization of public-private partnership toll road contracts with government guarantees. Transportation Research Part A: Policy and Practice, 117, 158–175. Xiong, W., & Zhang, X. (2014). Concession renegotiation models for projects developed through public-private partnerships. Journal of Construction Engineering and Management, 140(5). DOI: 10.4324/9781003453789-8 This chapter has been made available under a CC-BY-NC-ND 4.0 license. Abstract Applying Additive Manufacturing (AM) methodologies, such as 3D printing of concrete, might provide a mechanism to revolutionize the construction sector. However, the ambiguous nature of unit economics has deterred its more extensive integration. As such, this study first presents a deterministic model for estimating direct and indirect costs in AM and then advances a stochastic unit cost model by integrating uncertainty ranges. Using (Monte Carlo) Simulation Decomposition, this model is analyzed regarding probabilistic scenarios, the sensitivity of input factors, and uncertainty effects. The results confirm the existence of economies of scale and highlight AM’s potential for construction across a diverse range of scenarios. Managers, researchers, investors, and policymakers alike can use this model to interactively navigate the complexities of AM in the construction industry to inform decisions and drive technology development. As AM technology advances, the models can be iteratively refined and expanded, eventually improving unit economics, productivity, and profitability. Future research can then leverage such models to explore AM’s potential impact in construction, infrastructure, and housing projects. 1 Introduction The construction industry may elevate productivity by adopting advanced manufacturing techniques from other sectors, such as automation from automotive (Gann, 1996). However, the ongoing introduction of novel technologies in construction presents several challenges, for example, unclear user perception of robots (Walzer etal., 2023). Beyond understanding technological barriers and user needs, the economic implications of novel technology in the sector have drawn further attention in construction management scholarship (Kangari & Halpin, 1990; Tatum, 1986). Arguably, the emergence of Additive Manufacturing (AM) heralds a transformative phase in the Chapter 6 Printing homes Unit cost estimation for additive manufacturing in construction Alexander N. Walzer, Mariia Kozlova, and Julian Scott Yeomans Printing homes 117 construction industry by advancing novel methods of production that extend the boundaries of innovation (Berman, 2012). At the heart of this cutting-edge technology lies a digitally driven fabrication process which, through the layer-by-layer deposition of material – ranging from cementitious paste to steel, plastics, resins, or a blend of these – accurately maps the desired elements along a path dictated by a pre-approved 3D model (Gibson etal., 2015). The potential adoption of this technology within the construction industry is gaining interest, as it promises an array of unprecedented possibilities. Its offerings extend from enhanced design flexibility to potential cost efficiency and, perhaps, most significantly, a clear path toward sustainability (Wohlers & Caffrey, 2015; Zunino, 2023). The current innovation landscape in this field reveals a mix of pilot projects spearheaded by academia and industry worldwide. Large multinational corporations and well-funded start-ups are increasingly stepping into the arena, indicating a growing commitment to this technological revolution (Ford & Despeisse, 2016). However, closer examination is necessary to comprehend the profitability of AM in the construction industry. Estimating unit costs is a critical starting point for such analysis (Tucker, 1986) to evaluate whether AM could yield economies of scale, where increased production decreases per-unit costs (Besanko etal., 2009). “It’s really important to be profitable at the unit level – and take that as a first priority” (Eisenhardt, 2023, 19:40). Figure6.1 depicts a recent industry example of AM using 3D printing of cementitious materials (often referred to as 3D concrete printing, “3DCP”) by using an industrial robotic arm to produce infrastructure foundations (visible on the left). 1.1 Point of departure While previous investigations into productivity in emerging construction technologies exist (e.g. Garcia de Soto etal., 2018), the analysis tends to have been relatively simplified by neglecting numerous critical aspects integral to a comprehensive cost evaluation. Generally, productivity denotes the output volume an organization can generate per unit of input – labour, capital, and materials, among others (Syverson, 2011). Serving as a barometer of efficiency, it gauges how effectively a firm, an industry, or even an entire economy deploys its resources. For instance, a company might assess its productivity based on the number of units it manufactures per labour hour. Assuming other factors remain constant, heightened productivity could pave the way to reduced costs and augmented profits (Bloom & Van Reenen, 2010). Conversely, cost efficiency scrutinizes the relationship between input expenses and the value or quality of outputs. Aprocess can be classified as 220 Robert J. Moss, Mariia Kozlova, Anthony Corso, and Jef Caers 5.6 Accuracy analysis For POMDP problems that make a final decision, the accuracy is defined as how close the final decision was to the correct decision. For binary decisions, this is simply whether the agent took the correct action or not. For continuous decisions, this could be the difference between the final decision and the correct decision (e.g. distance to an unknown goal). For the mineral exploration POMDP, we treat the final decision as a classification task (i.e. whether to classify the orebody as a resource to mine or abandon). Based on the final decision, the accuracy is computed as: number of correct decisions accuracy =number of decisions The number of decisions is equivalent to the number of episodes (i.e. seeds) per model configuration. For a set of economical orebody states E, the number of correct decisions is: 11(|as=Îmine Ea)(+=abandon |)sEÏ The contour map analysis of the accuracy in Figure9.17 indicates that every model-fidelity configuration has accuracy above about 0.69 (where an accuracy of 0.5 would correspond to a random policy). Also evident in the contours is that the grid dimensions have the largest effect and that the mediumand low-fidelity state shapes have similar accuracy. Unsurprisingly, the highest-fidelity model configuration produces the highest accuracy. The sensitivity analysis in Table9.7 confirms that the model fidelities have little effect on the accuracy, all less than 1%. Table9.8 details the decomposition of accuracy based on the model fidelities with an explained variance of the decomposed output of 0.009. These results show that the accuracy ranges from 0.69 to 0.84 (where the latter corresponds to the highest fidelity case), matching the contour map analysis. This suggests that regardless of the choice of model fidelity, relatively high accuracy can be achieved. Table 9. 7 Sensitivity indices for accuracy Input First-order Second-order effect Combined effect sensitivity State shape Planning Grid index iterations dimensions State shape 0.002 —0.001 0.0002 0.002 Planning iterations 0.001 — — 0.0001 0.001 Grid dimensions 0.006 — — — 0.006 Model fidelity analysis using Simulation Decomposition 221 Figure 9.17 Contour maps of the accuracy in the final mine or abandon decision. (colour image is accessible via the link) 222 Robert J. Moss, Mariia Kozlova, Anthony Corso, and Jef Caers Table 9. 8 Decomposition of accuracy by state shape, planning iterations, and grid dimensions State shape Planning Grid Accuracy iterations dimensions Min Mean Max Probability Circle Low Low 0.00 0.76 1.00 7% High 0.00 0.81 1.00 15% High Low 0.00 0.74 1.00 4% High 0.00 0.79 1.00 7% Ellipse Low Low 0.00 0.74 1.00 7% High 0.00 0.79 1.00 15% High Low 0.00 0.69 1.00 4% High 0.00 0.79 1.00 7% Blob Low Low 0.00 0.75 1.00 7% High 0.00 0.83 1.00 15% High Low 0.00 0.80 1.00 4% High 0.00 0.84 1.00 7% 5.7 Seed vs . random sampling As described in Section4.2, two types of Monte Carlo data generation were studied: sampling seeds for a discrete set of input configurations and random sampling over a range of inputs. Results in Table9.9 suggest that the overall model fidelity sensitivity analysis conclusions are unaffected by the sampling scheme. Therefore, we chose to sample based on the seed strategy to allow us to visualize accuracy using contours. Figure9.18 compares two outputs (regret and runtime) using the different sampling schemes. The regret distributions are an example where the histograms (and supporting sensitivity indices) are nearly equivalent, and the runtimes are an example where the histograms are visually different (despite the sensitivity analysis closely matching). The runtime histograms for the random sampling scheme (Figure18b2) are smoother as they are computed over a finer range (i.e. defined by the x–y range in the contour plots). 6 Discussion The analysis presented in this chapter highlights the use of SimDec for model fidelity sensitivity of sequential planning performance. We introduced the POMDP model-fidelity framework (PMFF) and applied it to a real-world case study of critical mineral exploration. The results of this case study suggest that complex state modelling to accurately represent the subsurface may be less important than focusing on planning fidelity and environment fidelity, as shown in the sensitivity indices mean in Table9.10 (also shown in the overall results in Table9.1). The state shape model (i.e. state fidelity) had the largest sensitivity to the number of actions planning performance metric with Model fidelity analysis using Simulation Decomposition 223 Table 9. 9 Combined sensitivity indices for all outputs for the two sampling strategies tions Sampling 0.006 0.042 0.008 No. of ac Seed 0.008 0.052 0.003 Sampling 0.011 0.006 0.009 Bias Seed 0.001 0.002 0.007 Sampling 0.006 0.229 0.201 Runtime Seed 0.005 0.248 0.203 Simple random sampling b. Sampling 0.001 0.004 0.006 retgeR Seed 0.003 0.002 0.010 nture Sampling 0.001 0.005 0.005 scounted r Seed 0.0003 0.0002 0.0010 i.e. this analysis) DiInput/output hapeState s lanning iterations rid dimensions Seed sampling ( P G a. 224 Robert J. Moss, Mariia Kozlova, Anthony Corso, and Jef Caers Figure 9.18 Comparison of histograms obtained with seed sampling (a1 & a2) and random sampling (b1 & b2e similar esults ar). R for regret (a1 & b1) and different for runtime (a2 & b2). (colour image is accessible via the link) Model fidelity analysis using Simulation Decomposition 225 a sensitivity index of 0.008. Although relatively low, this fidelity resulted in reduced information gathering to converge the simple belief, thus leading to fewer actions per episode. Both the planning iterations (i.e. planning fidelity) and the grid dimensions (i.e. environment fidelity) had the largest sensitivities to runtime with sensitivity indices of 0.248 and 0.203, respectively. This runtime sensitivity is understood to come from two components of the planner: (1) the size of the action space is the size of the grid dimensions, thus leading to more actions to explore during planning, and (2) the particle filter belief updater uses importance resampling with conditional Gaussian simulations to condition the subsurface field on the ore measurements. Therefore, when the grid dimensions are larger, there are more actions and a larger subsurface to generate based on the observations. Also, one would expect the runtime to increase as a function of planning iterations. All three input fidelities had the lowest sensitivities to the discounted return with sensitivity indices of 0.0003, 0.0002, and 0.001, respectively. In POMDP planning and reinforcement learning, the discounted return is the primary metric used to determine the performance of a decision-making agent. This suggests that, across all input fidelities, the planning performance is not sensitive to model fidelity. 6.1 Applicability Demonstrating the use of SimDec to study the sensitivity of model choices in a POMDP has broader applicability outside the case study of mineral exploration. PMFF provides a general framework to be applicable to any POMDP. Other application areas could focus on the safety of POMDP planning algorithms, including studying POMDPs for autonomous driving (Sunberg & Kochenderfer, 2022) and carbon capture and storage (Wang etal., 2023). The sensitivity analysis could further analyze the system behaviour in terms of how the sensitivity in the inputs affects decision-making based on the actions taken. The combination of SimDec and PMFF could also be used not only for model performance analysis but also for model selection: fine-tuning the model behaviour for optimal decision-making based on the balance of multiple performance objectives. 6.2 Open-source code The source code for the general PMFF framework and the interface and experiment code used in this case study for the mineral exploration POMDP Table 9.10 Sensitivit y index totals Input/output Sensitivity indices mean State shape 0.019 Planning iterations 0.305 Grid dimensions 0.230 226 Robert J. Moss, Mariia Kozlova, Anthony Corso, and Jef Caers are available online.1 Open-source packages to run SimDec for other types of systems (using Monte Carlo data from simulated or real-measured situations) are available in Matlab, Python, Julia, R, and Excel.2 Acknowledgements This work is supported by grant 220178 from the Finnish Foundation for Economic Education, by grant OGP0155871 from the Natural Science and Engineering Research Council of Canada, and by the Stanford Institute for Human-Centered AI. Notes 1 https://github.com/sisl/POMDPModelFidelityFramework.jl. 2 https://github.com/Simulation-Decomposition. References Baffa, A. C., & Ciarlini, A. E. (2010). Modeling POMDPs for generating and simulating stock investment policies. ACM Symposium on Applied Computing (SAC), Sierre, Switzerland (pp.2394–2399). Balaban, E., Arnon, T., Shirley, M. H., Brisson, S. F., & Gao, A. (2018). A system health aware POMDP framework for planetary rover traverse evaluation and refinement. 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Online algorithms for POMDPs with continuous state, action, and observation spaces. International Conference on Automated Planning and Scheduling (ICAPS), Delft, The Netherlands (pp.259–263). Sunberg, Z. N., & Kochenderfer, M. J. (2022). Improving automated driving through POMDP planning with human internal states. IEEE Transactions on Intelligent Transportation Systems, 23(11), 20073–20083. Wang, Y., Zechner, M., Wen, G., Corso, A. L., Mern, J. M., Kochenderfer, M. J., & Caers, J. K. (2023). Optimizing carbon storage operations for long-term safety. arXiv preprint arXiv:2304.09352, p.14. https://arxiv.org/pdf/2304.09352 Wolf, T. B., & Kochenderfer, M. J. (2011). Aircraft collision avoidance using Monte Carlo real-time belief space search. Journal of Intelligent & Robotic Systems, 64, 277–298. DOI: 10.4324/9781003453789-13 This chapter has been made available under a CC-BY-NC-ND 4.0 license. Chapter 10 Upgrading the toolbox of techno-economic assessment with SimDec Power-to-X case Hannu Karjunen, Sini-Kaisu Kinnunen, Arto Laari, Antero Tervonen, Petteri Laaksonen, Mariia Kozlova, and Julian Scott Yeomans Abstract Power-to-X (P2X) technology holds great potential for decarbonization and sees an active growth of R&D, pilot projects, and scaling initiatives. Such projects are surrounded by tremendous uncertainty due to their long lifespan, dependency on market and policy evolution, and high upfront costs. Thus, the usage of adequate analytical tools is of paramount importance to design, evaluate, and execute P2X projects. In this chapter, we took one such techno-economic report for a P2X project and replicated its one-at-a-time sensitivity analysis and scenario analysis with the more powerful SimDec approach. The breadth of the newly acquired insights inspired the authors to upgrade the analysis to reflect present-day circumstances in order to determine what findings SimDec could contribute to the evolving situation. In all analyzed cases, SimDec showed an excellent capability of combining uncertainty and sensitivity analysis for deriving comprehensive actionable insights for investment projects. Keywords: P2X, power-to-X, SimDec, simulation, feasibility, capital investment, discounted cash flow model, net present value, global sensitivity analysis, uncertainty analysis 1 Introduction P2X (power-to-X) is an umbrella term for pathways which convert electricity into various commodities, such as fuels, chemicals, gases, heat, or even food. One subset of P2X is represented by electrofuels, also known as e-fuels. This class of fuels functions as a direct drop-in replacement for vehicles and other devices relying on internal combustion engines and liquid fuels. Instead of using fossil crude oil as the feedstock, e-fuels are synthetized from carbon dioxide and hydrogen. Notably, the hydrogen may be obtained from water The toolbox of techno-economic assessment with SimDec 229 electrolysis, which, in turn, is operated using renewable electricity. Synthesis processes for P2X products and e-fuels may take many forms and different reaction steps, such as the Sabatier reaction, hydrogenation using heterogeneous catalysts, and biosynthesis (Laaksonen etal., 2021). Feasibility studies related to P2X projects are highly relevant and interest is increasing regarding the profitability of these projects (Dahiru etal., 2022). However, the availability of reliable data using real cases is limited. This is partly because of confidentiality issues, but also because of uncertainties related to the key factors affecting the economic feasibility of P2X investments, such as end product price, expected decrease in production costs due to technological and manufacturing innovations, or general uncertainties in process design at preliminary design stage. Modelling the technical and economic feasibility of the new energy solutions is critical in order to promote the energy transition in a sustainable manner. Because previous cases with real data are limited, sensitivity analysis and simulations are important when evaluating the profitability of this kind of investment. This study aims to replicate and improve a previously conducted pre-feasibility study of an e-fuel plant (Laaksonen etal., 2021). Some of the challenges relating to the pre-feasibility study include (1) uncertainty in the product market prices and input electricity price, (2) numerous technical pathways producing a variety of products with different volumes, and (3) limited certainty in equipment cost estimates. The number of factors that need to be considered can quickly increase to such levels that it is hard to grasp which factors are relevant and which are not. In cost–benefit analysis and investment appraisal, usually only limited sensitivity analysis approaches are used, such as one-at-a-time sensitivity analysis (Botchway etal., 2023; Rahmanzadeh etal., 2023; Fang et al., 2023), when each input variable is changed individually, and neither synergetic managerial capabilities nor joint effects of multiple uncertainty sources can be captured (Kozlova, Lo Piano, etal., 2024). In a SCOPUS database search, out of over 200,000 results with the “cost–benefit” keyword, only 7% mention “sensitivity analysis” (in either title, keywords, or abstract), and only 0.05% contain “global sensitivity analysis”,1 a more sophisticated type where all inputs are varied at the same time (Kozlova, Lo Piano, etal., 2024). In addition to one-at-a-time analysis, some researchers perform uncertainty analysis by means of Monte Carlo simulation and display the distribution of resulting profitability indicators (Mombello etal., 2023; Gill-Wiehl etal., 2023). This type of analysis, however, lacks the knowledge of which input factors are more important and drive the output distribution one way or another. A recent methodological development, Simulation Decomposition (SimDec), combines the benefits of sophisticated sensitivity analysis and uncertainty analysis by revealing which inputs are important and how different combinations of them translate onto the output distribution (Kozlova & 236 Hannu Karjunen et al. where t is time index, n is the last year of analysis, i is yearly interest rate, and CF is cash flow. The initial investment cost (I) of technical installation, including engineering, occurs at the value point in time 0. The NPV is the difference between the present value of cash inflows and the present value of cash outflows from the year 1 to 20 (PV) less the present value of initial investment (I). NPVP=-VI (2) Numeric assumptions for the input variables of the model are presented in Appendix 1. Annual cash inflows are based on revenues from selling end-products (and by-products) and depend upon the annual production amounts, operation time, and the production ratio. The initial investment costs consist of technical investment costs, reserve, and working capital. Cash outflows of the model consist of expenses during operation. Aweighted average cost of capital (WACC) interest rate is employed and the cost of equity influences the interest rate. These input variables directly impact the discounted cash flow and the calculation of NPV. The model also computes specific investor cash flows when determining the investment IRR. Investor cash flow is calculated from the annual operating margin (equivalent to net cash flow) decreased by both debt amortization and debt interest costs. As cash flow depends on the profit, investor cash flow corresponds to the investment return from equity. The apportionment of subsidy, debt, and equity all impact the investor cash flow. CFtt =-operatingmargin debtamortizationt-debtinterestcosttst (3) 2.3 P revious sensitivity analysis studies Laaksonen etal. (2021) previously performed two forms of sensitivity analysis on the model: (1) a one-at-a-time sensitivity analysis of the base scenario, and (2) a selected scenario analysis that computed deterministic profitability indicators. 2.3.1 One -at-a-time sensitivity analysis The key variables together with their realistic ranges, were established by the participating researchers and experts (Table 10.1). The most critical factors were identified to be end product price (Gasoline price in Base scenario), Hydrogen price, Operation time, and Investment reserve. The various impacts on profitability of the uncertainty and risks were analyzed by examining changes to the model from varying one input at time. Table10.1 presents the results with starting values in the Base scenario. The toolbox of techno-economic assessment with SimDec 237 Table 10.1 One-at-a-time sensitivity analysis Base scenario Electricity price, €/MWh 20 30 40 50 IRR (investor) 18.0% 16.9% 15.8% 14.7% Hydrogen price, €/MWh 10 15 20 25 30 IRR (investor) 20.3% 15.8% 11.2% 6.3% 1.0% Investment reserve −30% −15% 0% 15% 30% IRR (investor) 52.0% 33.1% 22.6% 15.8% 10.9% Gasoline, €/t 1000 1200 1300 1400 1600 1800 IRR (investor) −3.3% 7.1% 11.5% 15.8% 24.0% 31.9% Debt interest rate 1% 2% 3% 4% 5% IRR (investor) 17.2% 15.8% 14.5% 13.2% 12.0% O&M 2% & 3% 3% & 4% 4% & 5% IRR (investor) 15.8% 11.6% 7.2% Operation time 6000 7000 8000 IRR (investor) 5.4% 10.8% 15.8% Investment subsidy (TEM) 30% 40% 50% IRR (investor) 12.4% 15.8% 20.1% Source: Updated from Laaksonen etal. (2021) (https://lutpub.lut.fi/bitstream/handle/10024/ 162597/P2X%20Joutseno%20Final%20Report.pdf?sequence=1&isAllowed=y), Figure4.5, p.66. 2.3.2 Scenario analysis Laaksonen etal. (2021) created five different scenarios to compare different strategies and alternatives (Table10.2). The first scenario is specified as the base scenario. The second scenario represents a variation in which electrolysis-sourced hydrogen is used instead of purchased by-product hydrogen. Since both of these scenarios employ an MTG-pathway approach, the final product is gasoline. The third scenario, named MTG, is a variation of the base scenario possessing more detailed cost parameters and other revised assumptions. The final two scenarios employ non-MTG technological approaches. In the MeOH case, production is halted at the methanol stage. The MTO-MOGD case initially converts methanol to olefins and then into gasoline and distillates. The MTG, MeOH, MTO-MOGD scenarios all require purchased hydrogen as opposed to hydrogen produced electrolytically. 3 SimDec analysis SimDec is first used to replicate the earlier sensitivity analysis, followed by a restructured sensitivity analysis with updated assumptions. 238 Hannu Karjunen et al. Table 10. 2 Scenario analysis Scenario Description Total Investment Investor IRR NPV, M€ investment, subsidy, M€ M€ Base scenario Initial first draft for 76.5 26.4 15.8% 30.1 plant profitability analysis. Base scenario Initial first draft for 116.0 40.2 --144.0 electrolysis plant profitability analysis using electrolyser-sourced hydrogen. MTG Basic route to drop82.8 28.6 24.7% 53.0 in-fuels. Compared to the base scenario, most critical changes are: • Upgraded product prices and investments • Utility consumptions revised • Catalyst renewal cost included MeOH Final product is 62.1 21.4 9.0% 7.2 methanol. • No drop-in-fuel synthesis in investment and catalyst cost • Product price assumed to be €400/t MTO-MOGD Alternative synthesis 99.3 34.4 23.0% 56.9 pathway including kerosene and diesel as end product. Source: Updated from Laaksonen etal. (2021). (https://lutpub.lut.fi/bitstream/handle/10024/ 162597/P2X%20Joutseno%20Final%20Report.pdf?sequence=1&isAllowed=y), Table4.3, p.67. 3.1 Monte Carlo simulation In Table10.3, the input variables for the SimDec Monte Carlo simulation are assumed to be uniformly distributed within the ranges of the values used in the previous sensitivity analysis study (i.e. Table10.1). The variation in all product prices is assumed to be the same as for the Gasoline price, ranging from 63% to 114%. The toolbox of techno-economic assessment with SimDec 239 from residual heat and oxygen were not studied in detail, as their influence would not eliminate the dominance of end product price, hydrogen price, and investment on profitability. Using these assumptions, the model was simulated 50,000 times, and the dataset was analyzed using the SimDec Matlab package.2 3.2 Simulation Decomposition The output was analyzed using both the simple binning approach for sensitivity indices computation and SimDec visualization (see Chapter2 of this book, Kozlova, Roy, etal. (2024)). The analysis was performed in four iterations: 1. Base scenario only (extension of earlier OAT analysis, Section2.3.1) 2. All scenarios (extension of earlier scenario analysis, Section2.3.2) 3. Currently realistic scenarios (updated uncertainty assumptions and new set of scenarios) 4. Methanol electrolysis scenario (currently under consideration as a pilot case) Table 10.3 Variation in input parameters Input variable Range Distribution Electricity price, €/MWh [20, 50] Uniform Hydrogen purchase price, €/MWh [10, 30] Uniform Product selling prices [63%, 114%] Uniform Methanol price, €/tn [253, 455] Uniform Gasoline price, €/tn [1000, 1800] Uniform Kerosine price, €/tn [1059, 1907] Uniform Diesel price, €/tn [1081, 1947] Uniform LPG, €/tn [323, 582] Uniform Total investment reserve Uniform Debt interest rate [1%, 5%] Uniform O&M costs Operations, % of actual revenue [2%, 4%] Uniform Maintenance, % of technical revenue [3%, 5%] Uniform Operation time, h/a [6000, 8000] Uniform Investment subsidy, share of [30%, 50%] Uniform technical investment Scenario {1} – Base scenario Discrete {2} – Base scenario electrolysis {3} – MTG 2.0 {4} – MeOH {5} – MTO-MOGD NPV was chosen as the output variable instead of IRR. The contributions 240 Hannu Karjunen et al. 3.2.1 Base scenario only (extension of OAT, Section2.3.1) The Base scenario is filtered from the simulated dataset, resulting in the extraction of 9,922 corresponding outputs from the 50,000 data points. Table10.4 shows the first-order sensitivity indices (individual influence of each input variable) to enable a ranking of the input variables by their relative importance. The sensitivity indices indicate that Gasoline price is the most influential variable, followed by Hydrogen price, and then by the Investment reserve. The second-order effects are all negligible. This finding implies that the model is additive and that these results are consistent with the previous OAT analysis (Table10.2). For the decomposition, the two most influential input variables, Gasoline price and Hydrogen price, are chosen. The resulting SimDec graph is shown in Figure10.3. Figure10.3 exhibits a monotonic relationship between the most influential input variables and the output. NPV increases with increasing Gasoline price, since it affects revenues, and decreases with Hydrogen price, since this contributes to costs. Most of the High sub-distribution portion of the Gasoline price lies in the positive NPV range, signifying the overall great potential for this technology. However, most of the influence over profitability resides in external market factors beyond the control of management. 3.2.2 All scenarios (extension of scenario analysis, Section2.3.2) Table10.5 shows the sensitivity indices computed for the individual scenarios over the entire dataset (with the new Scenario variable included as a separate input). Table 10.4 Sensitivity indices for Base scenario Input variable Sensitivity index Gasoline price 55% Hydrogen price 16% Investment reserve 16% Operation time 5% O&M 4% Investment subsidy 2% Debt interest rate 1% Electricity price 0% LPG Price 0% Kerosine Price 0% Diesel Price 0% Methanol Price 0% Total 100% The toolbox of techno-economic assessment with SimDec 241 Figure 10.3 Simulation Decomposition of NPV (Base scenario) by Gasoline price (55%) and Hydrogen price (16%). (colour image is accessible via the link) Colour Gasoline price, €/t Hydrogen price, €/MWh NPV, M€ Min Mean Max Probability Low [1,000, 1,267] Low [10, 17] −56 −11 45 11% Medium [17, 23] −70 −24 29 11% High [23, 30] −83 −37 13 11% Medium [1,267, 1,533] Low [10, 17] −39 15 83 11% Medium [17, 23] −47 157 11% High [23, 30] −67 −13 54 11% High [1,533, 1,800] Low [10, 17] −19 41 104 11% Medium [17, 23] −34 28 89 11% High [23, 30] −44 14 73 11% Table10.5 clearly indicates that the Scenario variable is the most influential parameter over the entire dataset. However, each scenario demonstrates its own relatively unique sensitivity profile and that product price has a significant effect on profitability for each scenario. In the Base electrolysis scenario, Electricity price is the most important factor, since the hydrogen is produced in-house and requires electricity. Investment reserve appears significant in the MTO-MOGD scenario, which can be partially explained by the larger absolute investment. 242 Hannu Karjunen et al. Figure10.4 decomposes the entire dataset by Scenario and Gasoline price. The corresponding sensitivity indices appear in the legend. The decomposition shows that the Base scenario electrolysis (yellow) is associated with a greater uncertainty (as it is wider) than other scenarios and is considerably shifted into the negative NPV range with only a negligible positive portion. All other scenarios essentially lie on top of each other with a similar profitability profile. The MeOH scenario exhibits a much lower upside potential than Base scenario, MTG, and MTO-MOGD, together with a negative expected mean (see mean NPV of the Medium Gasoline price in each scenario). The influence of Gasoline price provides a noticeable horizontal shift of shades in the Base scenario and MTG, while also possessing the highest sensitivity indices. The shift is noticeable in the Base scenario electrolysis, although less pronounced, given the value of its sensitivity index. In the MeOH and MTO-MOGD scenarios, the shaded sub-distributions clearly lie on top of each other, which provides a visual confirmation of their negligible sensitivity indices. The SimDec analysis visually increases the amount of insight over the earlier deterministic scenario analysis. Firstly, a very different importance profile of input variables is revealed in each scenario. Secondly, the distributions of the SimDec charts reinforce a distinct visual sense of the uncertainty exposure from each project. Thirdly, the decomposition further guides the decision-maker by providing a deeper comprehensive understanding of how different factors interact, thereby affecting the overall profitability of the investment. Table 10.5 Sensitivity indices for all scenarios and the entire dataset Scenario Base case Base case electrolysis MTG MeOH MTO-MOGD Altogether Electricity price30% 52% 1% 1% 0% 2% Hydrogen price416% 0% 15% 27% 17% 5% Investment reserve 16% 15% 16% 17% 27% 6% Gasoline price555% 24% 56% 0% 2% 7% Methanol Price 0% 0% 0% 46% 0% 8% Kerosine Price 0% 0% 0% 0% 8% 0% Diesel Price 0% 0% 0% 0% 21% 0% LPG Price 0% 0% 0% 0% 0% 0% Debt interest rate 1% 1% 1% 0% 1% 0% O&M 4% 4% 4% 4% 7% 2% Operation time 5% 0% 5% 2% 9% 1% Investment subsidy 2% 2% 2% 2% 3% 1% Scenario - - - - - 66% Sum 100% 97% 99% 99% 96% 97% The toolbox of techno-economic assessment with SimDec 243 3.2.3 Currently realistic scenarios After the sensitivity analysis replication, the model assumptions were updated to reflect current conditions. The following scenario updates were implemented: Figure 10.4 Simulation Decomposition of NPV by Scenario (66%) and Gasoline price (7%). (colour image is accessible via the link) Colour Scenario Gasoline price, €/t (sensitivity indices and states) NPV, M€ Min Mean Max Probability Base scenario electrolysis 24% Low [1,000, 1,267] −260 −143 −33 7% Medium (1,267, 1,533] −242 −119 −8 7% High (1,533, 1,800] −214 −92 22 7% MeOH 0% Low [1,000, 1,267] −95 −29 40 7% Medium (1,267, 1,533] −101 −29 42 6% High (1,533, 1,800] −99 −28 42 7% MTO-MOGD 2% Low [1,000, 1,267] −85 −6 94 7% Medium (1,267, 1,533] −91 −3 91 7% High (1,533, 1,800] −83 396 7% MTG 56% Low [1,000, 1,267] −100 −30 39 7% Medium (1,267, 1,533] −67 −1 77 6% High (1,533, 1,800] −41 26 101 7% Base scenario 55% Low [1,000, 1,267] −83 −24 45 7% Medium (1,267, 1,533] −64 183 6% High (1,533, 1,800] −44 28 104 7% 244 Hannu Karjunen et al. • Only electrolysis scenarios were considered. This assumption is related partly to local developments and also to reflect the view that hydrogen cannot be sourced reliably and in significant scale as a by-product from existing processes. Consequently, P2X processes must independently secure any hydrogen required. • One additional scenario is introduced. The MTG Electrolysis speculative scenario assumes a more efficient and less costly electrolyser. In addition, the oxygen obtained as a by-product from electrolysis is assumed to have a value of €20/MWh. • The variation of product selling prices is updated – to [400, 1000] €/tn for methanol and ±50% for everything else. • Electricity price is updated to a range of [30, 60] €/MWh. • The variation in Investment reserve is dropped and fixed at 15%. The variation of Total Investment cost is introduced instead within the range ±30%. • The Investment subsidy is changed to a binary variable with values [0, 40%] that represent the uncertainty in receiving the subsidy. Table10.6 provides the sensitivity indices calculated, as before, for the individual scenarios over the entire dataset. Once again, the Scenario variable is considered as a separate input. A number of differences can be observed in comparison with the previous case (Table10.5). Because only electrolysis cases have been considered, the Electricity price is uniformly influential across all scenarios, while, conversely, Hydrogen price is never important. However, the Investment subsidy becomes more influential after its binary variable modification. The scenario with the highest impact on Total investment, MTO-MOGD electrolysis, reveals the largest sensitivity to Investment subsidy. Product prices remain the most important profitability factor throughout. Figure10.5 shows the decomposition of investment profitability resulting from variables Scenario and Gasoline price. All electrolysis scenarios exhibit poor profitability profiles, with the majority of their sub-distributions falling into the negative NPV range. The considerably lowered sensitivity index for Scenario (12% here as opposed to 66% previously) is due to the rather-stacked appearances of the sub-distributions. The MTG Electrolysis speculative and MeOH Electrolysis demonstrate the highest upside potentials of all the scenarios. 3.2.4 Methanol electrolysis scenario In this section, the MeOH electrolysis case is scrutinized more closely. MeOH electrolysis represents the actual scenario chosen for the P2X construction project at Lappeenranta. According to Table10.6, the most influential input variables behind its profitability are Methanol price (56%) and Electricity price (25%). Consequently, Figure10.6 illustrates a decomposition based upon these two factors. The toolbox of techno-economic assessment with SimDec 245 Table 10.6 Sensitivity indices for the updated model with electrolysis-only scenarios Scenario Base case electrolysis MTG Electrolysis MeOH Electrolysis MTO-MOGD Electrolysis MTG Electrolysis speculative Altogether Electricity price 26% 24% 25% 29% 22% 22% Hydrogen price60% 0% 0% 0% 0% 0% Total investment 12% 10% 8% 17% 9% 13% Gasoline price 48% 51% 1% 3% 57% 27% Methanol price 0% 0% 56% 0% 0% 13% Kerosine price 0% 0% 1% 8% 0% 1% Diesel price 0% 0% 0% 23% 0% 3% LPG price 1% 0% 0% 1% 0% 0% Debt interest rate 1% 1% 1% 1% 1% 1% O&M 3% 2% 2% 3% 2% 2% Operation time 1% 0% 1% 0% 1% 0% Investment subsidy 10% 11% 8% 18% 9% 10% Scenario - - - - - 12% Sum 102% 101% 103% 103% 103% 103% 252 Hannu Karjunen et al. Appendix 1: Input variables of the DCF model (base scenario/ MTG) (Laaksonen etal., 2021) Input variable Numeric value Unit Source of info (assumption) Operation time 8,000 h/a Product quantities • Methanol 25,000 t/a Aspen modelling and • Gasoline 8,750/9,500 t/a case description • Diesel 0t/a (methanol quantity • Kerosine 0t/a fixed). • LPG 2,000/1,000 t/a • Purge stream 300/500 t/a • Oxygen 0t/a • Water 0t/a • Heat 0MWh/a Product selling prices • Methanol 0€/t Selling prices of end • Gasoline 1,400/1,583 €/t products are based • Diesel 1,500/1,677 €/t on market analysis • Kerosene 1,500/1,712 €/t and the knowledge • LPG 512 €/t of experts in the project. Production ratio (plant availability) • First year 50 % • Years 2–20 100 % Technical investment costs • Hydrogen 2.3/5.2 M€ Technical investment • Carbon dioxide 18.5 M€ costs are based on • MeOH synthesis 16.7 M€ budgetary offers and • MTG synthesis 19.3/17.6 M€ knowledge of expert • Auxiliary systems 1.0/2.0 M€ in • Other investment 4.5/6.6 M€ project team. costs* Reserve (%) 15 % Working capital addition 0.5 M€ (cash reserve) Financing • Investment subsidy 40 % • Debt 70 % • Equity 30 % Costs and expenses during operation (Continued) The toolbox of techno-economic assessment with SimDec 253 Input variable Numeric value Unit Source of info (assumption) • Operation costs (as 2 % Annual production percentage of actual amounts and annual revenue) consumptions of • Maintenance costs (as 3 % electricity, steam, percentage of techni-hydrogen, and carbon cal investment) dioxide are based on • Electricity 22,470/29,920 MWh Aspen modelling. consumption Prices of raw mate- • Electricity price 40 €/ rials (electricity, (including transfer fee) MWh steam, hydrogen, • Steam consumption 60,312/33,200 MWh and CO2) are based • Steam price 20 €/ on the knowledge of MWh experts in the project • H2 consumption 182,000 MWh and existing market • H2 price 15 €/ prices. MWh • Real estate tax (per-1.43 % centage of building investment, excluding equipment) • Insurance costs (as 0.25 % percentage of debt) • Administration costs 2 % (percentage of actual revenue) Other source data • Debt rate 2 % • Cost of equity 6 % • Change % of WACC 0%/a (weighted average cost of capital) • Rate of inflation 0 % • Income tax % 0 % • Number of years for 20 debt amortization • Straight-line deprecia-20 tion (years) • Residual value 0 € Note: LPG, liquid petroleum gas; WACC, weighted average cost of capital. *Other costs include electricity connection fees, infrastructure (roads, etc.), buildings, engineering, interest and expenses during construction, bank fees, land lease before the start-up, and permitting. Part IV Applications: Engineering DOI: 10.4324/9781003453789-15 This chapter has been made available under a CC-BY-NC-ND 4.0 license. Abstract The reliability of steel structures is a complex nonlinear phenomenon that depends on multiple external and technological factors. Accumulated fatigue can lead to a sudden collapse of construction, resulting in lost reputations, sunk investments, and even loss of lives. Thus, accurate models that capture fatigue damage accumulation are of paramount importance, as are the adequate methods that explain and communicate the complex input–output relationships of the model. In this study, we apply Simulation Decomposition (SimDec) to a fatigue assessment model developed for welded joints. SimDec exposed and communicated a three-dimensional heterogeneous effect in the model. The three input variables interact pairwise and affect the output in a nonlinear fashion conditioned to each other. Identifying such effects is critical when designing reliable steel structures. 1 Introduction Mechanical system components, such as that found in industry process equipment, bridges and infrastructures, vehicles and transportation equipment, machinery, and cranes, play an important role in the functioning of various industries. The mechanical design of these systems needs to satisfy the technical requirements (i.e. the functions or operations needed by an end-product). These functions can involve energy conversions, load-bearing capacities, and/or accessibility and dimensions to certain space. In addition to the technical requirements, the mechanical system must fulfil the structural safety and integrity during the service load actions. This is necessary in order to avoid loss of expensive structural assets and infrastructure or loss of human lives due to the sudden collapse or ruptures of mechanical components in the worst case. In the context of structural design and analysis, mechanical industries rely heavily on modelling and simulations to address Chapter 11 Capturing multi-dimensional nonlinear behaviour of a steel structure reliability model – global sensitivity analysis Antti Ahola, Mariia Kozlova, and Julian Scott Yeomans 258 Antti Ahola, Mariia Kozlova, and Julian Scott Yeomans such things as structural behaviour (strength, deflections, stability), user experience and control systems, and verification of various functionalities. Due to the increased computational resources, nowadays, these models are more frequently built on a numerical basis using commercially available software. Irrespective, it remains crucial to understand the underlying behaviour of those models in order to make effective decisions about the design of more resilient structures. In the context of structural applications, steel materials are widely used due to their excellent mechanical-performance-to-weight ratio, widespread availability, economical profitability, and manufacturability. Because of this, steel materials have been employed in many of the aforementioned fields. Amongst different failure criteria, fatigue is of paramount importance, particularly for those components subjected to cyclic and/or dynamic load conditions. Fatigue is a phenomenon in which a structural element experiences cracking under cyclic or fluctuating stress load conditions that are lower than the yield or, ultimately, strength of the material. Ahigh fraction of reported failures in steel components can be attributed to the fatigue phenomenon (Hobbacher, 2020). Several studies have even reported that more than 50% of failures in metallic components result from fatigue (Stephens et al., 2000). Some of these fatigue-originating failures have caused severe consequences: the crash of the first commercial passenger jet plane (De Havilland, 1954, US), the capsizing of the Alexander L. Kielland oil platform (1980, Norway), and the derailment of a high-speed train in Eschede (1998, Germany) claimed 21, 123, and 101 human lives, respectively. Compared to many other failure mechanisms, fatigue phenomenon can be complicated by multiple influencing factors. In addition, the uncertainty related to the predicted operating load conditions generally prevents an accurate assessment of fatigue and structural life cycle in engineering (Hultgren etal., 2021). Taking into account the fact that steel production currently generates more than 7% of the global CO2 emissions (World Steel Association, 2021), there is an escalating need to create more sustainable, high-performing structural steel applications. Steel structures usually incorporate welding as the joining method to create permanent connections. Welded connections are susceptible to fatigue failures as the process introduces geometrical discontinuities, tensile residual stresses, and potential flaws into the material. Over the past few decades, great efforts have been employed to establish design and analysis methodologies for fatigue assessments of welded connections. Acting stress, or strain amplitude, has been identified as one key parameter (Braun etal., 2022). Consequently, the vast majority of fatigue assessment approaches utilize applied stress-based methods that have also been documented in international product and (steel) structure standards (EN 1993-1-9, 2005). These methods have clearly established correlation between applied stress and life (S-N curves), for example, using Basquin or equivalent equations (Dowling, 2013). The model parameters of such equations are usually determined via Multi-dimensional nonlinear behaviour of a steel structure 259 experimental testing, i.e. fatigue testing is carried out on components. From empirical observations (applied stress versus life data), the model parameters are then statistically evaluated. Even though local stresses would be used, the parameters obtained are based on the stochastic models. At the high-cycle fatigue (HCF) regime, usually a Gaussian distribution of experimentally determined fatigue lives (at a certain load level) is obtained from which the characteristic design curves are then determined at the decided survival probability. Due to this, most fatigue approaches consider factors contributing to fatigue in a statistical manner, i.e. correction factors and different design curves are obtained for certain representative datasets. To adopt additional parameters governing fatigue performance of welded connections, such as material strength and residual stresses, a multiparametric 4R method has been proposed. Use of the method enables a consideration of the combined effects of the aforementioned parameters to improve the accuracy of fatigue assessments, particularly in the contexts of combined post-weld treatments (Ahola etal., 2021) and/or variable amplitude loads (Lipiäinen etal., 2023; Grönlund etal., 2024; Rohani Raftar etal., 2024). Sensitivity analysis of such complex models is an essential component of their design, analysis, and decision-making processes (Iooss etal., 2022). However, even advanced global sensitivity analysis techniques lack the means to identifying the shapes of the interactions in a model and to communicating their importance to decision-makers (Kozlova, Moss, etal., 2024). Simulation Decomposition (SimDec) is an approach that builds on global sensitivity analysis and extends it into a visualization of the most critical system behaviour (Kozlova, Roy, etal., 2024). In this chapter, we explore the added value of SimDec contrasted with previously done series of one-at-a-time sensitivity analyses for the 4R model. In the next section, the 4R method, its computational model, and a description of previously conducted sensitivity studies are summarized. Section3 describes the set-up of the Monte Carlo simulation, the computation of sensitivity indices, and the corresponding series of visualizations. The SimDec approach indicates a single main decomposition involving the three most influential variables, which interact pairwise, producing nested heterogeneous effects. Furthermore, the single-input decompositions are analyzed in conjunction with interaction decompositions using each pair of input variables. The chapter concludes with a comparison of the results to earlier sensitivity analysis studies and with a discussion of the added value provided by SimDec for this specific case. 2 Computational model 2.1 4R method The 4R method has been developed to construct more accurate assessments of effective stress in fatigue strength predictions. It is a multiparametric model 260 Antti Ahola, Mariia Kozlova, and Julian Scott Yeomans Figure 11.1 4R model description. Input parameters: (a) residual stress, (b) material’s ultimate strength at the heat-affected zone (HAZ), (c) applied stress ratio, and (d) weld toe radius, and cyclic behaviour at notch. (colour image is accessible via the link) that accounts for numerous parameters associated with the fatigue performance of welded connections. The 4R method considers residual stresses ( s res , Figure11.1a), material ultimate tensile strength (R m , Figure11.1b), applied stress ratio (R, Figure11.1c) of external loading, as well as geometrical weld quality via the weld toe radius (r true , Figure11.1d). Due to the applied four (bolded) “R”-related parameters, the approach was named the 4R method. The local stress ratio is applied in the mean stress-correction using the well-known Smith-Watson-Topper (SWT) equation (Smith etal., 1970). As a result, the mean stress-corrected effective stress is applied in the fatigue assessments using a conventional S-N (stress–life)–based correlation. Section2.2 describes the computation of cyclic stress behaviour in detail. 2.2 Computational model This section outlines the details of the computational model applied in the 4R method. The essence of the 4R method is to compute the local (cyclic) elastic– plastic behaviour at a fatigue-critical notch. The cyclic behaviour can be simulated based on the numerical model (e.g. using finite elements), but it is usually more feasible to conduct such analyses using analytical equations due Multi-dimensional nonlinear behaviour of a steel structure 261 to their complexity. As an output value, the mean stress-corrected reference effective notch stress range is computed as: ∆∆ kref ktruelocal true mres rRrR R , /, ,,= () − () 1 (1) where kref, is the mean stress-corrected (reference) local stress, k is linear-elastic notch stress obtained using an effective stress concept (i.e. either using a fictitious radius concept or theory of critical distance). Further definitions of the applicable stress concepts for welded connections and cut edges have been introduced in the previous works undertaken by Ahola etal. (2021) and Lipiäinen etal. (2023). The Rlocal value is obtained from the local cyclic behaviour based on the minimum and maximum stress as follows (see also Figure11.1): Rlocalmin max = / . (2) For a determination of local behaviour, both maximum and minimum stress is determined. The material behaviour is described using an elastic– plastic model. In this context, the well-known Ramberg–Osgood model can be employed (Dowling, 2013). For the monotonic load (first peak load of a first cycle), the material behaviour is formulated as: =+= () + () ep n EH// /1 , (3) where is the (total) strain, e is the elastic strain, p is the plastic strain, is the stress, E is the modulus of elasticity, and H and n are, respectively, the strength coefficient and strain hardening exponent of the material plastic behaviour. For the cyclic material behaviour, the kinematic hardening rule is assumed, and the cyclic material behaviour is formulated as: =+= () + () ep n EH// / 22 1, (4) where the variables are similar to equation (3) but described by the range ( ). To analytically compute the elastic–plastic behaviour, Neuber’s notch theory is applied to obtain plastic behaviour from elastic stresses (see Dowling (2013)). The upper bound (monotonic load) can be formulated as: maxreskmax max resk max E R E =+ () =+− () () () () / , 22 1 (5) 268 Antti Ahola, Mariia Kozlova, and Julian Scott Yeomans Figure11.3 depicts the distribution of the output stress, stretching from near zero to 900 MPa. The most influential input, residual stress, breaks down the output distribution into three parts according to its states, compressive & negligible (blue), medium (yellow), and high (green). Interestingly enough, whereas compressive & negligible residual stress results in a wide range of output values, medium and high residual stress focus the corresponding output values on a relatively constrained space. The influence of the second-important input variable, stress ratio, depends on the state of the first input variable, residual stress. If residual stress is medium or high, the stress ratio has a limited effect, which can be seen from the minor horizontal shift of shades of yellow and green. If residual stress is compressive & negligible, however, the stress ratio plays a significant role. The third important input variable, steel grade, divides the output attributed to compressive & negligible residual stress into two well-defined sub-distributions (dark-blue scenarios versus light-blue ones). Medium and high residual stress, however, is attributed to only a single steel grade, mild for the former and UHSS for the latter. Output stress values above 630 MPa can result only from three combinations of input factors: (1) compressive & negligible residual stress with medium & high stress ratio and UHSS steel grade, or (2, 3) high residual stress with both stress ratio levels and with only UHSS steel grade possible in such combination. The output stress values below 530 MPa can only be achieved when stress ratio is reversed and residual stress is either compressive & negligible or medium, no matter the steel grade. Mid values of the output stress are mostly comprised of the three scenarios, two with medium residual stress and one compressive & negligible one with low & high stress ratio and mild steel grade. The most uncertain (unpredictable or least under control) scenario is the one with compressive & negligible residual stress, low & high stress ratio, and UHSS steel grade (light-blue); it stretches over 90% of the range of the output. The decision-making implications turn out to be relatively straightforward. When the structural design involves high or medium residual stresses, the stress ratio does not play an important role in the estimation of the output stress. This is also in line with the general understanding of the residual stress effects on the fatigue behaviour of welded connections (Hobbacher, 2016). However, for structures with compressive and low residual stress, the accurate estimation of stress ratio or, if possible, the design of the operating conditions is critical. If the stress ratio appears to be high (or the operating conditions cannot be softened), the grade of steel becomes the next most important design parameter. Such decision logic is built based on the visually most prominent heterogeneities and illustrated by the decision tree in Figure11.4. To facilitate the perception for professionals, in the decision tree, normalized values are used. These are obtained by dividing the output reference stress kr,ef (equation (1)) by the applied nominal stress of norm=200 MPa. Multi-dimensional nonlinear behaviour of a steel structure 269 Figure 11.3 Main decomposition of the structural reliability model output by the most influential three input parameters, explaining 96% of the variance of the output (sum of their sensitivity indices) and portraying three second-order effects causing heterogeneous input–output relationship. The histogram is stacked and exposes the entire simulation data without overlapping. The share of data in each scenario (or the probability of scenario) is displayed in the rightmost column of the legend. (colour image is accessible via the link) Colour Residual stress, σres Stress ratio, R Steel grade, Rp0.2 Output stress, Δσk,ref Share of data Min Mean Max Compressive & negligible Reversed Mild 107 332 475 11% UHSS 11 264 515 11% Low & high Mild 320 467 591 22% UHSS 68 543 819 22% Medium Reversed Mild 383 456 524 6% UHSS NaN NaN NaN NaN Low & high Mild 422 499 622 12% UHSS NaN NaN NaN NaN High Reversed Mild NaN NaN NaN NaN UHSS 630 704 795 6% Low & high Mild NaN NaN NaN NaN UHSS 656 746 851 12% 270 Antti Ahola, Mariia Kozlova, and Julian Scott Yeomans Figure 11.4 Decision tree constructed based on the most prominent heterogeneity in the effects of input variables on the output stress (denoted with stress concentration factor K f,ref ). (colour image is accessible via the link) Multi-dimensional nonlinear behaviour of a steel structure 271 By doing this, a reference notch stress concentration factor is obtained. This factor directly implies the effects on the fatigue strength capacity. Kfr,,ef = kref /, norm (9) The decision tree presented in Figure11.4 should be taken with a grain of salt, however. Firstly, the resulting ranges of the output stress intersect, so the different branches of the tree do not lead to exclusive solutions, and as mentioned while discussing Figure11.3, different ranges of the output can be achieved with several scenarios. Secondly, the tree nodes are constructed from the most prominent heterogeneous effects across the entire range of the output. If one, however, would prefer to focus on a subset range or consider a limited set of applicable solutions (i.e. certain material strength), the decision logic needs to be reconstructed from Figure11.3 again and the resulting decision tree might change its structure. 3.3.2 Contr ibution of individual inputs In this subsection, the decompositions created for each input are shown to illustrate their individual contribution to the variance of the model output. Proceeding in order of significance (Table11.2), the first decomposition by residual stress is shown in Figure11.5. The range of the input variable is divided into four states; negative values fall into the compressive state, values up to 100 constitute the Negligible state, followed by medium and high, attributed to the different steel grade levels (Table11.1). From Figure 11.5, one can observe a rather peculiar effect. While the compressive and negligible states cover the majority of the model output value range, with Compressive stretching further into the lowest values, the medium and high states create narrow, distinct sub-distributions that explain the two peaks of the overall distribution of the model output. Such focused concentration of values in the medium and high states explains the high sensitivity index of 51%. The decomposition by stress ratio is depicted in Figure11.6. The range of the ratio values is divided into three states with all negative values constituting the reversed state, and the remainder divided according to the equal range principle. Figure11.6 demonstrates a gradual shift of the scenarios to the right, with the minimums of each scenario staying further apart than the maximums. Such a gradient effect signifies the monotonic type of the input’s effect. The decomposition by steel grade is opposite to monotonic (Figure11.7). Although the range of the steel grade values is divided into three states of equal sub-ranges, only the lower and the higher actually exist in the model, which constitute mild and UHSS steel grade, correspondingly. 272 Antti Ahola, Mariia Kozlova, and Julian Scott Yeomans Although steel grade receives a relatively low combined effect (only 10% according to Table11.2), the visualization in Figure11.7 reveals a clear distinction between the two steel grades. The mild steel grade occupies a relatively narrow area slightly shifted to the left on the graph, while the UHSS steel grade peaks in the right part of the distribution with its tail covering the entire range of the mild steel grade scenarios (and going even lower beyond them). Such an effect type that causes the difference in variation of the output but less so in the mean is poorly captured by variance-based sensitivity indices, since they are computed based on averages. SimDec visualization brings clarity and allows in-depth analysis of such effects based on the descriptive statistics of the corresponding scenarios (legend). Colour Residual stress, σres Output stress, Δσk,ref Share of data State Value, MPa Min Mean Max Compressive [−400, 0) 11 413 819 49% Negligible [0, 100) 322 503 819 16% Medium [100, 650) 383 481 622 17% High [650, 950] 630 732 851 17% Figure 11.5 Contribution of residual stress to the variance of the output: 51% of the variance is explained by the input variable. (colour image is accessible via the link) Multi-dimensional nonlinear behaviour of a steel structure 273 The factor with the least significance, fatigue notch factor, demonstrates appropriately little influence on the SimDec visualization as well (Figure11.8), where all three equally spaced states of this input largely intersect and lie on top of each other. Only a slight shift on the right edge of the distribution explains the computed 4% sensitivity index. 3.3.3 Interaction between residual stress and stress ratio Residual stress and stress ratio have the strongest interaction effect and together explain over 85% of the variance of the output. Each of the two variables is broken down into three states (the same as before for the residual Colour Stress ratio, R Output stress, Δσk,ref Share of data State Value Min Mean Max Reversed [−1.2, 0) 11 420 817 50% Low [0, 0.4) 219 522 825 21% High [0.4, 0.7] 393 602 851 29% Figure 11.6 Contribution of stress ratio to the variance of the output: 35% of the variance is explained by the input variable. (colour image is accessible via the link) 274 Antti Ahola, Mariia Kozlova, and Julian Scott Yeomans stress) and of equal ranges for the stress ratio (Table11.4). The resulting decomposition is shown in Figure11.9. Figure11.9 is very similar to Figure11.3 since the decomposition is constructed from the same two variables. Figure11.9 confirms the clear heterogeneous effect, which was also observed in Figure11.3. The effect of the second important variable, stress ratio, depends on the state of residual stress. When residual stress is medium or high, the stress ratio does not affect the output very much (slight horizontal shift of the shaded sub-distributions within the green and yellow parts). When the residual stress is compressive & negligible, however, the stress ratio has a much more pronounced impact (substantial horizontal shift of the blue-shaded sub-distributions). Colour Steel grade, Rp0.2 Output stress, Δσk,ref Share of data State Value, MPa Min Mean Max Mild [255, 523) 107 441 622 50% -[523, 792) NaN NaN NaN NaN UHSS [792, 1060] 11 548 851 50% Figure 11.7 Contribution of steel grade to the variance of the output: 10% of the variance is explained by the input variable. (colour image is accessible via the link) Multi-dimensional nonlinear behaviour of a steel structure 275 3.3.4 Interaction between steel grade and stress ratio The next strong interaction occurs between steel grade and stress ratio. The modelling of the steel grade values is done for the two case materials, S355 and S950 (Table11.1). Thus, the two earlier states are used for this input (Table11.5) and, for clarity of visualization, is chosen first for decomposition (Figure11.10). From Figure11.10, it can be observed that output values attributed to mild steel grade are concentrated in the middle (peaking at around 450 MPa), while the ones attributed to the UHSS steel grade are spread over the entire range of the output and form another peak around 700 MPa. The stress ratio has a visible monotonic impact in mild steel grade, and a messy effect in the UHSS steel grade. This appears as if monotonicity is repeated twice, which, Colour Fatigue notch factor, KfOutput stress, Δσk,ref Share of data State Value Min Mean Max Low [2.3, 2.6) 11 467 777 47% Medium [2.6, 2.8) 12 504 828 30% High [2.8, 3.1] 17 538 851 23% Figure 11.8 Contribution of fatigue notch factor to the variance of the output: 4% of the variance is explained by the input variable. (colour image is accessible via the link) 276 Antti Ahola, Mariia Kozlova, and Julian Scott Yeomans Table 11. 4 States formation of input variables for the supplementary decomposition by residual stress and stress ratio Residual stress Stress ratio State Min Max State Min Max Compressive & negligible −400 100 Reverse −1.2 −0.6 Medium 100 650 Low −0.6 0.1 High 650 950 High 0.1 0.7 Colour Residual stress, σres Stress ratio, R Output stress, Δσk,ref Share of data Min Mean Max Compressive & negligible Reverse 11 299 515 22% Low 68 417 650 13% High 237 541 819 30% Medium Reverse 383 446 524 6% Low 421 476 548 3% High 428 508 621 8% High Reverse 630 704 795 6% Low 656 724 816 3% High 669 754 851 8% Figure 11.9 Supplementary decomposition of the structural reliability model portraying the interaction effect between residual stress and stress ratio. Their combination explains 86% of the variance of the output. (colour image is accessible via the link) Multi-dimensional nonlinear behaviour of a steel structure 277 Table 11. 5 States formation of input variables for the supplementary decomposition by steel grade and stress ratio Steel grade Stress ratio Min Max Min Max Mild 255 657 Reverse −1.2 −0.6 UHSS 657 1,060 Low −0.6 0.1 - - - High 0.1 0.7 Colour Steel grade, Rp0.2 Stress ratio, R Output stress, Δσk,ref Share of data Min Mean Max Mild Reverse 107 370 524 17% Low 320 446 548 11% High 366 493 621 23% UHSS Reverse 11 413 795 16% Low 67 516 816 11% High 237 663 851 22% Figure 11.10 Supplementary decomposition of the structural reliability model portraying the weak 4% interaction effect between steel grade and stress ratio. Their combination explains 43% of the variance of the output. (colour image is accessible via the link) 284 Manuel García Pérez et al. The charged particles achieve circular trajectories thanks to dipolar magnetic fields that exert the necessary Lorentz forces that bend their paths according to the desired curvature. The target magnetic field is thus a function of the desired curvature (fixed by the length of the accelerator itself) and the mass of the particles (Ferracin, 2014). The Large Hadron Collider (LHC) at CERN has been using dipolar fields of about 8.33 T since 2008. The proposed Future Circular Collider (FCC), currently in viability study, aims for 100 TeV collisions with a 100km ring accelerator. Given this design choice, the necessary (dipolar) magnetic field is 16 T (Tommasini etal., 2017). 1.1.1 Basics of superconducting Nb3Sn Some materials/metals may exhibit superconducting behaviour under the right conditions. This behaviour is characterized by the absolute absence of electrical resistivity, thereby enabling large current densities (of the order of a few kA/mm2) to flow without any voltage drop or heat losses (Ferracin, 2014). For this condition to occur, cables of the material need to be maintained below a critical temperature (usually cryogenic, on the order of a few kelvin) and below a maximum external magnetic field. The superconducting dipole magnets in the LHC and many other particle accelerators currently use NbTi as superconducting material. However, this alloy is not capable of behaving as a superconductor at the magnetic fields required for the FCC operation. Therefore, Nb3Sn has been selected as the preferred superconducting material for the FCC (Schoerling etal., 2015), as it is capable of being superconductive over wider ranges of magnetic fields and temperatures (Figure12.1). In Figure12.1, the surfaces divide the space of magnetic field, temperature, current density into two regions. The material is superconducting if the working conditions fall in the region on the opposite side of this surface. From the observed data, it is clear that NbTi cannot generate magnetic fields of 16 T, as its peak field (at 0 K and no current) is 14.5 T. 1.1.2 Application on superconducting coils and magnets Some types of superconducting cables consist of multiple twisted strands (around 40) in the form of a tape. Such a shape, referred to as a Rutherford cable, proves especially practical for winding double-pancake coil. Each strand (of order 1mm in diameter) contains a variable number (from tens to hundreds depending upon the material and manufacturing process) of much smaller filaments with the superconducting material embedded in a stabilizing matrix of copper (see Figure12.2). The Rutherford cable is wound around a central piece referred to as the pole. Dedicated spacers, wedges, or fillers are often added to control and adjust the position of the windings. All of this ensemble (pole, cable windings, spacers/wedges, and fillers) is usually referred to as a coil (Figure12.3). Sensitivity analysis of a superconducting magnet design model 285 Figure 12.1 Critical surfaces of superconducting alloys approximated by the correlation of Bottura (1999) and fitted to data points of Godeke etal. (2007) and Ferracin (2017). (colour image is accessible via the link) 286 Manuel García Pérez et al. Figure 12.2 (a) View of an Nb 3 Sn Rutherford cable showing the strands and fiberglass insulation. (b) Close-up of the strands. (c) Cross-section of the eRMC/RMM Rutherford tape consisting of 40 strands. (d) Detailed view of one strand cross section showing the Cu matrix and the 120 filaments in a honeycomb pattern. (e) An eRMC coil being wound with the cable; the bottom pancake layer has been completed. (colour image is accessible via the link) Source: Izquierdo Bermúdez etal. (2018); courtesy of CERN. Sensitivity analysis of a superconducting magnet design model 287 Figure 12.3 CAD view of an eRMC (top) and RMM-type (bottom) coils. (colour image is accessible via the link) Source: Izquierdo Bermúdez etal. (2018); courtesy of CERN. 288 Manuel García Pérez et al. Superconducting coils are placed surrounding the region of interest, the bore. The electrical current passing through these coils generates the desired magnetic field. Surrounding the coils, a support structure ensures the necessary mechanical pre-stress and stability of the coils (Ferracin, 2017). This combination of superconducting coils and support structure is usually referred to as a superconducting magnet. Synchrotrons require a number of superconducting magnets in series to achieve the trajectory curvature of a beam passing through the bores. For instance, the FCC is expected to require around 4,600 14.3m long 16T dipole magnets (Schoerling & Zlobin, 2019), whereas the LHC utilizes 1,232 8.33T magnets. 1.2 The eRMC and RMM magnets 1.2.1 Coils and support structure The Enhanced Racetrack Model Coil (eRMC) and Racetrack Model Magnet (RMM) represent two superconducting magnets designed by the CERN R&D program. They were conceived to test the Nb3Sn superconducting alloy and its capabilities/challenges for the FCC as a part of a five-year (2016 to 2021) program (Tommasini etal., 2017). The eRMC magnet features two 1240mm long block double-pancake coils sitting on top of each other. The RMM magnet adds a third coil, featuring the 50mm bore inside its central pole that fits between the two eRMC coils (Izquierdo Bermúdez etal., 2017; Rochepault etal., 2018). Dedicated pads or pushers are used to compress the coils vertically and horizontally. These pads are bolted together, surrounding the coils, and G10 fiberglass plates serve as cushions. The entire ensemble is usually referred to as the coil pack (Figure12.4 and Figure12.5). Surrounding the coil pack are ferritic yokes which contribute to the target magnetic field. An aluminium external shell encloses the ensemble (Figure12.6 and Figure12.7). The support structure was analyzed and validated first with the use of dummy coils made of aluminium blocks (García Pérez etal., 2020). The final operation of both eRMC and RMM successfully achieved the 16.5T in the central pole and in a 50mm bore (respectively), as intended (Perez etal., 2022; Gautheron etal., 2023). 1.2.2 Magnet mechanical steps and power up The brittleness of the Nb3Sn superconducting material makes its application a major engineering challenge. The self-induced Lorentz forces tend to open any coil horizontally. These forces are of the form JB (where these vectors represent the current density in A/m² and the magnetic field in T, respectively). Since the magnetic field scales up with the current I, this cross product scales up with I². This results in very high internal forces trying to Sensitivity analysis of a superconducting magnet design model 289 Figure 12.4 CAD view of cross section of the eRMC coil pack, featuring the coils and the compressing pushers. (colour image is accessible via the link) 290 Manuel García Pérez et al. Figure 12.5 CAD view of cross section of the RMM coil pack featuring the RMM-type coil between two eRMC-type coils, with the bore in the centre, where the 16.5 T field is targeted. (colour image is accessible via the link) Sensitivity analysis of a superconducting magnet design model 291 Figure 12.6 oke and the shell held in placey the younded bwing the coil pack surr tion. (colour image is accessible via the link) Left: Picture of the end of the eRMC magnet sho by some shims. Right: CAD transversal cross-sec Source: Courtesy of CERN. 292 Manuel García Pérez et al. . (colour image is accessible via wing 3/8 of the magnetf view sho. Right: Cutofull assembly tesy of CERN. t: fef ); cour . L RMM magnet the link). on etal. (2022 12.7 Gauther re ce: uigF Sour Sensitivity analysis of a superconducting magnet design model 293 open the coil, since the superconducting current densities of the cables is of the order of 400–600 A/mm². The support structure must ensure that the coils are compressed enough in order to limit the displacements caused by these forces, but not to such an extent that the low material yield limits are reached. The compression is typically achieved in the following way. Firstly, dedicated bladders filled with pressurized water act upon the gaps in between the yoke halves and the pushers of the coil pack. This widens the gap sufficiently to allow shims of a specific width to be inserted manually. These shims are wider than the nominal slot gaps (Caspi etal., 2001). Once the bladder water pressure is relieved, these bladders can be extracted as the coil pack remains locked and compressed within the structure by the shims. Secondly, when the entire magnet is cooled down to cryogenic working temperatures, the external aluminium shell shrinks more than the rest of the materials, further raising the total compression. The magnet can then be powered to carry the current through the coils and generate the field, as the coils have been sufficiently preloaded (compressed) beforehand. 2 Computational model The magnet mechanical and magnetic behaviour are simulated by structural and magnetic finite-element models. These models provide useful parameters, variables, and conditions, such as the necessary horizontal interference (shimming) for a given target field, the material stress situation, or the necessary bladder pressure to insert the keys (shims). Throughout this text, the words keys/shims and interference/shimming are used interchangeably. Finite-element models work by creating a (typically simplified) geometry of some entity that is under study (Figure12.8). This geometry is then sub-divided into much smaller parts, referred to as elements (Figure12.9). Material properties and loads (such as weight, forces, frictions, contacts, currents, movement constraints) are defined on these elements. Thus, it is possible to approximate the solution to partial-differential equations on the whole domain assuming simplified variations (usually linear or quadratic) across each element. If the resolution is fine enough, the model can accurately capture the necessary details, variations, and gradients of the magnitudes in the more complex geometry. Figures12.8 to 12.10 show the buildup of the FEM model geometry, the discretization (mesh), and an example of output solution, respectively. In Figure12.8, the symmetries enable the calculation of only 1/4 of a transversal slice of the magnet. This is also why for the RMM, only one of the layers of the middle coil is modelled. The different parts have been coloured by material. An ANSYS Mechanical 2D finite-element model representing one cross-section of the RMM magnet is used for this study. Only 1/4 of the 300 Manuel García Pérez et al. Figure 12.11 BH cur ves of the ferritic materials utilized in the model (iron yoke and vertical pusher, and eRMC pole). (colour image is accessible via the link) 3 SimDec analysis SimDec combines the computation of global sensitivity indices (Kozlova etal., 2023) with a visualization technique based on the multivariable decomposition of a dataset (Kozlova, Moss, etal., 2024). Adetailed description of the algorithm, nuances of its usage, and instructions for interpreting its results can be found in Chapter2 (Kozlova, Roy, etal., 2024). In this section, the names of input and output variables appear in bold italic, and the states of input variables are shown in italic. 3.1 Sensitivity parameters Certain geometric aspects of the magnet have been parametrized and will be varied in a sensitivity study. The seven parameters are shown in Table12.2. The model variations were simulated on the CSC1 Puhti supercomputer, where each evaluation cycle required 13 minutes on average. Several potential evaluation levels were selected for each input variable (see Table12.1) leading to 15,360 possible data combinations analysis. Afull factorial experimental design, or grid search, was conducted over all the combinations of the data points. However, various input combinations result in degenerative geometries due to the yoke radius being too small or the size of the coil pack being too large. After discarding these cases portraying unrealistic geometries (Table12.3), 8,261 data points remained for further analysis. Sensitivity analysis of a superconducting magnet design model 301 Table 12. 2 Input parameters, their evaluation points, and physical meaning Input variables Evaluation points Default (design) value Aspect comparison of min and max values Cable height, cable_height {20.25, 21.75, 23.25, 24.75} 21.75mm Total number of cable windings, n_windings {240, 252, 264, 276, 288,} 264 Thickness of the shell, shell_thickness {40, 80, 120} 70mm (Continued) 302 Manuel García Pérez et al. Input variables Evaluation points Default (design) value Aspect comparison of min and max values Radius of the yoke, yoke_r {240, 280, 320, 360} 330mm Height of the vertical pusher, vpad_height {534, 539, 544, 549} 539mm Size of the horizontal shim, hkey_size {21, 23, 25, 27} 24mm Vertical location of the horizontal shim, hkey_position {18, 20, 22, 24} 22mm Table 12.2 (Continued) Sensitivity analysis of a superconducting magnet design model 303 Input variables Evaluation points Default (design) value Aspect comparison of min and max values Radius of the yoke, yoke_r {240, 280, 320, 360} 330mm Height of the vertical pusher, vpad_height {534, 539, 544, 549} 539mm Size of the horizontal shim, hkey_size {21, 23, 25, 27} 24mm Vertical location of the horizontal shim, hkey_position {18, 20, 22, 24} 22mm 3.2 Output parameters The model records 11 output values for operation viability, with 7 of them selected for further SimDec analysis (Table12.3). Current reflects the operating physical electrical current flowing through each of the windings. Limitations to these currents exist related to the superconductivity capabilities of Nb3Sn (see Figure12.1). The conductor area corresponds to the transversal cross section of the coil. It is proportional to the total number of windings and the cable height. The horizontal interference is the total value of the shimming with which the magnet is loaded at room temperature. The value reflects how much wider the horizontal key is than its gap. The maximum von Mises stress in the coil is one of the main outputs at each of the three steps of the mechanical submodel. This stress should stay below the coil degradation limits (see Table12.1) of 150 and 200 MPa. The bladder pressure required to insert the target shimming (horizontal interference) is also monitored. An excessive pressure can lead to bladder failures that render the room-temperature preload (shim insertion) unviable. These output variable considerations are summed up in Table12.3. Typically, the design parameters and proportions of these magnets entail complex multivariable optimizations involving magnetic field intensity, uniformity, feasibility, etc. For SimDec demonstration purposes, the design targets have been simplified to minimize the coil conductor area (one of main economic cost components of the magnets) while maintaining the mechanical stress levels below material failure at 16.5T operation and with a feasible horizontal shimming and bladder pressure. Table 12. 3 Output variables expected/viable ranges and consideration within the sensitivity study Output variable Adequate range Share of cases with fulfilled criterion Current, IUnder 14,000 [A] 100% Conductor transversal area (mm ) – (of 1/4 magnet due to symmetries) Horizontal interference, hintf Between 200 and 97% 2000 [ m] Coil von Mises stress room-tempera-Under 150 [MPa] 98% ture preload (1), coil_strees_1 Coil von Mises stress at cooldown (2), Under 200 [MPa] 100% coil_strees_2 Coil von Mises stress at powering (3), Under 200 [MPa] 92% coil_strees_3 Pressure of bladder, bladder_P Under 50 [MPa] 91% 304 Manuel García Pérez et al. 3.3 Sensitivity Indices Sensitivity indices are computed using the simple binning approach (Kozlova, Moss, etal., 2024) with open-source code available in Python, R, Julia, and Matlab. The combined sensitivity indices computed appear in Table12.4. These sensitivity indices capture the inherent complexity of the model in which each output is affected by different sets of the input variables. Table12.4 demonstrates that most of the outputs possess very different sensitivity profile.2 3.4 SimDec analysis of each model output This section presents a detailed analysis of the sensitivity indices for each model output supplemented with its corresponding SimDec visualizations. 3.4.1 Current The current necessary to achieve the design choice of 16.5 T at the bore naturally depends upon the coil surface area, which varies proportionally to the number of windings and the cable height. Sensitivity indices for current are presented in Table12.5. Inputs cable height and number of windings together explain 96% of the output variability, while the second-order effects are negligible. The negative values indicate correlation, which could arise from the absence of model output for certain combinations of these two inputs. The two most important input variables, number of windings and cable height, in the order of their importance, are chosen for the visual decomposition (Figure12.12). Table 12. 4 Sensitivity indices for selected model outputs Outputs Inputs current conductor area hintf coil_ stress_1 coil_ stress_2 coil_ stress_3 bladder_P cable_height 42% 57% 2% 15% 43% 55% 3% n_windings 54% 41% 3% 20% 5% 1% 1% shell_thickness 1% 1% 82% 19% 39% 12% 11% yoke_r 5% 1% 12% 43% 6% 18% 76% vpad_height 0% 0% 0% 1% 4% 2% 1% hkey_size 0% 0% 0% 0% 0% 1% 0% hkey_position 0% 0% 1% 0% 5% 14% 0% Total 103% 101% 100% 98% 102% 103% 92% Note: Values below 0.01 have been greyed out. The estimates are generally within 2–3% confidence interval. Sensitivity analysis of a superconducting magnet design model 305 Table 12. 5 Sensitivity indices for current Input variable First-order effect Second-order effect Combined sensitivity index cable_ height n_windings shell_ thickness yoke_r vpad _height hkey_ size hkey_ position cable_height 43% −3% 0% 2% 0% 0% 0% 42% n_windings 56% 0% −1% 0% 0% 0% 54% shell_thickness 1% 0% 0% 0% 0% 1% yoke_r 5% 0% 0% 0% 5% vpad_height 0% 0% 0% 0% hkey_size 0% 0% 0% hkey_position 0% 0% Total 105% 103% 306 Manuel García Pérez et al. Figure 12.12 Simulation Decomposition of the RMM superconducting magnet model for output variable current. (colour image is accessible via the link) Colour Number of windings Cable height [mm] Current [A] Min Mean Max Probability 240–252 20.25, 21.75 11,858 12,443 13,081 21% 23.25, 24.75 12,712 13,328 14,019 20% 264–288 20.25, 21.75 10,899 11,563 12,287 32% 23.25, 24.75 11,650 12,343 13,142 27% The graph appears dissected and barely resembles the more familiar-looking continuous histogram normally associated with probability distributions. This can be explained by the major influence of only two inputs on this output and by the discrete grid sampling that evaluates the model over only a limited set of points. Nevertheless, the decomposition still provides visual insight and is supported further by the descriptive statistics presented in the legend accompanying the SimDec graph. The pattern displayed in Figure12 Sensitivity analysis of a superconducting magnet design model 307 appears monotonic, with the cable current decreasing in number of windings and increasing in cable height. 3.4.2 Conductor area As the Conductor area is simply the product of cable height and number of windings, these two inputs should, in theory, fully explain the variability of this output. However, the method for computing sensitivity indices from simulated data is only an approximate one and can result in slight numeric noise. Some noise deviations can be observed in Table12.6, where the cable height and number of windings account for only 98% of the output variance (with the difference from 100% corresponding to numerical noise). The second-order effects for the conductor area are negligible, although a small interaction between the two inputs of 4% can be observed, which results from their multiplication in the model. Shell thickness exhibits a small correlation with cable height and number of windings. However, since these inputs are independent, this observed correlation is a result of the sample cleaning. The cable height and number of windings fully explain conductor area. Therefore, the corresponding SimDec histogram would consist of even fewer distinct values than for current in Figure12.12. Consequently, the need for a visualization is relinquished, and the tabular representation of the discrete output is presented in Table12.7, instead. The default RMM magnet features number of windings = 264, with 21.75mm cable height, resulting in 6,088mm² conductor area (Izquierdo Bermúdez, 2017). 3.4.3 Horizontal interference The horizontal interference is mostly affected by shell thickness, with a combined sensitivity index of 82%. The yoke radius shows a 12% influence, while all other input variables have a negligible effect on the output (see Table12.8). No pronounced second-order effects are detected, which is expected since an increased shell thickness ensures a much stronger cooldown compression. Consequently, less horizontal shimming is necessary to maintain the pole-coil contact pressure of 10 MPa. The shell thickness and yoke radius together explain 94% of the variation in the horizontal interference and are used to decompose its distribution in Figure12.13. Figure 12.13 demonstrates another monotonic relationship pattern in which the horizontal interference decreases in both shell thickness and yoke radius. This decrease is not linear. The difference of output averages for 80mm and 120mm (yellow and green) shell thicknesses is 188mm, whereas for 40mm and 80mm (blue and yellow), it is 589mm. Apart from the means, the output variance decreases with higher shell thickness. This 308 Manuel García Pérez et al. Table 12. 6 Sensitivity indices for conductor area Input variable First-order effect Second-order effect Combined sensitivity index cable_ height n_windings shell_ thickness yoke_r vpad _height hkey_ size hkey_ position cable_height 56% 4% −2% 0% 0% 0% 0% 57% n_windings 39% −1% 0% 0% 0% 0% 41% shell_thickness 3% 0% 0% 0% 0% 1% yoke_r 1% 0% 0% 0% 1% vpad_height 0% 0% 0% 0% hkey_size 0% 0% 0% hkey_position 0% 0% Total 99% 101% Sensitivity analysis of a superconducting magnet design model 309 Table 12. 7 Resulting conductor area from different combinations of cable height and number of windings n_windings 240 252 264 276 288 cable_height, [mm] 20.25 5143 5401 5658 5916 6174 21.75 5534 5811 6088 6365 6643 23.25 5924 6221 6518 6815 7112 24.25 6315 6631 6948 7264 7581 causes a larger overlap of the yellow and green scenarios with shell thickness 80mm and 120mm, respectively, and less with the blue one of shell thickness 40mm. These two patterns cause the blue scenario to stay further apart from the rest and forms a sort of cavity in the distribution. 3.4.4 von Mises coil stress at stage 1 (room-temperature preload) The von Mises coil stress at stage 1 is affected by several input variables, including yoke radius (43%), number of windings (20%), shell thickness (19%), and cable height (15%) (see Table12.9). The decomposition shown in Figure12.14 is done for the three most influential input variables that together explain 82% of variability of the output. Examining the SimDec graph, one can observe that coil stress at stage 1 decreases with yoke radius and shell thickness, but increases with the number of windings. Interestingly, the combination of shell thickness of 40mm and yoke radius of 280mm results in noticeably dissected sub-distributions of the output. This observation can be explained by the cable height input variable being excluded from the decomposition. If the designer wished to navigate into a disparate part of the sub-distributions, the decomposition analysis could be easily repeated for that respective part of the dataset. 3.4.5 von Mises coil stress at stage 2 (cooldown) The von Mises coil stress at stage 2 exhibits a different sensitivity profile compared to that of stage 1. At stage 2, cable height and shell thickness play the important role, whereas the influence of other input variables is negligible (see Table12.10). The decomposition by these two most important input variables explains 82% of the output variability and shows non-monotonic patterns of relationships (Figure 12.15). The lowest shell thickness (40 mm) results in 316 Manuel García Pérez et al. Figure 12.15 Simulation Decomposition of the RMM superconducting magnet model for output variable coil stress at stage 2. (colour image is accessible via the link) Colour Cable height [mm] Shell thickness [mm] Coil stress 2 [MPa] Min Mean Max Probability 20.25 40 165.0 169.6 173.5 9% 80 170.1 175.1 179.3 9% 120 172.5 176.5 180.9 9% 21.75 40 158.8 164.2 169.4 9% 80 165.5 169.7 174.6 10% 120 166.9 171.2 175.8 8% 23.25 40 157.3 162.4 167.3 6% 80 163.2 167.9 177.8 10% 120 163.7 169.0 177.5 9% 24.75 40 160.6 164.3 167.2 3% 80 165.0 171.8 184.0 9% 120 166.7 173.2 184.6 9% Sensitivity analysis of a superconducting magnet design model 317 Table 12.1 1 Sensitivity indices for coil stress at stage 3 Input variable First-order effect Second-order effect Combined sensitivity index cable_ height n_ windings shell_ thickness yoke_r vpad _height hkey_ size hkey_ position cable_height 39% 1% 6% 1% 2% 0% 22% 55% n_windings 0% 0% 0% 0% 0% 0% 1% shell_thickness 9% 0% 0% 0% 1% 12% yoke_r 17% 0% 0% 1% 18% vpad_height 1% 0% 0% 2% hkey_size 0% 0% 1% hkey_position 2% 14% Total 68% 103% 318 Manuel García Pérez et al. Table 12.1 2 Effects of the hkey position and cable height on maximum von Mises stresses in the coil Situation Aspect Coil stress at cooldown (2) Coil stress at powering (3) Legend [MPa] a) default b) taller cable by 2mm per layer; position of the horizontal shim lowered c) taller cable, position of the horizontal shim maintained at the center of the coil Note: The maps of the stress highlight the position and value (in MPa) of the maximum. Note how when the shim is not well centred (situation b), the stress focuses and peaks on the corner of the bottom layer of windings. (colour image is accessible via the link) Sensitivity analysis of a superconducting magnet design model 319 Figure 12.16 Simulation Decomposition of the RMM superconducting magnet model for output variable coil stress at stage 3. (colour image is accessible via the link) Colour Cable height [mm] Yoke radius [mm] Coil stress at stage 3 [MPa] Min Mean Max Probability 20.25 280 142.6 150.1 158.0 9% 320 144.8 152.2 160.3 8% 360 147.3 154.3 162.3 10% 21.75 280 137.7 143.7 149.8 8% 320 139.8 146.2 151.8 9% 360 141.5 148.4 155.1 9% 23.25 280 136.3 141.7 146.9 8% 320 138.2 144.5 151.0 9% 360 139.9 147.0 155.8 9% 24.75 280 137.8 143.3 150.5 5% 320 137.8 145.8 156.6 8% 360 139.7 149.2 159.9 8% 320 Manuel García Pérez et al. Table 12.1 3 Sensitivity indices for bladder pressure Input variable First-order effect Second-order effect Combined sensitivity index cable_ height n_ windings shell_ thickness yoke_r vpad _height hkey_ size hkey_ position cable_height 1% 1% 1% 1% 0% 0% 0% 3% n_windings 0% 0% 1% 0% 0% 0% 1% shell_thickness 9% 2% 0% 0% 0% 11% yoke_r 74% 0% 0% 0% 76% vpad_height 1% 0% 0% 1% hkey_size 0% 0% 0% hkey_position 0% 0% Total 85% 92% Note: Values below 0.01 are greyed out. Sensitivity analysis of a superconducting magnet design model 321 Figure 12.17 Simulation Decomposition of the RMM superconducting magnet model for output variable bladder pressure. Colour Yoke_r [mm] Shell_ thickness [mm] Bladder pressure [MPa] Min Mean Max Probability 280 40 28.2 54.5 62.4 8% 80 42.2 49.5 70.0 11% 120 48.9 55.8 69.5 10% 320 40 20.5 37.3 45.0 9% 80 30.5 35.1 49.4 13% 120 36.8 42.8 53.6 12% 360 40 15.9 25.3 33.1 9% 80 23.2 27.0 36.3 14% 120 25.1 35.2 37.1 13% 322 Manuel García Pérez et al. ness kshell thicof oups coded gr ) -colourbladder pressure with (colour image is accessible via the link to erence fontal interzhori pairs consistent to Figure12.17. of plot e radius ter yok Scat and 12.18 reuigF Sensitivity analysis of a superconducting magnet design model 323 due to the fact that a smaller yoke offers less bladder area, thereby requiring a higher pressure in order to exert a given force. The shell thickness, however, portrays a diamond-shape relationship with bladder pressure, with high and low shell thickness values resulting in intermediate values of bladder pressure and the medium value of shell thickness leading to high and low bladder pressure. This pattern occurs for each value of yoke radius and would not be visible without further decomposition. Figure12.18 highlights an investigation of the relationship between bladder pressure and horizontal interference while preserving the decomposition and colouring logic from Figure12.17. Some trends in Figure12.18 reveal a pattern that shows the required pressure growing linearly with the interference (the diagonally-stretched clusters of points). As intuitively expected, the wider the targeted gap, the higher the required pressure to open that gap. This linear dependency possesses a slope that grows in conjunction with the shell thickness due to inherent rigidity. Conversely, other patterns exhibit no pressure dependence (horizontally stretched clusters), more often than not within the same colour (representing a fixed value of yoke and shell sizes). Due to the nested nature of the effects (patterns occurring inside the groups of different combinations of input variables), only one group is analyzed separately. This analysis is performed by fixing the yoke radius at 360mm, the shell thickness at 40mm, and the number of windings at 264. The resulting set of 152 data points was examined using SimDec. The sensitivity indices indicate that cable height now explains 86% of bladder pressure variation (compared with a negligible 3% for the full dataset, Table12.13), hkey position explains 8.5%, hkey size explains 6%, vpad height explains 3%, and all other variables have no influence (i.e. 0%). The decomposition of bladder pressure is performed by another output of interest, horizontal interference, and the most influencing input, cable height (see Figure12.19). Figure12.19 shows that the horizontal interference and cable height for the selected portion of data are highly correlated (observe the many missing scenarios in the legend). Low interference with low cable height results in low values of bladder pressure, and vice versa. Moreover, lower values of the interference result in more compact ranges of bladder pressure, indicating that those points in which increasing interference does not seem to increase the bladder pressure tend to occur more frequently with the lowest values of cable height – effectively making the coil pack smaller and less rigid. Unfortunately, no physical explanation to explain this trend has yet been established. 324 Manuel García Pérez et al. Figure 12.19 Decomposition of bladder pressure on a limited dataset (yoke radius=360mm, shell thickness=40mm, and number of windings=264) by cable height that explains 86% of variation of the bladder pressure, and by another output of interest – horizontal interference. Absent due to correlation and rarely occurring scenarios are marked with grey font colour. (colour image is accessible via the link) Colour Hintf [μm] Cable_ height [mm] Bladder pressure [MPa] Min Mean Max Probability [1.35, 1.45] 20.25 35.3 36.3 37.5 29% 21.75 37.0 37.6 38.2 3% 23.25 24.75 (1.45. 1.55] 20.25 36.5 36.5 36.5 1% 21.75 36.6 37.4 38.3 26% 23.25 38.7 39.0 39.3 4% 24.75 (1.55, 1.70] 20.25 21.75 23.25 38.0 39.2 40.3 23% 24.75 40.2 41.0 41.8 15% Sensitivity analysis of a superconducting magnet design model 325 4 Conclusions This chapter has presented an application of SimDec to a computationally complex magnetic and mechanical, finite-element model of a superconducting magnet. This model possesses numerous outputs of interest and SimDec has been used to show that the sensitivity profiles of these variables differ quite considerably. SimDec provided a convenient way to exhaustively analyze the mechanical behaviour of the magnets and their support structure. Among other results, the findings confirmed that the aluminium shell thickness is a major contributor to the cooldown compression with respect to necessary shimming and coil stresses, and that the relative height of the vertical pusher actually has little effect on most parameters, in line with the existing knowledge in this field. One unexpected and yet unexplained pattern between bladder pressure, interference, and cable height is revealed by SimDec. Consequently, SimDec could be considered a powerful ancillary tool for determining interdependencies and synergies of the parameters in superconducting magnet. From a broader perspective, SimDec has demonstrated a novel way for conducting a sensitivity analysis on a complex system with multiple outputs of interest and has led to the exposure of previously concealed heterogeneous effects. Acknowledgements The work is supported by grant 220178 from the Finnish Foundation for Economic Education and by grant OGP0155871 from the Natural Sciences and Engineering Research Council of Canada. The authors are grateful to staff of the Magnet Design and Technology of the Department of Technology of CERN for the valuable feedback and the finite-element mAPDL models of the RMM magnet. Notes 1 CSC (IT Center for Science) is a Finnish centre of expertise in information technology owned by the Finnish state and higher education institutions (https://csc.fi/web/ guest). 2 https://github.com/Simulation-Decomposition. References Bottura, L. (1999, September26–October2). A practical fit for the critical surface of NbTi. Proceedings of 16th International Conference on Magnetic Technology, Ponte Vedra Beach, USA. Caspi, S., Gourlay, S., Hafalia, R., Lietzke, A., ONeill, J., Taylor, C., & Jackson, A. (2001, March). The use of pressurized bladders for stress control of superconducting magnets. IEEE Transactions of Applied Superconductivity, 11(1), 2272–2275. Ferracin, P. (2014). Superconductivity and superconducting magnets for the LHC upgrade. Course Lectures. https://indico.cern.ch/event/318566/attachments/612965/843302/140824_summer-students_I_final.pdf 332 Anna Sidorenko et al. (Klein, 2015; Walczak, etal., 2012; Klein, 2008). Consequently, this underlying level of duress leads to a high chance of negatively impacting the overall quality of the decision-making process (Keinan, 1987, p.639). As a result, a “safer” option is commonly opted for instead of the optimal or most strategically beneficial one (Wagner & Morisi, 2019; Badre etal., 2012). Humans are both limited by their cognitive structures (i.e. mental processing abilities) and, at the same time, overwhelmed by multifaceted and often discordant factors. All of the conditions mentioned can induce stress (Phillips-Wren & Adya, 2020; Marsden etal., 2006; Lerch & Harter, 2001; Hwang & Lin, 1999). In turn, stress contributes to critical information being overlooked and discarded when critical decisions must be taken (Van Bruggen etal., 1998). Technological support and data visualization tools can help in overcoming stress-related cognitive flaws and boost the rationality of the decision-making process (Walczak etal., 2012). People encounter complex dilemmas at both organizational and individual levels. Managers decide on the strategic development of a company and, outside of their work roles, individuals also must make choices. While some decisions are trivial, other decisions will shape major future life trajectories. Under either scenario, the criteria that we need to consider make it challenging to grasp the full picture for reaching a comprehensive solution. Accessible and user-friendly modelling and visualization have the potential to assist the decision-making process and to reduce/mitigate the negative impact from inhibiting factors. In this chapter, we examine six situations based on real events where the speed of the decision-making is not pressing. Their gravity and complexity, nonetheless, hinder the ability to derive straightforward solutions without employing analytical tools to support the process. The context of individual decision-making implies a divergent quality of risk factors in comparison to organizational ones. This is not the same thing as facing the negative repercussions arising from poor decision-making. To counteract potential risks, SimDec is used to aid in the problem analysis and solution discovery that the six protagonists embarked upon (Kozlova, Moss, Caers, etal., 2024). 2 Cases We consider six decision-making situations that were recently encountered either directly by the authors or indirectly by their friends/family (see Table13.1). The Savings case demonstrates how the annuity function translates a small monthly deposit into a future significant gain. Similarly, the Language learning case is based on a power curve that transforms hours of studying into a future mastery level. These two cases demonstrate that even a very basic, simple function can be studied with SimDec. The SimDec analysis provides valuable insights for decision-making that can prove indispensable when even more sources of variation of uncertainty are included. New level of personal decision-making 333 The Mortgage and Fat percentage cases represent situations where two different model outputs are analyzed: loan term in years and interest expenses in euros in the former case, and body weight in kilos and fat percentages in the latter. The two outputs in the Mortgage case display the exact same dependency structure and the distribution shape, so one can use the same policy to affect both outputs equally. Conversely, the fat percentage case demonstrates how a model can produce two different output distributions with different dependency profiles from the same inputs. As a result, a compromise strategy should be considered for pursuing the two conflicting objectives, or the objectives themselves may need to be reconsidered. The Country and Car choice cases show how multi-criteria decision-making problems, either based on objective data or subjective rating scales, can be advanced from simple ordering tasks to a more in-depth investigation of the uncertainty behind each alternative and an understanding of the driving forces behind it. All of the models are publicly available, together with the SimDec implementation in various languages (Kozlova, Moss, Roy, etal., 2024).1 2.1 Savings Timo leads a frugal lifestyle and wants to convince his roommate of its benefits. Every penny matters and even small, but regular savings, can make a difference. The power of regular savings lies in the “miracle” of compound interest. Compound interest takes into account not only the sum of all previous savings, but also the interest already accrued on them (i.e. interest is earned on the interest). The more money that is saved, the faster the growth of the total balance. Furthermore, the longer the money has been saved, the Table 13.1 Modelling construc t for the six cases considered Case Model Description Savings Annuity Deciding on monthly saving amount by examining the future value of savings Country MCDM with a Choosing a country of residence based on perchoice rating scale sonal perceptions of different criteria Mortgage Cash flow model Computing the loan duration and interest expenses under uncertain interest rate Language Power function Exploring how much time should be committed learning to language learning based on experimentally derived learning curve function Car choice MCDM with Deciding on whether to buy a car, and which, measured data based on their cost and technical characteristics Fat per-Addition & Understanding the effect of lean and fat tissue centage multiplication reduction on body fat percentage Note: MCDM stands for multi-criteria decision-making. 334 Anna Sidorenko et al. more prominent the results of the accelerating growth The future value of savings that compound can be determined by the following annuity formula: () n 11+r− FV =P (1) r where FV is the future value of the balance, P is a regular savings payment, r is the interest rate, and n is a number of time periods The interest rate should match the units for the number of periods (i e annual for years, monthly for months) The question of interest is how much difference would monthly instalments of €100, €200, or €300 make over a five-year savings challenge in which the uncertain interest rate varies uniformly between 0% and 10% To address this question, the simple annuity formula (1) with the discrete monthly savings amount and uniformly distributed interest rate was simulated 3,000 times to produce the distribution of future (in five years) value of savings2 (see Figure13 1) Figure13 1 shows that monthly instalments of €300 result in nearly four times more capital after five years in comparison to instalments of €100 (see darker portion of the distribution on the right compared to the light-coloured portion on the left) The higher savings are naturally more sensitive to the Figure 13.1 Distribution of f uture savings for different monthly saving amount (marked with different colours) and uncertain interest rate. (colour image is accessible via the link) New level of personal decision-making 335 variations in the interest rate, which results in the wider distribution along the X-axis. Higher rates in the €300 scenario generate much more lucrative outcomes in comparison to the lower monthly savings scenarios. The heights of the three distinct distribution parts do not hold much meaning and are only different because the same number of observations (same area) are distributed on intervals of different width. By further examining the annuity formula (1), one could notice that the future value of savings is linearly dependent on the regular payments, linearly dependent on the interest rate (but the two are multiplied reinforcing the effect of each other, which is shown in Figure13.1), but exponentially affected by the number of periods (Figure13.2). Thus, the duration of savings is the key to wealthy future, if only one can remain persistent over the long term. By letting the savings period vary between 1 and 50years, the distribution of future savings changes dramatically. Savings are now not only impacted by the exponential effect of the number but also reinforced by the interaction between the monthly saving amount and the interest rate (Figure13.3). Saving for 17years or less barely produces a hundred thousand euros even under the most favourable conditions (high interest rate and high €300 monthly payment) – see red scenarios. Only those who can sustain 50years of monthly savings can dare to dream of becoming a millionaire – see the long tail of the green scenarios. Figure 13.2 Future value of €200 monthly savings at 5% interest rate as an exponent of the number of years. (colour image is accessible via the link) 336 Anna Sidorenko et al. Figure 13.3 Future savings value under uncertain interest rate (0–10%), variable duration (1–50years), and monthly payment (€100, €200, or €300). The X-axis is truncated; the maximum amount of savings approaches seven million euros when all factors are favourable. (colour image is accessible via the link) New level of personal decision-making 337 2.2 Country choice On 24 February 2022, the lives of millions were shattered when Russia invaded Ukraine and started a full-scale war. Air strikes, explosions, martial law, total panic, and the brutality of warfare – that is how a family from Kyiv ended up in Finland. Three sisters, all under 25, with the youngest sibling being 15, were taken in by a close relative. Ayear has since passed - Finnish language courses have been taken, and residency documents have been filed. While the middle sibling moved back to Kyiv, Daria, the oldest, found herself contemplating her future. Does she want to stay in Finland, return to Ukraine, move somewhere else, and if so, to where? Daria decided that if she were to move somewhere else, it had to be to an English-speaking country – the USA or the UK – where the language barrier would not be a substantive concern. With these four options in mind, Daria was confronted with another dilemma: How could she make the choice, and what factors should she account for? Since she had a younger sibling to care for, Daria decided that the most important criterion for her was to guarantee her sister’s well-being. Apart from that, Daria developed six criteria that reflected her life values: job availability, language barrier, well-being, climate, healthcare, and quality of food. The criteria were weighted by their perceived importance. Each criterion for each country was scored on a scale from 0 to 100, in a range that reflected the subjective assessment, from the most negative to the most positive outcomes (Table13.2.) Wider score ranges reflect the level of unpredictability and volatility of a criterion perception. Given the climate diversity in the USA, the level of satisfaction varies significantly if an exact location for potential residence is not specified. However, narrow ranges can still denote unpredictability as in the first criteria score for Ukraine. In this instance, Daria offered a score of 50 out of 100, positioning their assessment right in the middle of the range. Table 13.2 Inputs to multicriteria decision-making problem: choosing a country based on seven criteria Criterion Importance Finland Ukraine UK USA Min Max Min Max Min Max Min Max Peace of mind 100 % 80 100 50 50 20 60 060 about sister Job availability 100 % 020 50 70 050 050 Language comfort 90 % 40 50 90 100 50 60 70 80 Wellbeing 100 % 80 100 050 10 60 0100 Climate 80 % 10 70 50 100 060 0100 Medicine 80 % 70 75 70 80 0100 40 50 Food 80 % 020 100 100 020 050 [Document text truncated for crawler view.]