Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Zhao, Yuchen
Master of Applied Science - MASc
University of British Columbia
Recycling is essential for reducing the environmental impact of plastic waste and is a key part of sustainable supply chains. However, optimizing these networks is challenging due to uncertainties in input parameters like demand and return rates, requiring solutions to enhance efficiency and resilience in Closed-Loop Supply Chains (CLSC). This research focuses on designing robust CLSC networks to improve resource efficiency, reduce costs, and maximize recycling rates, addressing the need for more resilient and sustainable systems. This research develops a novel approach using Data-Driven Robust Optimization (DDRO), which incorporates historical data into the optimization process to effectively deal with uncertainties such as fluctuating demand and variable return rates of recycled materials. A key innovation in this research is the introduction of a novel concept in DDRO approaches, focusing on the construction of balanced uncertainty sets. A ”balanced uncertainty set” is defined as one where the boundaries maintain nearly equal distances in all directions to the uncovered data samples, ensuring a more equitable representation of uncertainty. To achieve balanced uncertainty sets across varying conservatism levels, a single-Kernel learning-based approach is developed, wherein the weights of the basis kernel matrices are adjusted based on the data’ dispersion in each direction. Additionally, this thesis introduces a Robust Fairness (RF) index to evaluate and compare the balance of different uncertainty sets. A data-driven algorithm is further developed to efficiently compute the RF index. Numerical results demonstrate that the proposed DDRO approach in this thesis generates more balanced uncertainty sets compared to competing methods. In addition, the proposed data-driven algorithm can effectively compute the RF index and compare uncertainty sets in terms of balance, without introducing computational complexities.
ROBUST OPTIMIZATION FOR PLASTIC CLOSED-LOOP SUPPLY CHAINS: ADVANCING DATA-DRIVEN MODELS WITH KERNEL WEIGHT ADJUSTMENT AND FAIRNESS METRICS by Yuchen Zhao B.A.Sc., University of British Columbia, 2022 A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE COLLEGE OF GRADUATE STUDIES (Mechanical Engineering) Master of Applied Science (M.A.Sc.) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) November 2024 © Yuchen Zhao, 2024 ii The following individuals certify that they have read, and recommend to the College of Graduate Studies for acceptance, a thesis/dissertation entitled: Robust Optimization for Plastic Closed-Loop Supply Chains: Advancing Data-Driven Models With Kernel Weight Adjustment And Fairness Metrics submitted by Yuchen Zhao in partial fulfillment of the requirements of the degree of Master of Applied Science. Dr. Tosarkani, School of Engineering Supervisor Dr. Chen, School of Engineering Supervisory Committee Member Dr. Ardestani-Jaafari, School of Engineering Supervisory Committee Member Dr. Zarifi, School of Engineering University Examiner iii Abstract Recycling is essential for reducing the environmental impact of plastic waste and is a key part of sustainable supply chains. However, optimizing these networks is challenging due to uncertainties in input parameters like demand and return rates, requiring solutions to enhance efficiency and resilience in Closed-Loop Supply Chains (CLSC). This research focuses on designing robust CLSC networks to improve resource efficiency, reduce costs, and maximize recycling rates, addressing the need for more resilient and sustainable systems. This research develops a novel approach using Data-Driven Robust Optimization (DDRO), which incorporates historical data into the optimization process to effectively deal with uncertainties such as fluctuating demand and variable return rates of recycled materials. A key innovation in this research is the introduction of a novel concept in DDRO approaches, focusing on the construction of balanced uncertainty sets. A ”balanced uncertainty set” is defined as one where the boundaries maintain nearly equal distances in all directions to the uncovered data samples, ensuring a more equitable representation of uncertainty. To achieve balanced uncertainty sets across varying conservatism levels, a single-Kernel learning-based approach is developed, wherein the weights of the basis kernel matrices are adjusted based on the data’ dispersion in each direction. Additionally, this thesis introduces a Robust Fairness (RF) index to evaluate and compare the balance of different uncertainty sets. A data-driven algorithm is further developed to efficiently compute the RF index. Numerical results demonstrate that the proposed DDRO approach in this thesis generates more balanced uncertainty sets compared to competing methods. In addition, the proposed data-driven algorithm can effectively compute the RF index and compare uncertainty sets in terms of balance, without introducing computational complexities. iv Lay Summary This research focuses on designing a Closed-Loop Supply Chain (CLSC) network by selecting optimal locations for manufacturers, suppliers, recycling centers, and collection centers. The aim is to enhance recycling within the plastic industry and reduce environmental impact. However, there are some challenges such as uncertainty of some input parameters, such as fluctuations in demand values and return rates that highly affect the number and capacity of required facilities. To address this challenge a data-driven optimization framework is developed which utilizes the available historical data related to uncertain parameters to effectively address uncertainty and design a CLSC network. The outcomes of this research contribute to the development of sustainable and efficient recycling networks, promoting a circular economy and addressing pressing environmental challenges. v Preface This thesis was conducted under the supervision of Dr. Babak Mohamadpour Tosarkani at the Industrial & Systems Engineering (ISE) Research Group. I was responsible for the design and execution of the research, the performance of the research tasks, and the data analysis. This work reflects my individual efforts in these areas, with guidance and support from my supervisor. Generative artificial intelligence (AI) was not used in any aspect of the core research described in this thesis. The analysis and modelling focus on existing data and mathematical formulations, conducted without reliance on AI-generated outputs. vi Table of Contents ROBUST OPTIMIZATION FOR PLASTIC CLOSED-LOOP SUPPLY CHAINS: ............. i ADVANCING DATA-DRIVEN MODELS WITH KERNEL WEIGHT ADJUSTMENT .... i AND FAIRNESS METRICS ........................................................................................................ i Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ......................................................................................................................... vi List of Tables .............................................................................................................................. viii List of Figures ............................................................................................................................... ix List of Abbreviations ................................................................................................................... xi Glossary ...................................................................................................................................... xiii Acknowledgements .................................................................................................................... xvi Dedication .................................................................................................................................. xvii Chapter 1: Introduction ........................................................................................................... 1 1.1 Motivation ................................................................................................................... 1 1.2 Research Objectives .................................................................................................... 4 1.3 Research Methodologies and Contributions ............................................................... 7 1.4 Thesis Outline ........................................................................................................... 10 Chapter 2: Literature Review ................................................................................................ 12 2.1 Deterministic and Multi-Objective Models in CLSC ............................................... 12 vii 2.2 Optimization under uncertainty in CLSC ................................................................. 15 2.3 Data-driven Robust Optimization ............................................................................. 18 Chapter 3: Designing Deterministic Model for Plastic CLSC ............................................ 29 3.1 Introduction ............................................................................................................... 29 3.2 Problem Definition.................................................................................................... 29 3.3 Mathematical model.................................................................................................. 31 3.4 Application of The Proposed Model and Solution Approach ................................... 36 3.5 Sensitivity Analysis .................................................................................................. 37 3.6 Conclusion and limitations ....................................................................................... 40 Chapter 4: Data-Driven Approach for A Plastic Closed-Loop Supply Chain .................. 41 4.1 An overview of data-driven robust optimization approach ...................................... 41 4.2 OC-SVM with Kernel Weight Adjustment............................................................... 46 4.3 Fairness Index ........................................................................................................... 47 4.4 Numerical Results of the DDRO Approach.............................................................. 51 Chapter 5: Conclusion ............................................................................................................ 79 5.1 Research Summary ................................................................................................... 79 5.2 Research Contributions ............................................................................................. 81 5.3 Research Limitations ................................................................................................ 83 5.4 Future Research ........................................................................................................ 85 Bibliography .................................................................................................................................88 viii List of Tables Table 2.1 Summary of Related Works on CLSC Network Design .............................................. 25 Table 3.1 Values of Parameters Applied to Solve the Proposed Model ....................................... 36 Table 3.2 Comparison of Nine Scenarios of Deterministic Model ............................................... 37 Table 4.1 Comparison of Nine Scenarios of Deterministic Model Regarding Disposal Fraction 54 Table 4.2 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand for DDRO ....................................................................................................................... 56 Table 4.3 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Return for DDRO .......................................................................................................................... 57 Table 4.4 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand and Return for DDRO .................................................................................................... 58 Table 4.5 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Return for New DDRO ................................................................................................................. 62 Table 4.6 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand for New DDRO .............................................................................................................. 63 Table 4.7 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand and Return for New DDRO ............................................................................................ 64 Table 4.8 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand and Return between Original DDRO and New DDRO .................................................. 65 Table 4.9 Comparing OC-SVM and OC-SVM with KWA Approaches in Constructing Balanced Uncertainty Sets Using RF Index ................................................................................................. 69 Table 4.10 Comparison of Seven Scenarios of the Deterministic Model with Node and Link Disruptions .................................................................................................................................... 78 ix List of Figures Figure 3.1 Demonstration of CLSC Network ............................................................................... 30 Figure 3.2 The Configuration of the Plastic CLSC Network for Different Scenarios .................. 39 Figure 4.1 Overall Procedure of the DDRO Approach to Address Parameter Uncertainty ......... 45 Figure 4.2 Procedure for Calculating the Distance of Each SV from the Uncertainty Set Boundary ....................................................................................................................................... 50 Figure 4.3 DDRO Deterministic Model Visualizations ................................................................ 59 Figure 4.4 Comparison of Supporting Vectors for Evenly Distributed Data Set between Original DDRO and New DDRO................................................................................................................ 60 Figure 4.5 Comparison of Supporting Vectors for Ellipse Distributed Data Set Between Original DDRO and New DDRO................................................................................................................ 61 Figure 4.6 Comparison of Operating Facilities Between OC-SVM and OC-SVM with KWA Approaches ................................................................................................................................... 67 Figure 4.7 Comparison of Expected Costs Between OC-SVM and OC-SVM with KWA Approaches ................................................................................................................................... 67 Figure 4.8 Comparison of the OC-SVM and OC-SVM with KWA Approaches in Constructing Uncertainty Sets with 80% Coverage ........................................................................................... 68 Figure 4.9 Optimal Configuration of the CLSN Using DDRO with 70% Coverage for Scenario 1....................................................................................................................................................... 70 Figure 4.10 Disrupted Link of the CLSN Using DDRO with 70% Coverage for Scenario 2 ...... 72 Figure 4.11 Disrupted Node of the CLSN Using DDRO with 70% Coverage for Scenario 3 and Scenario 6...................................................................................................................................... 73 x Figure 4.12 Disrupted Link and Node of the CLSN Using DDRO with 70% Coverage for Scenario 4...................................................................................................................................... 74 Figure 4.13 Disrupted Link of the CLSN Using DDRO with 70% Coverage for Scenario 5 ...... 76 Figure 4.14 Disrupted Link and Node of the CLSN Using DDRO with 70% Coverage for Scenario 7...................................................................................................................................... 77 xi List of Abbreviations ANP Analytic Network Process BSV Boundary Support Vector CLSC Closed-Loop Supply Chain CLSND Closed-Loop Supply Network Design CSVC Conservative Support Vector Clustering CVaR Conditional Value at Risk DDRO Data-Driven Robust Optimization GHG Greenhouse Gas HDPE High Density Polyethylene ID Inlier Data Points KWA Kernel Weight Adjustment MCDM Multi-Criteria Decision-Making MILP Mixed-Integer Linear Programming MIP Mixed-Integer Program MKL Multiple-Kernel Learning OC-SVM One-Class Support Vector Machine PCA Principal Component Analysis PET Polyethylene Terephthalate RF Robust Fairness xii RFIC Robust Fairness Index Calculator RO Robust Optimization SC Supply Chain SCM Supply Chain Management SKL Single-Kernel Learning SV Support Vector SVC Support Vector Clustering SVM Support Vector Machine WGIK Weighted Generalized Intersection Kernel xiii Glossary Supply Chain Management: It manages the flow of goods, services, and information across the entire supply chain, from raw materials to the final customer. Closed-Loop Supply Chain: An SC that integrates forward logistics with reverse logistics focuses on a product’s entire lifecycle, including production, distribution, consumption, and the return of products for recycling, remanufacturing, or disposal. Reverse Logistics: The process of moving goods from their destination back to the manufacturer for return, recycling, refurbishing, or disposal. One-Class Support Vector Machine: A variant of SVM is used for anomaly detection. It is trained on data belonging to a single class and is used to identify outliers. However, OC-SVM is used to construct data-driven uncertainty sets, in this research. Kernel Weight Adjustment: KWA is a modification of the traditional SVM approach that adjusts the weights of different kernels within the model, based on the dispersion degree of data samples in each principal direction. This adjustment aims to improve the model’s performance, particularly in scenarios where it is crucial to handle uncertainty. xiv Robust Fairness Index: RF is a metric proposed to evaluate the balance of an uncertainty set by measuring the cumulative distance of all support vectors from the boundary of that uncertainty set. A lower RF index indicates a more balanced and robust uncertainty set. Data-driven Robust Optimization Approach: An optimization approach that uses data to construct data-driven uncertainty sets using machine learning algorithms, and hedge against the worst case within the uncertainty set. These approaches are developed to improve the feasibility of robust solutions and reduce over-conservatism. Uncertainty Set: A set of possible values for uncertain parameters within an optimization model ensures that the solution remains feasible and effective for even the worst case within the uncertainty set. Conservatism degree: In robust optimization, the conservatism degree indicates how strongly a solution is protected against uncertainty. While a highly conservative solution can perform reliably under diverse scenarios, it may also significantly increase overall costs. Support Vector: In this thesis, Support Vectors (SVs) refer to data samples that lie on or outside the boundaries of the uncertainty sets. Boundary Support Vector: In the context of Support Vector Machines (SVM), Boundary Support Vectors (BSVs) are the data points that lie closest to the boundary of the uncertainty sets. xv Inlier Data Points: IDs refer to the data samples that fall within the boundary of the uncertainty sets. The greater the number of IDs, the more conservative the uncertainty set becomes. Regularization Parameter (𝒗): A parameter used in SVM models to control the trade-off between maximizing the margin and minimizing classification errors influences the model’s generalization ability. In this thesis, it is used to tune the conservatism degree of the DDRO approach, and it determines the fraction of data covered by the uncertainty set. MILP Model: MILP is a mathematical optimization technique used to solve problems involving both continuous and discrete variables. Acknowledgements xvi Acknowledgements I want to express my deepest gratitude to my supervisor, Dr. Tosarkani, whose guidance, encouragement, and invaluable insights have been instrumental throughout this research journey. Your support and constructive feedback have greatly enriched my work, and I am profoundly thankful for that. I also extend my heartfelt thanks to the members of my committee, Dr. Qian Chen, Dr. Amir Ardestani-Jaafari, and Dr. Mohammad Hossein Zarifi, for their time, expertise, and thoughtful critiques that have significantly shaped the direction and quality of this thesis. This research would not have been possible without the generous financial support provided by UBC. This support, which allowed me to fully dedicate myself to this project, has been instrumental in its success. I am also profoundly grateful to the research team members, especially Ghiasvand M. and Yousefi S. for their camaraderie, insightful discussions, and the countless hours they spent solving my problems together. Your support and have made this journey all the more rewarding. To my family, especially my parents, Sheng Z. and Bilin L., your unwavering belief in me and your endless support, both emotionally and financially, have been the foundation upon which I have built my academic pursuits. Your emotional support, particularly during my years of education, has been invaluable. I could not have reached this milestone without your love and encouragement. Finally, I thank everyone who has supported me in one way or another during this challenging yet fulfilling journey. Your contributions, whether big or small, have made a lasting impact on me and my work. xvii Dedication To my esteemed advisor, Dr. Tosarkani, for your unwavering guidance and support throughout this journey.1 Chapter 1: Introduction 1.1 Motivation The production and use of polymers are essential for manufacturing many products critical to modern life, but this reliance has significant negative impacts. The polymer production and use processes contribute to habitat destruction, fragmentation, and modification, posing a threat to wildlife (Hardesty et al., 2015). The waste generated by polymer products worsens the problem, causing a harmful environmental impact. Notably, an estimated 4.8 to 12.7 million metric tons of plastic entered the world’s oceans in 2010, resulting in harm to marine life, ecosystem disruption, and coastal degradation (Jambeck et al., 2015). Promoting the recycling and reuse of end-of-life products is essential for conserving valuable and costly raw materials utilized in their production (Hasan Shuvo et al., 2024). In addition, recycling aims to mitigate the severe environmental impact of polymer products, fostering economic opportunities and promoting a sustainable, circular economy (Huysman et al., 2015). In 2017, the Prime Minister of Canada took a stance on addressing the issue of single-use plastics, advocating for a transition towards zero plastic waste. The study by Walker and Xanthos in 2018 highlights Canada’s commitment to combating plastic pollution. The research emphasizes the significance of recycling plastic waste. It underscores Canada’s leadership in global initiatives, mainly through G7 efforts focused on plastic recycling. Proposed measures in the study include implementing a nationwide ban on single-use plastics and formulating recycling strategies for domestic single-use plastic items. A notable example of plastic recycling is in soft drink and mineral water bottles, especially PET. PET is more convenient and cost-effective than glass bottles, resulting in widespread consumption and a noticeable annual increase in PET use (Şimşek et al., 2018). Unlike organic materials, PET 2 bottles persist for an extended period without biodegrading, posing environmental concerns due to their space occupancy (Lebreton et al., 2017). Unfortunately, consumers frequently discard PET bottles carelessly, with little financial motivation for collection and recycling. The gross recycling rate of PET bottles in the USA was only 28.4% in 2016 (Taniguchi et al., 2019). Canadians discard over 3 million tons of plastic waste yearly, with only 9% recycled (Government of Canada, 2023). The rest is either incinerated, landfilled, or contaminates the environment, contributing to a significant economic burden and environmental threats, including marine litter and harm to wildlife. As PET bottles are derived from oil compounds, abandoning them in nature leads to the wasteful depletion of oil resources (Vural Gursel et al., 2021). Reusing these bottles aligns with sustainable resource utilization, reducing the demand for new raw materials, production costs, and environmental pollution (Bharadwaj et al., 2024). The recycling process of consumed PET bottles also benefits various industries, including carpet manufacturing (Yasin et al., 2024), new bottle production, and the creation of granules and fibres for applications in blanket production (R. Zhang et al., 2020). Many soft drinks and mineral water bottles that are not adequately recycled contribute to environmental waste (Paydar and Olfati, 2018). Plastic recycling faces economic challenges, progressing at a modest pace with material losses occurring throughout the process, from collection to sorting and from mechanical recycling of high-grade fractions to chemical recycling of low-grade fractions. Moreover, microplastics in the environment, which may escape collection and recycling efforts, pose a significant environmental and health concern (Lange, 2021b). Moreover, determining the optimal location, capacity, and quantity of various facilities, such as collection centers and recycling plants, as well as optimizing the material flow between these facilities, is critical to the economic efficiency of the recycling process and essential for minimizing total costs (Uekert et al., 2023). 3 The CLSC for plastic recycling has emerged as a promising solution to address these challenges. A CLSC involves a circular system encompassing production, distribution, and recovery processes (Govindan et al., 2015). This structure goes beyond the traditional forward SC elements to include the reverse logistics network (Berlin et al., 2022). The reverse logistics network is responsible for collecting, sorting, processing, recycling, remanufacturing, and redistributing or disposing products (Van Engeland et al., 2020). Introducing these reverse logistics processes is crucial in transforming a linear SC into a CLSC (Ramanathan et al., 2023). The success of CLSC relies on overcoming various obstacles, with a crucial factor being the development of a resilient network and infrastructure for recycling facilities. This network must efficiently collect and classify plastic waste from diverse sources, including households, industries, and commercial establishments (Lange, 2021a). The efficient treatment of plastic waste within the CLSC must have a well-established network and infrastructure of recycling facilities. However, the availability and accessibility of such infrastructure may vary across different regions, necessitating careful planning and investment to ensure the effectiveness of the CLSC. The viability of the CLSC is also intricately tied to market demand for recycled plastic products and PET. Therefore, creating a sustainable network for PET requires collaborative efforts among stakeholders, including manufacturers, retailers, and consumers. By leveraging appropriate methodologies and decision-making techniques, such as Mixed-Integer Linear Programming (MILP) and multi-objective programming, the CLSC can navigate these challenges and realize its potential to create a robust circular economy for plastics, reducing waste and minimizing environmental impact. Emphasizing the significant role of CLSC in plastic recycling, it is imperative to recognize the substantial challenges of uncertainty in designing such networks, particularly demand and return rates (Govindan et al., 2020). The future value of demand and return rates is still being determined, 4 which complicates the optimal design and operation of CLSCs and underscores the need for practical solutions. Fortunately, various optimization under uncertainty approaches, such as robust optimization, stochastic programming, and fuzzy programming, can effectively address these challenges, enhancing the resilience and effectiveness of CLSC design and management (H. Peng et al., 2020). Conventional optimization under uncertainty approaches rely on predefined assumptions about uncertain parameters to address uncertainty. However, they must leverage the available data to extract valuable information regarding uncertain parameters. Consequently, decisions from conventional approaches to deal with uncertainty may result in over-conservative or sub-optimal solutions (Ning and You, 2019). It highlights the advantage of utilizing available data to accurately extract the proper distribution of uncertain parameters, ensuring feasibility and mitigating over-conservatism of the optimal solutions. Data-driven optimization approaches are developed to fill the gap in optimization under uncertainty approaches to integrate machine learning algorithms with optimization problems to extract useful information from historical data and effectively deal with uncertainty (Bertsimas et al., 2018). 1.2 Research Objectives Recycling is a pivotal strategy for achieving a future with zero plastic waste (Awogbemi et al., 2022). Minimizing waste by maximizing the production of new materials from recycling is highly dependent on a well-designed and coordinated Closed-Loop Supply Chain (CLSC) network. This study aims to design a resilient CLSC network for the plastic industry while effectively addressing uncertainties inherent in the system. In a plastic CLSC network, end-of-life and end-of-use products are returned by customers to collection centers. These collection centers deliver the 5 collected products to recycling centers, where recyclable materials are processed and returned to manufacturers for reintegration into the production cycle. Conversely, non-cyclable materials are sent to incineration facilities for energy recovery or other uses. The manufacturers then use the recycled materials to produce new products, closing the loop within the circular economy. The design of a CLSC network involves critical decisions regarding facility location, capacity, number of facilities, and the flow of products between them—all of which significantly impact the network's performance. However, designing an efficient CLSC network presents several challenges, one of the primary ones being the uncertainty in key input parameters such as demand forecasts, return rates, and disposal fractions. These uncertainties directly influence decisions regarding facility capacities, quantities, locations, and inter-facility flows. As a result, the CLSC network must be designed to accommodate uncertain market demands, ensure resilience against disruptions, and optimize recycling processes. This research addresses these challenges by developing a data-driven system for evaluating and designing robust CLSC networks in the plastic industry. Specifically, the study will focus on: 1. Designing a robust optimization framework to generate balanced uncertainty sets that effectively handle variations in key input parameters, such as demand and return rates. 2. We are developing a system for analyzing and comparing different robust optimization approaches, particularly evaluating the effectiveness of the Kernel Weight Adjustment (KWA) approach compared to standard methods. 3. We establish metrics for assessing robustness, ensuring the uncertainty modelling process is fair, accurate, and practical for real-world applications. A critical part of the study involves leveraging historical data to construct data-driven models that account for variability in uncertain parameters. The research will also introduce a standardized 6 Robust Fairness Index (RFIC) as a novel metric for evaluating and balancing uncertainty sets. This index will benchmark the robustness of different data-driven optimization approaches, such as the standard One-Class Support Vector Machine (OC-SVM) and the enhanced OC-SVM with KWA. The thesis addresses the following research questions to achieve these objectives: 1. How can the optimal location of facilities and the flow between them be determined to minimize total costs in a CLSC network? 2. How can a resilient CLSC network be designed to withstand fluctuations in uncertain parameters effectively? 3. How can historical data be leveraged to model uncertainties in key input parameters, such as demand and return rates, in a CLSC network for the plastic industry? 4. How can fair uncertainty modelling be ensured within a data-driven decision-making framework for CLSC network design? 5. What metrics can be developed to evaluate the robustness and performance of different data-driven approaches? This research develops a data-driven robust optimization (DDRO) framework to answer these questions and design resilient CLSC networks that can accommodate fluctuations in demand, return rates, and disposal fractions. The framework incorporates balanced uncertainty sets generated through KWA alongside a system for evaluating the robustness and fairness of these uncertainty sets. Additionally, a standardized performance index will be introduced to compare the KWA approach against baseline methods, comprehensively evaluating their strengths in enhancing CLSC network resilience. 7 1.3 Research Methodologies and Contributions This section presents the research methodologies employed to develop a data-driven decision-making framework for designing a CLSC network in the plastic industry. Subsequently, the overall contributions of this research will be outlined. 1.3.1 Mixed-Integer Linear Programming MILP is an effective tool for optimizing decisions based on predefined objective functions. These decisions often involve inventory management, production planning, logistics, location-allocation, and recycling processes within a CLSC network. MILP is also particularly effective in handling decisions that require a balance between discrete choices or binary and integer decisions, such as the location and number of facilities, and continuous decisions, such as the flow of products and costs. Its strength lies in simultaneously optimizing these diverse decision variables, ensuring efficient and effective SC operations. In this research, a MILP model will be used to design a resilient CLSC network in the plastic industry. MILP is a valuable tool for addressing uncertainty in SCM. Critical parameters such as market demand or resource availability are often subject to variability. MILP models can be extended to incorporate optimization under uncertainty approaches, such as robust optimization or stochastic programming, to account for these uncertainties. By doing so, they provide reliable solutions that are resilient to fluctuations and ensure the SC operates effectively under varying conditions. 1.3.2 Data-Driven Robust Optimization (DDRO) Conventional optimization under uncertainty approaches, such as robust optimization and stochastic programming, are widely used in literature to deal with parameter uncertainty. The 8 uncertainty set is the heart of robust optimization and contains all possible future realizations of uncertain parameters. The main idea of robust optimization is optimizing the network against the worst-case scenario within the uncertainty set (Gabrel et al., 2014). Therefore, the feasibility, conservatism degree and computational tractability of robust optimization approaches depend on the quality of the uncertainty set. A quality uncertainty set is a set that encompasses all areas of uncertainty with a high possibility of occurrence of future realization (Ning and You, 2019). The approaches effectively handle uncertainty when limited information regarding the uncertain parameters is available. However, these approaches may lead to overly conservative solutions. Conventional robust optimization uses a predefined shape to construct uncertainty sets. It does not employ available historical data to tune the uncertainty set. It cannot handle uncertain parameters with complex distributions (Shang et al., 2017a). Therefore, data-driven optimization frameworks have been developed to tackle challenges with conventional approaches. Data-driven optimization is a framework in which the uncertainty model is formulated based on data, allowing the data to “speak” for themselves within the optimization algorithm (Ning and You, 2019). Data-driven optimization approaches extract useful information from data and integrate it with optimization under uncertainty approaches to obtain reliable and data-driven solutions. DDRO and DRO are subsets of data-driven optimization frameworks. DRO is a specific form of data-driven stochastic programming that deals with uncertainty through a family of probability distributions known as the ambiguity set, which captures the uncertainty data (Lin et al., 2022). This approach aims for optimal decisions under the worst-case distribution within this set, thereby offering performance guarantees for out-of-sample performance. DRO stands out for its ability to incorporate partial distributional information learned from uncertainty data into the optimization process, improving out-of-sample performance (van Parys et al., 2021). It also 9 maintains computational tractability, with some problems being solvable precisely in polynomial time without resorting to approximation schemes. DDRO is also the data-driven version of robust optimization. DDRO employs machine learning algorithms to construct data-driven uncertainty sets encompassing regions with historical data while effectively excluding low-density areas, thus reducing over-conservatism (Ning and You, 2019). Another advantage of DDRO approaches is their ability to construct data-driven polyhedral uncertainty sets, which result in tractable robust formulations. Additionally, the degree of conservatism can be easily adjusted, for example, by using a regularization parameter to control the fraction of data captured within the uncertainty set (Shang et al., 2017a). This research will employ a DDRO framework to address uncertainty in the CLSC network design. This approach ensures feasibility, avoids excessive conservatism, and maintains the computational tractability of the MILP model. 1.3.3 Contributions and Research significance The primary aim of this study is to understand, evaluate, and enhance the operations within the plastic CLSC. The benefits of this study extend to multiple stakeholders within the plastic industry, including manufacturers, distributors, consumers, and policymakers. Through improved understanding and strategies, the study aims to reduce plastic waste, enhance resource efficiency, and promote economic benefits within the SC. Numerous frameworks have been developed in the literature for designing CLSC networks and mitigating uncertainty using traditional optimization under uncertainty approaches. However, the potential of leveraging historical data to enhance the optimality of these approaches is still in its early stages. Furthermore, the application of DDRO in addressing uncertainty and building resilient CLSC networks is in its infancy. 10 Moreover, despite the availability of practical DDRO approaches in the literature for addressing uncertainty, more attention must be paid to creating balanced uncertainty sets. A balanced uncertainty set is equitably spread in all directions relative to the available data, with uniform reduction across each direction. In other words, the distance from the available data to the boundary of the uncertainty set is consistent in all directions. An imbalanced uncertainty set can lead to varying conservatism across different optimization problems. Therefore, to ensure that data-driven uncertainty sets are balanced and that the DDRO approach can be applied across different optimization problems with greater reliability, this research develops a balanced DDRO framework. Additionally, this research develops a fairness index algorithm to evaluate and compare the balance of different uncertainty sets. The existing literature on DDRO approaches must include appropriate metrics for comparing data-driven uncertainty sets. The fairness index introduced in this study provides a systematic method to assess and compare the balance of various data-driven uncertainty sets, ensuring more consistent and reliable optimization outcomes. 1.4 Thesis Outline The structure of this thesis is as follows: Chapter 2 presents a comprehensive literature review on CLSC network design and DDRO approaches. Chapter 3 details the problem statement and provides the mathematical formulation for the CLSC network design along with the numerical results of the deterministic model. Chapter 4 explains the DDRO approach for handling uncertainty, including an enhanced version for constructing balanced uncertainty sets. Additionally, the development of the fairness index for comparing data-driven uncertainty sets is discussed. Finally, 11 Chapter 5 summarizes the key findings, contributions, and limitations and proposes potential directions for future work. 12 Chapter 2: Literature Review This section aims to conduct a literature review and investigate the application of mathematical optimization techniques in plastic recycling. We focus on four main approaches: deterministic models, multi-objective models, stochastic programming, and data-driven robust optimization. 2.1 Deterministic and Multi-Objective Models in CLSC The pivotal role of deterministic models in enhancing plastic recycling processes within CLSC is our starting point. These mathematical models offer invaluable insights into optimizing facility locations, production levels, transportation, and inventory management in the recycling context. Various studies have developed deterministic models for designing CLSC networks. Liao et al. (2020) conducted a comprehensive literature review on CLSCs and Reverse Logistics, focusing on recent trends and the state-of-the-art in these fields. They highlighted the importance of these areas in addressing environmental issues, such as pollution and waste management, and the role of effective planning and control in maximizing profit and recapturing usefulness in the SC. Kazancoglu et al. (2022) developed a multi-objective optimization model for designing a green dual-channel CLSC network, focusing on minimizing economic costs and environmental impacts, such as CO2 emissions and particulate matter. Their study highlights the integration of e-commerce channels within the SC, providing valuable insights for optimizing network flow while considering home appliance industry sustainability. Pazhani et al. (2021) proposed a multi-period, multi-product CLSC model to optimize logistics, production, and inventory decisions. Their research contributes by providing a relaxation approach for designing complex real-world SCs. Gharibi and Abdollahzadeh (2021) developed a MILP model to design a circular economy-led CLSC within a green reverse logistics network. Their contribution lies in integrating green SCM 13 factors and using the Simultaneous Data Envelopment Analysis (SDEA) method to enhance the efficiency of disassembly centers. Gharibi and Abdollahzadeh’s (2021) study encompasses Reverse Logistics (RL) and CLSC management, emphasizing mathematical models, supplier selection, and the circular economy. The importance of sustainable waste management, mainly electronic and electrical waste, is emphasized. It advocates using MILP models to solve CLSC problems, particularly in RL network design. It offers strategies and models to improve the efficiency and sustainability of CLSC, such as dual-objective RL network design to balance economic benefits and environmental impact. Gharibi and Abdollahzadeh (2021) underscored the circular economy's role in driving economic growth through resource consumption and waste recycling. Managing recyclable waste is crucial for environmental sustainability. Santander et al. (2020) propose a MILP model to design a CLSC network for local and distributed plastic recycling, specifically targeting 3D printing applications. Their research contributes to the literature by demonstrating the economic and environmental feasibility of distributed plastic recycling networks, using a case study to highlight the benefits of localized recycling in reducing CO2 emissions and lowering costs compared to traditional centralized systems. Rajak et al. (2022) developed a multi-objective MILP model for designing a sustainable CLSC network focusing on remanufacturing steering columns. They aim to incorporate remanufacturing with reverse logistics into the SC to achieve sustainability by satisfying the Sustainable Development Goals. The model is designed to handle the uncertainty involved in the SC and determine the optimum quantity of raw materials to be bought from the suppliers based on the number of recyclable parts and disposed parts while maximizing the industry's profit. 14 The use of multi-objective models represents a significant shift in the plastic recycling procedures at CLSC. These advanced models offer dynamic capabilities essential for promoting sustainability by accounting for multiple conflicting objectives. In plastic recycling, multi-objective models provide a nuanced decision-making approach that optimizes facility locations, production levels, transportation, and inventory management while considering environmental impact, economic efficiency, and social factors. The study conducted by Allehashemi et al. (2022) and Li and Li (2023) both focus on multiple objectives to optimize the CLSC network. Allehashemi et al. (2022) concentrated on minimizing the total cost, maximizing customer satisfaction, and reducing carbon emissions from transportation and fossil fuel usage. Incorporating customer satisfaction and emissions reduction marks a novel approach in the existing body of CLSC literature, expanding the traditional focus beyond mere cost minimization. Li and Li, (2023) aimed not only for cost and emission reduction but also for customer satisfaction, timely delivery, and product quality assurance. The distinctive contribution of their paper is the adjustment of weights assigned to each objective, aligning with the specific needs and priorities of involved customers and enterprises. Tosarkani and Amin (2018) and Lashkari and Zhang (2014) facilitated the fine-tuning of the model to various situations, providing a dynamic balance between cost and quality objectives. Hasan et al. (2023) developed a nonlinear mathematical model to minimize the total cost, energy consumption, CO2 emission, and waste generation of SCs, focusing on disruption risk. Hashmi et al. (2022) also developed a multi-objective model to design a CLSC network and optimize the distribution plan to minimize the transportation and other operational costs and carbon footprint. Dehshiri et al. (2023) identified critical objectives in CLSC design, including cost minimization, carbon emissions reduction, and enhanced responsiveness. 15 They emphasized balancing economic efficiency with environmental sustainability and operational agility to develop a resilient and sustainable CLSC. 2.2 Optimization under uncertainty in CLSC CLSC networks face various uncertainties that complicate their optimization and efficient management. Specifically, the future values of key input parameters, such as demand and return rates, are unpredictable and challenging to forecast accurately. To this end, several methods are available to optimize under uncertainty and mitigate the associated risks. In this section, we delve into stochastic programming and robust optimization and explore DDRO in the subsequent section. Stochastic programming has emerged as an practical approach for handling uncertainties and refining models by assigning probabilities to different scenarios (Nabavizadeh et al., 2024). Two-stage stochastic programming is a widely used methodology for addressing uncertainty in decision-making processes. This approach divides decisions into two categories: here-and-now decisions, which are made before the uncertainty is revealed, and wait-and-see decisions, which are made after the uncertainty has been observed. This structure allows for the development of more flexible and adaptive strategies by enabling decision-makers to hedge against various possible future scenarios, thereby optimizing outcomes in uncertain environments (Popela et al., 2014). In a subsequent expansion of the uncertainties within the CLSC network, Badri et al. (2017) applied stochastic programming to address uncertainties in the CLSC network design. They develop a multi-period, multi-product stochastic model that covers production, transportation, and inventory planning. The authors use a two-stage stochastic programming approach introduced by Dantzig (1955). This approach allows for proper decision-making in stochastic environments and has been applied to various problems. Ren et al. (2020) developed a GIS-based green SC model to 16 assess the effects of carbon price uncertainty on plastic recycling within a CLSC. It addresses uncertainty by incorporating a two-stage stochastic programming model that uses a geometric Brownian motion process to simulate carbon price variations, allowing for the evaluation of different scenarios and their impact on the SC’s design and operation. Das et al. (2022) extended the application of stochastic programming to the design of CLSCs for reusable packaging materials, specifically focusing on wooden pallets. They introduced a risk-averse two-stage stochastic programming model that integrates CVaR to manage uncertainties in demand and the availability of used pallets for refurbishment or reuse. Tao et al. (2020) expanded on these approaches by focusing on the integrated problem of strategic network planning and operation for sustainable CLSCs. They developed a two-stage stochastic programming model that addresses demand uncertainty, retail market configuration, transportation mode selection, and carbon emission control policies. Biçe and Batun (2021) tackle uncertainty in demand, return rates, and the quality of returned products by employing a two-stage stochastic MIP model. They utilize the L-shaped method within an iterative and branch-and-cut framework to improve computational efficiency, emphasizing the significant impact of demand uncertainty on the overall system. Mohammadi et al. (2020) developed a multi-stage stochastic programming model to design a sustainable CLSC network that incorporates financial decisions like investments and loans. The approach deals with the uncertainty by modelling demand and return rates on investment as stochastic parameters, enabling the evaluation of different scenarios and ensuring robust decision-making across multiple periods. Shafiee Roudbari et al. (2023) applied stochastic programming as a two-stage stochastic MILP model to design a CLSC network. The first stage of the model decides on the location of facilities. The second stage decides on the allocation of customers to Distribution and Collection Centers. It 17 calculates the optimum flow of material between facilities. The model also considers uncertainty in critical parameters like the quantity and quality of returned products and the demand for recovered products. Subbaih and Ramasamy (2022) developed an optimization framework for a multi-echelon SC infused with considerations pivotal for sustainability. Their study successfully navigates towards minimizing the joint expected total costs, all within the operational boundaries defined by stochastic constraints. Subbaih and Ramasamy (2022) and Fazli-Khalaf et al. (2017) also used stochastic programming to design a CLSC network that manages uncertainty by creating a stochastic scenario-based programming model. Using stochastic chance-constrained programming methods, the model minimizes the expected network design costs across different uncertainty scenarios. Kamyabi et al. (2022) propose a multi-objective optimization model under uncertainty, focusing on environmental and economic objectives within a CLSC. They use a scenario-based approach to address uncertainties in demand, return, and recovery rates, providing insights into balancing cost minimization with environmental impact. In addition to stochastic programming, robust optimization and fuzzy programming are widely used to deal with uncertainty (Tordecilla et al., 2021). Ebrahimi and Bagheri (2022) developed a robust multi-objective optimization model for designing a CLSC network in the plastic bottle industry, focusing on minimizing costs, supply risk, and maximizing customer satisfaction. The approach deals with uncertainty by employing robust optimization techniques combined with a multi-choice goal programming method, ensuring the model’s resilience under varying conditions and uncertainties in SC parameters. Borajee et al. (2023) also presented a chance-constrained optimization model for a multi-echelon, multi-product, closed-loop SC that accounts for brand diversity and demand uncertainty. Their research introduces an accelerated Benders decomposition algorithm to efficiently solve the complex model, making a significant contribution by enhancing 18 the computational efficiency of solving large-scale stochastic SC problems under uncertainty. Each of these methodologies—stochastic programming, robust optimization, and fuzzy programming—offers distinct advantages in handling uncertainty. As a result, hybrid approaches that combine these methods have been developed to leverage their respective strengths for more effective decision-making under uncertainty. Gholizadeh et al. (2020) proposed a robust stochastic fuzzy rule that uses fuzzy numbers of triangular likelihood ratios to manage uncertainty. This approach considers average probable value, possible changeability, and random probabilities. The authors argued that this robust stochastic model is more preferred than a definite model in achieving risk-taking results and is less sensitive to constraint violations. Moheb-Alizadeh et al. (2021) further contribute by addressing the uncertainty in supply, demand, and remanufacturing costs within CLSCs for automotive parts, using a hybrid robust-stochastic optimization model to ensure resilient decision-making. Tosarkani et al. (2024) also integrated fuzzy theory with scenario-based stochastic programming to deal with uncertain demand and return rates in reverse logistics network design in the plastic industry (Tosarkani et al., 2024). 2.3 Data-driven Robust Optimization As mentioned earlier, stochastic programming is widely used in the literature to deal with parameter uncertainty. However, implementing stochastic programming approaches presents some challenges. Firstly, it relies on the probability distribution of random variables, which can be challenging due to the limited availability of historical data in many real-world problems. Additionally, in scenario-based stochastic programming, the probability distributions are represented through different scenarios, leading to computationally complex and large-scale 19 problems as the number of scenarios increases exponentially with uncertain parameters (Mohseni and Pishvaee, 2020a). RO is often used as an alternative to overcome the limitations of stochastic programming. RO does not require precise probability distributions; instead, it models uncertainty using predefined uncertainty sets based on the maximum and minimum possible values of uncertain parameters. The uncertainty set is the heart of RO and contains all possible future realizations of uncertain parameters. RO reduces computational complexity and provides solutions that remain feasible under all possible realizations of uncertainty within the defined set, making it a more robust choice when dealing with no historical data available. Various uncertainty sets have been introduced in the literature to address parameter uncertainty, including the Box set (Soyster, 1973), Ellipsoidal set (Ben-Tal and Nemirovski, 1999), Polyhedral set t, and the combined Polyhedral-Ellipsoidal set (Bertsimas et al., 2004). These approaches effectively deal with uncertainty when there is little to no data regarding the uncertain parameters. However, the issue with conventional RO approaches is their reliance on predefined shapes for uncertainty space, often assuming that the uncertainty set spreads symmetrically from the center in different directions, thereby failing to capture the complexity of uncertain parameters that may have more complex distributions (Ghiasvand and Rahmani, 2023a). In the last decade, technological advancements have led to the rapid collection and storage of vast data. Additionally, improvements in computational capabilities and machine learning algorithms have led to the development of a new paradigm known as data-driven optimization, which leverages available data to enhance decision-making in uncertain environments. The data-driven optimization framework can be divided into three categories: 1) data-driven stochastic programming and distributionally robust optimization, 2) data-driven chance-constrained 20 programming, and 3) DDRO. In DDRO approaches, machine learning algorithms or statistical methods are employed to construct data-driven uncertainty sets encompassing regions containing historical data while excluding areas with no data. This ensures that the resulting robust solutions are feasible while avoiding overly conservative outcomes. DDRO is highly effective in SC design and related areas where data is available. Data-driven approaches can be applied to various problems to deal with uncertainty, including SC design (Mohseni and Pishvaee, 2020a), joint price-inventory management (Hu et al., 2015), healthcare operations (Lotfi et al., 2022), and water allocation (Yao et al., 2022). Hejazi and Khorshidvand (2023) have also implemented a DDRO approach to handle uncertainties in a CLSC effectively. This approach considers the SC’s technological, economic, and environmental aspects, making it a valuable tool for navigating uncertain environments. By utilizing the DDRO approach, a resilient SC that can quickly adapt to changes in data patterns can be established, resulting in more dependable decision-making in the face of uncertainty. Gao et al. (2023) have utilized the DDRO approach to optimize a Dual-Channel CLSC while effectively addressing demand uncertainty. By implementing the DDRO approach, they have created a robust model that can handle the variability in customer demand, a crucial factor in the SC network. Various machine learning or statistical approaches have been developed or used in the literature to construct data-driven uncertainty sets. Bertsimas et al. (2018) introduced a schema for constructing data-driven uncertainty sets by leveraging statistical hypothesis tests. The proposed approach creates flexible, computationally tractable models outperforming traditional methods, especially in data-rich applications like portfolio management. Jiao et al. (2018) employed a data-driven approach incorporating Kullback-Leibler (K-L) divergence into the uncertainty sets. By including the K-L divergence, the authors measured the distance between the estimated 21 distribution of uncertainties (obtained from historical data) and the real distribution. It allows for a more accurate and data-driven robust optimization model that accounts for the value of data in decision-making processes. Zhang et al. (2018a) focus on addressing the challenge of correlated uncertainties in industrial scheduling. The authors introduce copulas to capture the joint probability distribution of uncertainties and propose robust formulations that reduce conservatism by eliminating unnecessary scenarios. Zhang et al. (2018b) proposed a robust optimization approach for petrochemical scheduling. They utilized probability density contours to define uncertainty sets, enabling more accurate and adaptive scheduling under uncertainty and removing low-density areas. The approach is validated through a rolling-horizon strategy that integrates real-time data, improving the robustness and efficiency of scheduling in a real-world ethylene production process. Gumte et al. (2021) also presented a novel approach that integrates machine learning with RO to address uncertainty in SC planning. The authors introduced a neuro-fuzzy clustering mechanism to more accurately sample the uncertain parameter space, leading to better performance than traditional methods like box and ellipsoidal sets. This approach is efficient in scenarios with sparse and discrete data, improving the reliability and efficiency of SC models under uncertainty. Zhang et al. (2022) addressed uncertainty in nonlinear optimization by proposing a DDRO framework that constructs compact, convex uncertainty sets using statistical limit techniques and PCA. To manage large perturbations, they developed an iterative algorithm that refines the solution through multiple linearization, effectively solving the nonlinear adversarial problem by ensuring robustness and computational efficiency across varying uncertainty levels. Various DDRO approaches have been developed to construct data-driven polyhedral uncertainty sets that improve the feasibility of robust solutions and tractability of robust counterpart models. Ning and You (2018) introduce a robust optimization framework that 22 leverages PCA and kernel smoothing methods to construct uncertainty sets. This approach effectively captures correlations between uncertain parameters. It addresses distributional asymmetry, resulting in less conservative and more computationally tractable robust counterpart models. Shang et al. (2017b) introduced a novel DDRO optimization approach using a WGIK-based SVC method in another study. This approach constructs uncertainty sets by capturing the distributional geometry of uncertain data through a piecewise linear kernel, leading to less conservative and computationally tractable robust counterpart models. Additionally, the approach allows for fine-tuning the conservatism of the uncertainty set through a regularization parameter, which controls the fraction of data included within the uncertainty set. WGIK-based SVC may result in over-conservative solutions in case of high-dimensional uncertainty due to the course of dimensionality of machine learning approaches. To this end, Shang and You (2018) proposed a DDRO approach tailored for high-dimensional uncertainty. They proposed a novel method combining PCA for dimension reduction with WGIK-based SVC to construct a flexible uncertainty set in the principal subspace, using traditional polyhedral sets in the residual subspace. This approach reduces conservatism and ensures computational efficiency, making it ideal for complex, high-dimensional datasets. Therefore, Mohseni and Pishvaee (2020b) developed WGIK-based CSVC to deal with uncertainty when limited data is available. The WGIK-based CSVC incorporates fuzzy support neighbourhoods around data points with high local density values, making it more conservative than the WGIK-based SVC approach in specific directions. It reduces reliance on historical data, providing a distinct advantage when data availability is limited. WGIK-based SVC is tailored for datasets with a single cluster, but real-world data may consist of multiple clusters. To account for datasets with multiple clusters, Shen et al. (2022) proposed a stochastic DDRO framework that integrates 23 machine learning and data-driven methods to handle uncertainties in renewable energy sources like solar and wind. The framework uses fuzzy clustering to obtain different clusters within a dataset. It employs WGIK-based SVC for each cluster to construct uncertainty sets, optimizing economic and environmental objectives for industrial energy systems. Constructing data-driven uncertainty sets for limited data availability may result in infeasible or sub-optimal solutions since the limited available data may need to reflect the proper distribution of uncertain parameters. In another study, Goerigk and Kurtz (2023) developed a DDRO framework by employing unsupervised deep learning, One-Class Deep Neural Networks, to construct non-convex uncertainty sets. They addressed the nonlinear adversarial problem by formulating it as a convex quadratic mixed-integer program, allowing for robust solutions through an iterative scenario generation process. This approach effectively manages the complexity of high-dimensional uncertainty and datasets with multiple clusters. The WGIK-based SVC approach can result in over-conservatism when the historical data of uncertain parameters exhibit an asymmetric distribution. Han et al. (2021a) developed a DDRO approach called SKL- and MKL-based OC-SVM to tackle this issue. In the MKL-based setting, various kernel matrices are constructed, each representing different directions of the uncertainty space. Upon solving the MKL-based OC-SVM model, optimal kernel matrices will be selected to construct convex and non-conservative uncertainty sets for even datasets with asymmetric distributions. The number of variables and constraints in the SKL- and MKL-based OC-SVM approach, and consequently the computational tractability of the robust counterpart model, depends on the number of selected kernel matrices and support vectors. In addition, as the regularization parameter of the SKL- and MKL-based OC-SVM approaches increases (decreasing the conservatism degree), more support vectors are generated, which results in a complex robust 24 counterpart model. To tackle this, Ghiasvand et al. (2024) developed a two-stage clustering algorithm to improve the computational tractability of the robust counterpart models by eliminating unnecessary support vectors. Supplementary data (e.g., local density information, proximity to predicted values, and occurrence time) regarding the available historical data may improve decision-making and avoid over-conservative solutions. Ghiasvand and Rahmani (2023) developed a Weighted DDRO framework that considers the weight of each historical data. The data point with a higher weight will have a higher probability of being encompassed by the data-driven uncertainty set. They also developed distance-based and density-based approaches to calculate the weight of data samples. In the distance-based approach, data samples closer to the predicted point are assigned higher weights, leading to the movement of uncertainty sets towards these predicted points, effectively covering the areas around them. The density-based approach aims to assign higher weights to data samples in high-density regions, constructing uncertainty sets that prioritize covering high-density areas while avoiding low-density regions. Despite the advantages of developed DDRO approaches in the literature, the balance of constructed uncertainty sets across different directions still needs to be examined. Ensuring that the reduction rate of uncertainty sets is consistent across all directions is crucial for maintaining a uniform level of robustness across various optimization problems and directions. Additionally, developing indices to compare uncertainty sets constructed by different approaches has yet to be explored. The application of data-driven optimization approaches in designing CLSC networks within the plastic industry remains unexplored.25 Table 2.1 Summary of Related Works on CLSC Network Design Authors Multi-Product Multi-Period Objective Data-Driven Uncertainty Type of Industry Solution Approach MahmoumGonbadi et al. (2021) √ √ Costs - Demand, Returns, Costs - MILP, Fuzzy Programming Liao et al. (2020) - √ Costs √ - Citrus fruit crates MILP Allehashemi et al. (2022) √ √ Costs - Demand, Return Cellphone MILP, Fuzzy Programming Li and Li (2023) √ - Costs, Carbon Satisfaction - Demand, Return Cold chain products Multi-objective Programming Gharibi and Abdollahzadeh’s (2021) √ - Profitability Quantity Efficiency - Demand, Return, Cost, Supplier E-waste Multi-objective Programming Tosarkani and Amin (2018) √ √ Profitability - All parameters Battery Fuzzy Programming Fuzzy ANP 26 Table 2.1 Summary of Related Works on CLSC Network Design Authors Multi-Product Multi-Period Objective Data-Driven Uncertainty Type of Industry Solution Approach Real Location Hasan et al. (2023) - - Costs, Energy Carbon - - Wood Multi-objective Programming - Hashmi et al. (2022) √ - Costs, Carbon Satisfaction √ Demands, Inaccurate forecasting Shoes, Saddles, Bags Fuzzy Programming - Dehshiri et al. (2023) √ √ Costs, Carbon Time - Demand, Capacity Stone paper Multi-objective Fuzzy programming √ Lashkari and Zhang (2014) √ √ Costs, Carbon - - Lead-acid battery Multi-objective Programming - Jiao et al. (2018) - √ - √ Disposal rate Demand Durable products Chance Constraints Programming - Ren et al. (2020) √ √ Costs √ Carbon price Plastic materials Stochastic Programming √ Das et al. (2022) - - Costs √ Demand Initial inventory Wooden pallets Stochastic programming CVaR √ 27 Table 2.1 Summary of Related Works on CLSC Network Design Authors Multi-Product Multi-Period Objective Data-Driven Uncertainty Type of Industry Solution Approach Real Location Kamyabi et al. (2022) - √ Profitability √ Demand Carbon tax Various MILP √ Tao et al. (2020) - √ Costs - Demand, Supply Carbon emission Return rate - Stochastic MILP Model √ Biçe and Batun (2021) - √ Profitability - Product quality Return - Two-Stage Stochastic Programming - Moheb-Alizadeh et al. (2021) √ √ Profitability Carbon - Demand, Return Capacities, Fixed costs Laptop Stochastic MINLP Programming √ Badri et al. (2017) √ √ Costs - Demand Return - Two-Stage Stochastic Programming - Gholizadeh et al. (2020) √ √ Costs Carbon √ Quantity of product Dairy Product Robust, Stochastic Fuzzy approach √ Mohammadi et al. (2020) √ √ Profitability - Demand Return Plastic Multi-Stage Stochastic Programming √ 28 Table 2.1 Summary of Related Works on CLSC Network Design Authors Multi-Product Multi-Period Objective Data-Driven Uncertainty Type of Industry Solution Approach Real Location Shafiee Roudbari et al. (2023) - √ Profitability - Quantity of return Garment Two-Stage Stochastic Programming √ Gumte et al., (2021) √ - Costs √ Production costs Demand, Price - DDRO, MILP - Mohseni and Pishvaee, (2020) - √ Costs √ Quantity of product Biodiesel DDRO, MILP, CSVC √ Lotfi et al., (2022) √ √ Costs √ Capacity Emission - DDRO, MILP - Gao et al. (2023) - √ Costs √ All parameters Electrical products DDRO, MILP - Hejazi and Khorshidvand (2023) √ - Costs Carbon - All parameters - DDRO, MILP - Proposed model √ - Cost √ Demand, Return Capacity Plastic Bottle SKL-based OC-SVM with KWA, MILP √ 29 Chapter 3: Designing Deterministic Model for Plastic CLSC 3.1 Introduction The following chapter presents a deterministic model for designing a Plastic CLSC network. The model is intended for a multi-echelon plastic CLSC, considering multiple suppliers, plants, retailers, markets, plastic recovery centers, and disposal centers. The objective is to determine the optimal number and location of plants, distribution centers, and recovery centers, the number of plastic components to be purchased, and the number of products to be produced and distributed to achieve the maximum profit for the plastic CLSC. The scenario-based analysis is conducted based on different market demand scenarios, and the returned plastic products are used to evaluate the model’s performance. 3.2 Problem Definition The purpose of this section is primarily to define the research problem and provide an overview of the solution approach proposed in this research. Figure 3.1 shows a multi-echelon CLSC network with multiple echelons and products in the context of a plastic bottle. The proposed CLSC includes suppliers, plants, distributors, and markets in the forward flow. In the reverse flow, plastic recovery centers and disposal centers are incorporated. The structure of the CLSC integrates both the forward and reverse flows. Suppliers provide the main components of the plastic, namely PET and HDPE, to the plants, which use them to produce various plastic products. These products are then distributed to retailers aiming to meet market demand while minimizing inventory and transportation costs. Customers purchase plastic products, and some of these products are subsequently returned to the plastic recovery centers. The returned plastic products are disassembled at the recovery centers into their main components. Some PET and HDPE 30 components can be recycled and returned to the plants for reuse. At the same time, any unrecoverable parts are sent to the disposal center for appropriate handling. Figure 3.1 Demonstration of CLSC Network This study develops a structured approach to optimize the CLSC network for plastic bottle production and recycling. A deterministic MILP model is formulated, capturing both forward and reverse flows within the CLSC to minimize total costs, including production, transportation, and processing costs at recovery and disposal centers, while meeting market demand for plastic products. A DDRO approach is developed to address uncertainties in parameters such as demand, supply, and recycling rates. This approach hedges against uncertainty through a data-driven uncertainty set containing possible future realizations of the uncertain parameters. It ensures the optimized CLSC design remains resilient and effective even under worst-case scenarios. Sensitivity analysis is conducted to evaluate the impact of parameter variations on the optimal solution, providing insights into the robustness and flexibility of the proposed CLSC design. The comprehensive approach taken in this research supports informed decision-making and strategy formulation for sustainable plastic bottle production and recycling. 31 3.3 Mathematical model In this section, the deterministic MILP model for designing a CLSC network for plastic products is proposed and explained. First, the mathematical notations for the MILP model are introduced. Subsequently, the objective function and the constraints of the MILP model are provided. Sets: 𝐼: Set of plastic items (1… 𝑖 … 𝐼) 𝑆: Set of Suppliers (1…𝑠 …𝑆) 𝑅: Set of Manufacturers and re-Manufacturers (1…𝑟 …𝑅) 𝐷: Set of Distribution Centers (1…𝑑 …𝐷) 𝑀: Set of Markets (1…𝑚…𝑀) 𝐶: Set of Recovery Centers (1…𝑐 …𝐶) 𝐹: Set of Incineration Facility (1…𝑓 …𝐹) Parameters: 𝑃𝐶𝑖: Production cost of plastic item 𝑖 𝑇𝑆𝑅𝑖𝑠𝑟: Transportation cost of plastic item𝑖 per km from Supplier 𝑠 to Manufacturer 𝑟 𝑇𝑅𝐷𝑖𝑟𝑑: Transportation cost of plastic item 𝑖 per km from Manufacturer 𝑟 to Distribution Center 𝑟 𝑇𝐷𝑀𝑖𝑑𝑚: Transportation cost of plastic item 𝑖 per km from Distribution Center 𝑑 to Market 𝑚 𝑇𝑀𝐶𝑖𝑚𝑐: Transportation cost of plastic item 𝑖 per km from Market 𝑚 to Recovery Center 𝑐 𝑇𝐶𝑅𝑖𝑐𝑟: Transportation cost of plastic item 𝑖 per km from Recovery Center 𝑐 to Manufacturer 𝑟 𝑇𝐶𝐹𝑖𝑐𝑓: Transportation cost of plastic item 𝑖 per km from Recovery Center 𝑐 to Incineration Facility 𝑓 32 𝐹𝐶𝑆𝑠: Fixed cost for opening Supplier 𝑠 𝐹𝐶𝐶𝑐: Fixed cost for opening Recovery Center 𝑐 𝐹𝐶𝑅𝑟: Fixed cost for opening Manufacturer 𝑟 𝐹𝐶𝐷𝑑: Fixed cost for opening Distribution Center 𝑑 𝐶𝑆𝐻𝑖: Cost of shortage for Product 𝑖 𝑆𝐶𝑖: Cost saving of plastic item 𝑖 𝐷𝐶𝑖: Disposal cost of plastic item 𝑖 𝐶𝑆𝑠𝑖: Capacity of Supplier 𝑠 for plastic item 𝑖 𝐶𝐶𝑐𝑖: Capacity of Recovery Center 𝑐 for plastic item 𝑖 𝐶𝑅𝑟𝑖: Capacity of Manufacturer 𝑟 for plastic item 𝑖 𝐶𝐷𝑑𝑖: Capacity of Distribution Center 𝑑 for product 𝑖 𝐷𝐶𝑅𝑐𝑟: Distance between Recovery Center 𝑐 and Manufacturer 𝑟 𝐷𝑆𝑅𝑠𝑟: Distance between Supplier 𝑠 and Manufacturer 𝑟 𝐷𝑅𝐷𝑟𝑑: Distance between Manufacturer 𝑟 and Distribution Center 𝑑 𝐷𝐷𝑀𝑑𝑚: Distance between Distribution Center 𝑑 and Market 𝑚 𝐷𝑀𝐶𝑚𝑐: Distance between Market 𝑚 and Recovery Center 𝑐 𝐷𝐶𝐹𝑐𝑓: Distance between Recovery Center 𝑐 and Incineration Facility 𝑓 𝐷𝑓𝑖: Minimum disposal fraction of plastic item 𝑖 𝐷𝑀𝑚𝑖 : Demand of Market 𝑚 for plastic item 𝑖 𝑅𝑀𝑚𝑖: Return of Market 𝑚 for plastic item 𝑖 33 Variables: 𝑄𝐶𝑅𝑐𝑟𝑖 Quantity of returned plastic item 𝑖 from Recovery Center 𝑐 to Manufacturer 𝑟 𝑄𝑅𝐷𝑟𝑑𝑖 Quantity of returned plastic item 𝑖 from Manufacturer 𝑟 to Distribution Center 𝑑 𝑄𝐷𝑀𝑑𝑚𝑖 Quantity of returned plastic item 𝑖 from Distribution Center 𝑑 to Market 𝑚 𝑄𝑀𝐶𝑚𝑐𝑖 Quantity of returned plastic item 𝑖 from Market 𝑚 to Recovery Center 𝑐 𝑄𝑆𝑅𝑠𝑟𝑖 Quantity of returned plastic item 𝑖 from Supplier 𝑠 to Manufacturer 𝑟 𝑄𝐶𝐹𝑐𝑓𝑖 Quantity of returned plastic item 𝑖 from Recovery Center 𝑐 to Incineration Facility 𝑓 𝑆𝐻𝑚𝑖: Shortage cost of Demand for Market m and Product i 𝐵𝐶𝑐 1, if a Recovery Center 𝑐 is located and set up; 0, otherwise 𝐵𝑅𝑟 1, if a Manufacturer 𝑟 is located and set up; 0, otherwise 𝐵𝐷𝑑 1, if a Distribution Center 𝑑 is located and set up; 0, otherwise 𝐵𝑆𝑠 1, if a Supplier 𝑠 is located and set up; 0, otherwise 𝐵𝑆𝑠 1, if a Supplier 𝑠 is located and set up; 0, otherwise The Objective function and the constraints of the MILP model are as follows: 𝑀𝑖𝑛 𝑍1 = ∑𝐹𝐶𝐶𝑐 × 𝐵𝐶𝑐𝑐+ ∑𝐹𝐶𝑅𝑟 × 𝐵𝑅𝑟𝑟+ ∑𝐹𝐶𝐷𝑑 × 𝐵𝐷𝑑𝑑+ ∑𝐹𝐶𝑆𝑠 × 𝐵𝑆𝑠𝑠+ ∑∑∑(𝑃𝐶𝑖 + 𝑇𝑆𝑅𝑖 × 𝐷𝑆𝑅𝑠𝑟) × 𝑄𝑆𝑅𝑠𝑟𝑖𝑖𝑟𝑠+ ∑∑∑𝑇𝑅𝐷𝑖 × 𝐷𝑅𝐷𝑟𝑑 × 𝑄𝑅𝐷𝑟𝑑𝑖𝑖𝑑𝑟+ ∑∑∑𝑇𝐷𝑀𝑖 × 𝐷𝐷𝑀𝑑𝑚 × 𝑄𝐷𝑀𝑑𝑚𝑖𝑖𝑚𝑑+ ∑∑∑𝑇𝑀𝐶𝑖 × 𝐷𝑀𝐶𝑚𝑐 × 𝑄𝑀𝐶𝑚𝑐𝑖𝑖𝑐𝑚+ ∑∑∑(−𝑆𝐶𝑖 + 𝑇𝐶𝑅𝑖 × 𝐷𝐶𝑅𝑐𝑟) × 𝑄𝐶𝑅𝑐𝑟𝑖𝑖𝑟𝑐+ ∑∑∑(𝐷𝐶𝑖 + 𝑇𝐶𝐹𝑖 × 𝐷𝐶𝐹𝑐𝑓) × 𝑄𝐶𝐹𝑐𝑓𝑖𝑖𝑓𝑐+ ∑∑𝑆𝐻𝑚𝑖𝑖𝑚× 𝐶𝑆𝐻𝑖 (3.1) 34 Subject to: ∑𝑄𝐷𝑀𝑑𝑚𝑖𝑑≥ 𝐷𝑀𝑚𝑖 − 𝑆𝐻𝑚𝑖 ∀ 𝑚 ∈ 𝑀, 𝑖 ∈ 𝐼 (3.2) ∑𝑄𝑀𝐶𝑚𝑐𝑖𝑚= ∑𝑄𝐶𝑅𝑐𝑟𝑖𝑟+ ∑𝑄𝐶𝐹𝑐𝑓𝑖𝑟 ∀ 𝑐 ∈ 𝐶, 𝑖 ∈ 𝐼 (3.3) ∑𝑄𝑀𝐶𝑚𝑐𝑖𝑐= 𝑅𝑀𝑚𝑖 ∀ 𝑚 ∈ 𝑀, 𝑖 ∈ 𝐼 (3.4) ∑𝑄𝐷𝑀𝑑𝑚𝑖𝑚= ∑𝑄𝑅𝐷𝑟𝑑𝑖𝑟 ∀ 𝑐 ∈ 𝐶, 𝑖 ∈ 𝐼 (3.5) ∑𝑄𝑀𝐶𝑚𝑐𝑖𝑚× 𝐷𝐹𝑖 = ∑𝑄𝐶𝐹𝑐𝑓𝑖𝑓 ∀ 𝑐 ∈ 𝐶, 𝑖 ∈ 𝐼 (3.6) ∑∑𝑄𝐶𝑅𝑐𝑟𝑖𝑖𝑐+ ∑∑𝑄𝑆𝑅𝑠𝑟𝑖𝑖𝑠≤ ∑𝐶𝑅𝑟𝑖𝑖× 𝐵𝑅𝑟 ∀ 𝑟 ∈ 𝑅 (3.7) ∑∑𝑄𝑀𝐶𝑚𝑐𝑖𝑖𝑚≤ ∑𝐶𝐶𝑐𝑖𝑖× 𝐵𝐶𝑐 ∀ 𝑐 ∈ 𝐶 (3.8) ∑∑𝑄𝑆𝑅𝑠𝑟𝑖𝑖𝑟≤ ∑𝐶𝑆𝑠𝑖𝑖× 𝐵𝑆𝑠 ∀ 𝑠 ∈ 𝑆 (3.9) ∑∑𝑄𝑅𝐷𝑟𝑑𝑖𝑖𝑟≤ ∑𝐶𝐷𝑑𝑖𝑖× 𝐵𝐷𝑑 ∀ 𝑑 ∈ 𝐷 (3.10) ∑∑𝑄𝑅𝐷𝑟𝑑𝑖𝑖𝑑= ∑∑𝑄𝐶𝑅𝑐𝑟𝑖𝑖𝑐+ ∑∑𝑄𝑆𝑅𝑠𝑟𝑖𝑖𝑠 ∀ 𝑟 ∈ 𝑅 (3.11) 𝑙𝑐 , 𝑚𝑟, 𝑛𝑑 ∈ {0,1} ∀ 𝑐, 𝑟, 𝑑 (3.12) 𝑒𝑐𝑟𝑖, 𝑓𝑟𝑑𝑖 , 𝑔𝑑𝑚𝑖, ℎ𝑚𝑐𝑖 , 𝑗𝑠𝑟𝑖 , 𝑘𝑐𝑠𝑖 > 0 ∀ 𝑐, 𝑟, 𝑑, 𝑚, 𝑐, 𝑠 (3.13) Equation (3.1) represents the objective function to minimize the total costs. The costs of CLSC networks include the costs of associates to suppliers, the fixed cost of opening distribution centers, recovery centers, and manufacturers, operational costs, transportation costs, and shortage costs while considering the cost savings from the recycling process. CLSC revenues are earned by 35 selling new and repaired products, recovered material, and carbon credits. Constraints (3.2) ensure that the quantity of products dispatched from retailers to demand markets meets or exceeds customer demand, accounting for potential shortages if the market demand cannot be fully satisfied. Constraints (3.3) indicate that the number of returned products sent from the markets to the recovery centers should equal the total number of products sent from the recovery centers to the incineration facility and manufacturers. Constraints (3.4) state that the total number of products sent from the markets to the recovery centers must equal the number of returned products, implying that the inputs and outputs should be balanced. Constraints (3.5) ensure that the forward flow from the distributor to the demand market equals the forward flow from the manufacturer to the distributor, ideally with no losses during transportation. Constraints (3.6) consider a minimum disposal fraction for the returned products. It restricts the number of products returned from the market to the collection centers to be greater than or equal to the quantity of products sent to the disposal centers, as some products will be directed to the plants for recovery. Constraints (3.7), (3.8), (3.9), and (3.10) represent the capacities of the corresponding facilities: manufacturers, recovery centers, distribution centers, and suppliers. If a facility is located, the binary variables 𝐵𝐶𝑐, 𝐵𝑅𝑟 , 𝐵𝐷𝑑, and 𝐵𝑆𝑠 will be 1. Constraints (3.11) indicate that the quantity of items sent from suppliers and collection centers to manufacturers should be greater or equal to the number of items shipped to distributors. This process can be summarized as the balance between inflow and outflow. The decision variables mainly consist of two categories, which are the quantity variables and the binary variables. Constraints (3.12) are binary constraints that limit the value of the decision variables associated with manufacturers, distribution centers, and recovery centers to 0 or 1. Constraints (3.13) imply that the quantity of the products in both forward and backward flows must be non-negative. 36 3.4 Application of The Proposed Model and Solution Approach The application of the proposed MILP model will be tested on a real-world case study with real locations in Vancouver, Canada. Twenty-two areas in Vancouver are considered to be in the demand market. Table 3.1 shows that five suppliers, six plants, seven retailers, ten collection centers, and one disposal center have been considered. Google Maps has been utilized to calculate the distances among the echelons. Similar to Tosarkani et al. (2024), the demand value of the product in the market has been assumed to be one percent of the population of each municipality region based on the 2011 census of Canada. The return value of the market is also ten percent of each market demand. The transportation costs have been unified as 0.5412 per kilometer, and the fixed costs for operating the facilities have also been set to 2000. Table 3.1 Values of Parameters Applied to Solve the Proposed Model Index Definition Number I Number of plastic items or products 3 S Number of Suppliers 5 R Number of manufacturers 6 M Number of market or demand points 22 D Number of distribution centers 7 C Number of recovery centers 10 F Number of Incineration Facilities 1 The mathematical model is developed using Python within the Pyomo optimization environment and solved using the Gurobi solver. It is solved in 0.11 seconds. There are 314 constraints, 1549 single variables, 28 binary variables, and 7074 non-zero coefficients. Upon solving the deterministic model, the optimal objective function value, representing total costs, is 716,288.37 units. Furthermore, the proposed CLSC network selects three manufacturers, three suppliers, four collection centers, and four distribution centers to satisfy the deterministic demand. 37 3.5 Sensitivity Analysis This section analyzes the impact of input parameters—market demand and return rates—on the overall configuration of the CLSC network and the value of the objective function. A scenario-based analysis is utilized to assess how variations in the value of demand and return impact the total costs of the proposed plastic CLSC network. Scenario 1 assumes the same market demand and returned product values as the deterministic model in 3.4. Every possible combination involving a 10 percent adjustment in market demand and returned products is examined. As presented in Table 3.2, the outcomes are compared to those of Scenario 1. Table 3.2 Comparison of Nine Scenarios of Deterministic Model Scenarios Demand Return Total Expected Cost Change % #R #S #C #D 1 (Base) No change No change 716,288 0.00% 3 3 4 4 2 10% increase 10% decrease 789,430 10.21% 4 3 4 4 3 10% decrease 10% increase 650,129 -9.24% 3 2 4 4 4 No change 10% increase 728,268 -1.67% 3 3 4 4 5 No change 10% decrease 704,309 1.67% 3 3 4 4 6 10% increase No change 801,409 11.88% 4 3 4 4 7 10% decrease No change 638,340 -10.88% 3 2 4 4 8 10% increase 10% increase 813,389 13.56% 4 3 4 4 9 10% decrease 10% decrease 627,158 -12.44% 3 3 4 4 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors The ‘Change’ column in Table 3.2 indicates the variation in the objective function for each scenario relative to the baseline scenario. The numerical results in these tables underline that the total costs of the plastic CLSC’s network are notably sensitive to changes in market demand and 38 returned products. A 10 percent increase in market demand corresponds to an 11.88 percent increment in CLSC costs. In addition, the number of manufacturers has increased to meet the market demand. A 10 percent reduction in demand results in a 10.88 percent decrease in CLSC costs. Notably, the influence of market demand changes on CLSC costs surpasses that of returned products. Figure 3.2 visualizes the status of the facilities for each scenario. Across nine different scenarios, five distinct configurations are developed based on the deterministic model for the CLSC network. Figure 3.2a illustrates scenarios 1, 4, 5, and 9, where demand remains unchanged or decreases by 10%. It can be observed from the other images that increments in demand significantly impact the operational facilities, with a 10% increase in demand leading to a drastic change in the facility configurations. For instance, as shown in Figure 3.2b and Figure 3.2d, although scenarios 2, 6, and 8 have the same number of facilities, their locations differ: scenario 2 places a facility in the east, while scenarios 6 and 8 place a facility in the north. Figure 3.2c and Figure 3.2e depict scenarios 3 and 7, which are independent and do not share similarities with the other scenarios. 39 a. Scenario 1, 4, 5, 9 b. Scenario 2 c. Scenario 3 d. Scenario 6, 8 a. Scenario 7 Figure 3.2 The Configuration of the Plastic CLSC Network for Different Scenarios 40 3.6 Conclusion and limitations Chapter 3 presented a deterministic optimization model for a multi-echelon plastic CLSC, which aimed to maximize the profit of the SC by determining the optimal locations and quantities of facilities and products. The model incorporated various echelons, including suppliers, manufacturers, retailers, recovery centers, and disposal centers, and was formulated as a MILP problem to minimize the total costs incurred in the SC while meeting the market demand and handling the reverse flow of returned plastic products. The proposed MILP model was demonstrated through a real-world case study in Vancouver, Canada, involving twenty-two demand markets, five suppliers, six plants, seven retailers, ten recovery centers, and one disposal center. The model was efficiently solved using the Gurobi solver, demonstrating its practical applicability with a solution time of 0.11 seconds. Sensitivity analysis was conducted to evaluate the impact of different parameters on the CLSC network. The analysis revealed that the network’s total costs are notably sensitive to changes in market demand and returned products. A vital limitation of the deterministic model is its inability to effectively address various uncertainties, including fluctuating demands, unexpected disruptions, and variations in market conditions. Operating under the assumption of perfect information, the model tends to overlook the dynamic and unpredictable nature of the SC environment. Consequently, decisions from the deterministic model may result in infeasible or sub-optimal solutions. Therefore, optimization under uncertainty approaches is needed for the proposed MILP model to deal with uncertainty and improve the adaptability of the CLSC network. A data-driven optimization framework is proposed in the next chapter to deal with uncertainty in different parameters (i.e., demand values, return rate, and disposal fraction). 41 Chapter 4: Data-Driven Approach for A Plastic Closed-Loop Supply Chain In this chapter, a balanced DDRO approach is developed to address the uncertainty of the proposed MILP model in Chapter 3:. First, a data-driven optimization method is proposed. Then, the DDRO framework is extended to enhance balance and equity across various dimensions. Subsequently, a data-driven algorithm is introduced to compare different uncertainty sets regarding balance and equity. Finally, the numerical results are presented. 4.1 An overview of data-driven robust optimization approach Han et al. (2021a) developed a SKL- and MKL-based OC-SVM approach for parameter uncertainty. The strength of this approach lies in its ability to construct compact uncertainty sets, leading to a tractable robust formulation. This section explains the methodology proposed by Han et al. (2021a) is explained. OC-SVM estimates support for unknown distribution of historical observations {𝑑𝑖|𝑖 = 1,… ,𝑁} (Han et al., 2021b; Schölkopf and Smola, 2001). The dual formulation of the OC-SVM is as follows: minα ∑∑αi. αj. K(di, dj)𝑁𝑗=1𝑁𝑖=1𝑠. 𝑡. 0 ≤ αi ≤1Nv, i = 1,2,3, … , N∑αi𝑁𝑖=1= 1 (4.1) Dual variable 𝛼 in (4.1) determines the condition of each historical observation 𝑑𝑖. In Model (4.1), N is equal to the number of available historical observations. Regularization parameter 𝑣 ∈[0,1) also determines the fraction of data covered by the uncertainty set. Upon solving the OC-42 SVM model (4.1), Support Vectors (SVs), Boundary Support Vectors (BSVs), and Inlier Data points (IDs) are defined according to Eqs. (4.2) to (4.4). SV = {𝑑𝑖|α𝑖∗ > 0, ∀ i = 1,… , N} (4.2) 𝐵𝑆𝑉 = {𝑑𝑖|0 < α𝑖∗ <1𝑁𝑣 , ∀ 𝑖 = 1,… , N} (4.3) ID = {𝑑𝑖|α𝑖∗ = 0, ∀ i = 1,… , N} (4.4) SVs are data points that lie on the boundary or outside the uncertainty set. BSVs also refer to those data samples that lie approximately on the boundary of the uncertainty sets. IDs are also the data samples that are encompassed by the uncertainty set. K(. , . ) in (4.1) is equal to ∑ 𝜋𝑚. 𝑘𝑚𝑚 , where 𝜋𝑚 is the coefficient of the basis kernel matrix 𝑘𝑚 . A piecewise linear concave kernel function is used to construct the uncertainty set and define each basis kernel matrix as (4.5): 𝐾𝑚(𝑑𝑖, 𝑑𝑗) = 1 − |𝑞𝑚𝑇 (𝑑𝑖 − 𝑑𝑗)𝑐𝑚. 𝑘| ∀ i, j = 1,… , N, ∀ m = 1,… ,M (4.5) Each kernel function differs in values of 𝑞𝑚, 𝐶𝑚 , 𝑎𝑛𝑑 𝑘. Normalization factor 𝐶𝑚 determines dispersion degree of data along with projection direction 𝑞𝑚. It is calculated as Eq. (4.6): 𝐶𝑚 = 𝑚𝑎𝑥𝑖𝑒{𝑞𝑚𝑇 (𝑑𝑖)} − 𝑚𝑎𝑥𝑖1−𝑒{𝑞𝑚𝑇 (𝑑𝑖)} (4.6) In Eq. (4.6), 𝑚𝑎𝑥𝑖𝑒{} indicates the maximum value that is merely less than 𝑒% of the set {.}. The value of e should be as low as 0.01. In Eq. (4.5), k is a scale factor. The positive definiteness of the all kernel matrices depend on the value of k to be sufficiently large. The value of k is calculated in Eq. (4.7): 𝑘 = 𝑚𝑎𝑥𝑚=1,…,𝑀 𝑆𝑚𝐶𝑚 , 𝑆𝑚 = 𝑚𝑎𝑥𝑢𝑖,𝑢𝑗∈𝐷 |𝑞𝑚𝑇 (𝑢𝑖 − 𝑢𝑗)| (4.7) 43 Afterwards, the only thing that needs to be done is the construction of eigenvectors through the covariance matrix presented in Eq. (4.8) (Shang et al., 2017c). 𝐶𝑜𝑣 =1𝑁 − 1[∑𝑢𝑖(𝑢𝑖)𝑇− (∑𝑢𝑖𝑁𝑖=1)(∑𝑢𝑖𝑁𝑖=1)𝑇𝑁𝑖=1] (4.8) After obtaining the COV matrix, the eigenvectors of this matrix are extracted as direction vectors 𝑞𝑚 and the kernel matrix of these vectors is calculated according to Eq. (4.5). Then the kernel matrix coefficients 𝜋𝑚 become equal to 1/M, and the final kernel matrix is calculated according to ∑ 𝜋𝑚. 𝑘𝑚𝑚 . Finally, the SKL-based uncertainty set is built by the following equation: 𝑈(𝐷) = {𝑑𝑖| ∑ α𝑖∗ ∑ 𝜋𝑚∗𝑘∈𝑆𝐾 |𝑞𝑚𝑇 (𝑑𝑖 − 𝑑𝑗)𝑐𝑚. 𝑘| ≤ 1 − 𝑝∗𝑖∈𝑆𝑉 } (4.9) The value of 𝑝∗ is also determined by Eq. (4.10). 𝑝∗ =1|𝐵𝑆𝑉|∑ ∑ α𝑖∗𝑖∈𝑆𝑉∑ 𝜋𝑚∗ . 𝐾𝑚(𝑑𝑖, 𝑑𝑗)𝑚∈𝑆𝑘𝑗∈𝐵𝑆𝑉 (4.10) 4.1.1 Robust counterpart model To demonstrate the computational tractability of the MKL-based uncertainty set, consider the RO problem as Model (4.11). min𝑥 𝐶𝑇 . 𝑥 𝑎𝑇𝑥 ≤ 𝑏 ∀ 𝑎 ∈ 𝑈(𝐷) (4.11) In model (4.11), C is the coefficient of the objective function, x is the decision variable, and 𝑎 is the left-hand side (LHS) coefficient matrix of the model, which is uncertain. This constraint can be presented as the worst-case scenario: 44 𝑚𝑎𝑥𝑎∈𝑈(𝐷) 𝑎𝑇 . 𝑥 ≤ 𝑏 (4.12) To eliminate the 𝑙1-norms in the uncertainty set {𝑈(𝐷)}, the auxiliary variables {θ𝑖𝑚} are introduced with the uncertainty set (4.9), where the uncertainty set is formed in the form of a series of linear inequalities. Therefore, the LHS coefficients of the Eq. (4.12) are demonstrated in the form of the linear programming Model (4.13): 𝑚𝑎𝑥𝑑,Θ 𝑑𝑇 . 𝑥 𝑠. 𝑡. − θ𝑖𝑚 ≤𝑞𝑚𝑇(𝑑 − 𝑑𝑖)𝑐𝑚. 𝑘≤ θ𝑖𝑚, ∀ 𝑖 ∈ 𝑆𝑉, 𝑘 ∈ 𝑆𝐾 ∑ α𝑖∗𝑖∈𝑆𝑉∑ 𝜋𝑚∗ . θ𝑖𝑚 ≤ 1 − 𝑝∗ 𝑚∈𝑆𝑘 (4.13) The dual problem can be reformulated as problem (4.14) by introducing the dual variables 𝜇 and 𝜆: ∑ ∑ (𝜇𝑖𝑚 − 𝜆𝑖𝑚) ×𝑞𝑚𝑇(𝑑𝑖)𝑐𝑚. 𝑘𝑘∈𝑆𝐾𝑖∈𝑆𝑉 + 𝜂 × (1 − 𝑝∗) ≤ 𝑏 ∑ ∑ (𝜇𝑖𝑚 − 𝜆𝑖𝑚) ×𝑞𝑚𝑐𝑚. 𝑘𝑘∈𝑆𝐾𝑛∈𝑆𝑉= 𝑥 𝜇𝑛𝑘 + λ𝑛𝑘 = 𝜂 × α𝑖∗ × 𝜋𝑚∗ ∀ 𝑛 ∈ 𝑆𝑉, 𝑘 ∈ 𝑆𝐾 𝜇𝑛𝑘, 𝜆𝑛𝑘 ≥ 0 ∀ 𝑛 ∈ 𝑆𝑉, 𝑘 ∈ 𝑆𝐾 𝜂 ≥ 0 (4.14) Therefore, the robust counterpart of Eq. (3.2) can be replaced with Eq. (4.15) to account for uncertainty in demand parameter. ∑ ∑ (𝜇𝑛𝑘𝑚𝑖 − 𝜆𝑛𝑘𝑚𝑖) × ℎ𝑑𝑛𝑘 + 𝜂 × (1 − 𝑝∗)𝑘∈𝑆𝐾𝑛∈𝑆𝑉≤ ∑𝑄𝐷𝑀𝑑𝑚𝑖𝑑+ 𝑆𝐻𝑚𝑖 ∀ 𝑚 ∈ 𝑀, 𝑖 ∈ 𝐼 45 ∑ ∑ (𝜇𝑛𝑘𝑚𝑖 − 𝜆𝑛𝑘𝑚𝑖) × ℎ𝑘𝐾∈𝑆𝐾𝑛∈𝑆𝑉= 1 ∀ 𝑚 ∈ 𝑀, 𝑖 ∈ 𝐼 𝜇𝑛𝑘𝑚𝑖 + 𝜆𝑛𝑘𝑚𝑖 = 𝜂 × 𝛼𝑛∗ × 𝜋𝑘∗ , ∀ 𝑛 ∈ 𝑆𝑉 , 𝑘 ∈ 𝑆𝐾 𝜇𝑛𝑘𝑚𝑖, 𝜆𝑛𝑘𝑚𝑖 ≥ 0 ∀ 𝑛 ∈ 𝑆𝑉 , 𝑘 ∈ 𝑆𝐾 𝜂 ≥ 0 (4.15) 4.1.2 The overall procedure of DDRO approach The overall framework for constructing data-driven uncertainty sets using SKL-based OC-SVM approach is provided in Fig. 4.1. Figure 4.1 Overall Procedure of the DDRO Approach to Address Parameter Uncertainty 46 As shown in Fig. 4.1, the first step is collecting historical observations of uncertain parameters. Next, the desired conservatism degree should be tuned. As mentioned earlier, the conservatism degree is tuned by regularization parameter 𝑣 which controls the fraction of data covered by the uncertainty set. Next, the covariance matrix of data is calculated using Eq. (4.8). Following that, the eigenvectors of the covariance matrix is calculated and considered as direction vectors. Afterward, the values of 𝐶𝑚 and 𝑘 are calculated using Eqs. (4.6) and (4.7). Subsequently, calculated basis kernel for each direction vector using Eq. (4.5). In the following, set an equal weight to each basis kernel as 𝜋𝑚 = 1/𝑀. Then, calculate the final kernel matrix using Eq. (4.16). 𝐾 = ∑𝜋𝑚. 𝑘𝑚𝑚 (4.16) Upon calculation of the final kernel matrix (K), the SKL-based OC-SVM model will be solved to obtain the value of 𝛼𝑖. The next step involves determining SVs, BSVs, and IDs. Afterward, the value of 𝑝 is calculated using Eq. (4.10). Then, the uncertainty set will be constructed, and robust counterpart models will be developed. Finally, upon solving the robust counterpart models, robust solutions will be obtained. 4.2 OC-SVM with Kernel Weight Adjustment With the SKL-based OC-SVM approach, each basis kernel matrix corresponds to a specific direction vector. Additionally, all basis kernel matrices are equally weighted. Using this approach, the dispersion degree along with each direction vector is ignored when weighting kernel matrices and creating uncertainty sets. This may lead to an inaccurate representation of the uncertainty set. This is because when reducing the conservative degree, the reduction rate of uncertainty sets in 47 different directions is not based on the dispersion degree and the data range along with each direction vector. In this section, a new approach is developed as OC-SVM with KWA. For this purpose, the dispersion factor of each direction vector {𝐶𝑚} (Eq. (4.6)) is employed. In the dispersion factor {𝐶𝑚}, all data samples are projected into each direction vector {𝑞𝑚}.Afterwards, the maximum and minimum values of projected data will be determined with respect to the 𝑒% quantile that disregards outliers. The distance between maximum and minimum values determines the dispersion degree along with each direction vector. After dispersion degree calculations, the inversed value of dispersion degrees is defined using Eq. (4.17): 𝐶𝑚𝑖𝑛𝑣 = 1/𝐶𝑚∀𝑚 = 1,… ,𝑀 (4.17) The reason behind using inverse values in the weighting of kernel matrices is to ensure that matrices with smaller ranges receive greater weight. This adjustment is crucial for equalizing the rate at which the boundary of the uncertainty set is reduced from each direction. Finally, the weight of each basis kernel matrix is calculated using Eq. (4.18): 𝜋𝑚 =𝐶𝑚𝑖𝑛𝑣∑ 𝐶𝑚𝑖𝑛𝑣𝑀𝑚=1 (4.18) 4.3 Fairness Index This research proposes an improvement for the OC-SVM approach. The uncertainty sets derived from the OC-SVM approach and its improved version (OC-SVM with KWA) are compared regarding their effectiveness in handling uncertainty within the proposed CLSND problem. The proposed analysis compares the constructed uncertainty sets toward a specific direction. To further evaluate the capability of the enhanced OC-SVM approach in different 48 directions, a Robust Fairness (RF) index is proposed in this section. The overall idea of the RF index is to calculate the distance of all SVs from the boundary of the uncertainty set. The aggregation of the distance of all SVs from the uncertainty boundary determines the RF index of that uncertainty set. A lower total distance of SVs from the boundary of the uncertainty set indicates a more balanced uncertainty set. A balanced uncertainty set proposed in this research has approximately identical boundary reductions in various directions. It allows for a more robust and equitable handling of uncertainties, ensuring that the proposed uncertainty set results in the same level of conservatism for different optimization under uncertainty problems. The proposed RF index can also be used to compare different DDRO approaches with the same level of conservatism in terms of constructing balanced uncertainty sets. The RFIC algorithm (Algorithm 1) is developed to calculate the cumulative distance of all SVs from the boundary of the uncertainty set. The first step of the RFIC algorithm is to detect outliers and remove them from the set of SVs using lines 1 and 2 of the RFIC algorithm. The next step is to calculate each SV's distance from the uncertainty set's boundary. Using lines 5 and 6 of the RFIC algorithm, a direction vector is generated along the direction from the centroid of all samples to the SV. Following this, a set of points is created, starting from the centroid of the data samples and extending towards the scenario vector (SV). The score value {S} of each generated point is then calculated. The point with a score closer to 0 is selected as the boundary point. Eq. (4.19) can be used to determine the situation of a new data sample within the uncertainty set. In Eq. (4.19), the 𝜌∗ is calculated using Eq. (4.4) and specifies the average score value of BSVs the lie on the boundary of the uncertainty set. d in Eq. (4.19) also is the point whose score value {𝑆} is calculated. Suppose the value of S is approximately close to 0. In that case, it means that the point is situated near the boundary of the uncertainty set. 49 𝑆 = ∑ ∑ 𝛼𝑖∗. 𝜋𝑚. 𝐾𝑚(𝑑, 𝑑𝑖)𝑖∈𝑆𝑉𝑚∈𝑆𝐾− 𝜌∗ (4.19) Subsequently, the Euclidean distance of each SV from the boundary point is calculated. Finally, the cumulative distance of all SVs is calculated and considered as the RF index of that uncertainty set. The RFIC algorithm can calculate the RF index for various uncertainty sets with the same level of conservatism. Table 4.1 Robust Fairness Index Calculator (RFIC) Input: Dataset 𝐷𝑆𝑉, 𝑃𝑚𝑒𝑎𝑛, Regularization parameter 𝑣, 𝑠𝑡𝑒𝑝 1. Apply SKL-based OC-SVM approach on dataset 𝐷𝑆𝑉 using 𝑣 2. Determine SVs using Eq. (4.17) and exclude them from dataset 𝐷𝑆𝑉 3. 𝐷𝐼𝑆𝑠𝑣 ← 0 4. for 𝑑𝑠𝑣𝑖𝑛 𝐷𝑆𝑉 do: 5. ?⃗? ← 𝑑𝑠𝑣 − 𝑃𝑚𝑒𝑎𝑛 6. 𝑞 ← ?⃗? /|?⃗? | 7. 𝐶𝑜𝑢𝑛𝑡𝑒𝑟 ← 0 8. while 𝐶𝑜𝑢𝑛𝑡𝑒𝑟 == 0 do: 9. Ṗ ← 𝑃𝑚𝑒𝑎𝑛 + 𝑞 × 𝜏 10. 𝜏 ← 𝜏 + 𝑠𝑡𝑒𝑝 11. Calculate the value of 𝑆 for point Ṗ using Eq. (4.19) 12. if 𝑆 < 𝑒𝑝𝑠 do: 13. 𝐶𝑜𝑢𝑛𝑡𝑒𝑟 ← 1 14. 𝐵𝑃 ← Point 15. end if 16. end while 17. 𝑑𝑖𝑠𝑠𝑣 ← ‖𝑑𝑠𝑣 − 𝐵𝑃‖ 18. 𝐷𝐼𝑆𝑠𝑣 ← 𝐷𝐼𝑆𝑠𝑣 + 𝑑𝑖𝑠𝑠𝑣 19. end for Output: mean and SD of distance all SVs from the boundary of the uncertainty set 50 A lower RF value indicates a more balanced uncertainty set, which signifies that the uncertainty set is more uniformly constructed, resulting in shorter distances with SVs in all directions. This uniformity ensures that the uncertainty set is evenly distributed around its centroid, enhancing its robustness and reliability in representing uncertainty. Figure 4.2 also illustrates the different steps involved in calculating the distance of one SV from the boundary of the uncertainty set. a. Detecting SVs b. Determining Direction Vector a. Identifying Boundary Point b. Calculating Euclidean Distance Figure 4.2 Procedure for Calculating the Distance of Each SV from the Uncertainty Set Boundary Figure 4.2.a shows the first step, determining the SVs and identifying the SV whose distance to the boundary of the uncertainty set needs to be calculated. Figure 4.2.b displays the direction vector connecting the center of all data points to that SV. Figure 4.2.c shows the calculated boundary 51 point on that direction vector. Finally, Fig. 4.2.d depicts the Euclidean distance of the SV from the boundary of the uncertainty set. 4.4 Numerical Results of the DDRO Approach This section provides the numerical results of employing proposed DDRO approaches, SKL-based OC-SVM, and OC-SVM with KWA, which were proposed in this research to deal with uncertainty and design a robust CLSN for plastic bottles. In this section, synthetic data, such as historical data related to uncertain parameters, is generated due to the unavailability of real-world data. Therefore, this section explains the data generation process to generate random data from historical observations of uncertain parameters. Following the impact of conservatism degrees on the optimal CLSN and objective function of the optimization models is investigated using both OC-SVM and OC-SVM with KWA approaches. Following that, both OC-SVM and OC-SAM approaches are compared in terms of creating balanced uncertainty sets. Finally, a scenario-based analysis is provided to assess the resiliency of the resulting CLSN when disruptions occur and provide some managerial insights. The optimization model and DDRO approaches are coded in Python using Gurobipy solver, and all numerical results are run on a computer with Intel(R) Core i9-13980HX CPU 2.2GHz and 32.0 GB RAM personal computer. 4.4.1 Methodology for Synthetic Data Generation Historical data related to an uncertain parameter should be available to employ the proposed data-driven approach to dealing with parameter uncertainty. However, accessing real-world historical data about uncertain parameters could be challenging. As a result, this section explains 52 the data generation process that generates random data as historical observations of uncertain parameters for subsequent analysis and numerical results. To generate historical data for the DDRO approach to dealing with parameter uncertainty, we employ a method like the one proposed by Goerigk and Kurtz (2023) in their work. We use the Scikit-learn library to generate random covariance matrices. Using the generated covariance matrices, we generated mixed multivariate average data. The meaning of all generated data is then adjusted to represent the deterministic value of all uncertain parameters. The mean values of all uncertain parameters can be found in the deterministic section for their nominal values. The distribution of the generated samples is flexible, preferably elliptical or circular. This adaptability ensures that the synthetic data can be effectively used in various scenarios. The synthetic data produced through this methodology serves as the foundation for evaluating diverse scenarios and assessing the efficacy of different strategies within CLSND. It enables informed decision-making and strategy formulation to pursue a circular economy for plastics by accurately capturing the variability and complexities of real-world SC dynamics. 4.4.2 The Impact of Disposal Fraction on the Overall Configuration of the CLSC This section analyses the impact of the conservatism degree of the DDRO approach on the overall configuration of the CLSN and the optimal objective value of the objective function. To conduct this analysis, we generate 500 data samples as historical observations of the uncertain disposal fraction, ranging from 0.01 to 0.45, using Gaussian distributions described in Section 4.5.1. We consider nine different values for the conservatism degree (regularization parameter 𝑣) as different scenarios for this experiment. Additionally, we establish a base scenario with a value 53 of 0.01 for the regularization parameter 𝑣, which covers almost 99% of all data samples within the data-driven uncertainty set. Subsequently, we construct data-driven uncertainty sets for each scenario to address the uncertain disposal fraction parameter for the CLSN explained in section 3. We compare the objective value of all scenarios with the base scenario. The numerical results of this experiment are provided in Table 4.1. As shown in Table 4.1, the optimal objective value of the SCN optimization model experiences a significant decrease of 6.52% when the value of the regularization parameter 𝑣 increases from 0.01 to 0.9. This change in the regularization parameter directly impacts the data-driven uncertainty set's conservatism degree and the uncertainty set's fraction of data coverage. Consequently, a higher value of v leads to a lower value of the disposal fraction, resulting in more returned products being used in the recycling process and producing new plastic products. This, in turn, reduces the total costs of the SCN, leading to a lower objective value. It is essential to note that while a larger value of the regularization parameter can lead to cost reductions, it may also result in infeasible solutions. Therefore, the conservatism degree should be tuned carefully based on the decision-maker's preferences. This decision involves balancing the trade-off between the total costs of the SCN and the feasibility of the robust solution, a crucial aspect of the analysis. 54 Table 4.1 Comparison of Nine Scenarios of Deterministic Model Regarding Disposal Fraction Scenarios Disposal Regularization Total Expected Cost Change % R S C D 1 (Base) 0.01 1,968,068 0.00% 6 4 5 7 2 0.1 1,935,513 -1.65% 6 4 5 7 3 0.2 1,920,643 -2.41% 6 4 5 7 4 0.3 1,904,158 -3.25% 6 4 5 7 5 0.4 1,896,296 -3.65% 6 4 4 7 6 0.5 1,888,738 -4.03% 6 4 4 7 7 0.6 1,883,121 -4.32% 6 4 4 7 8 0.7 1,869,942 -4.99% 6 4 4 7 9 0.8 1,861,957 -5.39% 6 4 4 7 10 0.9 1,839,674 -6.52% 6 4 4 7 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors The expected total cost in the deterministic model is consistently lower than or equal to the model in all scenarios except when DF=1 during the analysis of the disposal fraction. This difference is because the deterministic model relies on exact demand and return values, making it less adaptable to real-world situations. In contrast, the DDRO approach, with its ability to create a data-driven uncertainty set that covers all possible future values of uncertain parameters and deals with the worst-case scenario within the uncertainty set, is more adaptable to real-world situations and changes in the value of uncertain parameters. This adaptability of the DDRO approach is a key advantage, as it ensures that suboptimal or infeasible solutions are not generated in scenarios where fluctuations and uncertainties are present. The optimal value of the deterministic model is lower than all the scenarios in Table 4.1. However, the solution of the deterministic model may result in infeasible solutions. 55 4.4.3 Impact of Conservatism Degree of the Uncertain Demand and Return Parameters This section delves into an in-depth examination of the impact of the regularization parameter 𝑉 (conservatism degree) on the optimal objective value and configuration of the CLSC network. For this experiment, historical data for uncertain parameters such as demand and return rates were generated to facilitate the DDRO approach to addressing parameter uncertainty within CLSC networks. The methodology explained in Section 4.5.1 is employed to generate data as historical observations of uncertain parameters. Table 4.2 presents the numerical results assessing the impact of the regularization parameter on uncertain demand, highlighting its effect on the optimal configuration of the CLSC and the total costs. The numerical results of Table 4.2 highlight the pivotal role of the conservatism degree of the uncertain demand in the optimal configuration of the CLSC network. Adjusting the conservative degree of the uncertain demand parameter from 0.01 to 0.9 reveals significant variations in total expected costs and facility configurations. As the regularization parameter increases, reflecting a reduced degree of conservatism in the demand parameter, there is a consistent decrease in the total expected costs within the SC network. However, it is essential to note that adopting lower values for uncertain demand parameters increases the likelihood that the optimal solution becomes infeasible. The conservatism degree decreases as the regularization parameter increases in the SKL- or MKL-based OC-SVM approach due to the fraction of data covered by the uncertainty set equals (1 − 𝑣) × 100% of the data. Therefore, the lower the value of 𝑣 , the more data the uncertainty set covers, increasing the conservatism degree. By decreasing the conservatism degree, the total expected costs will be reduced. However, the trade-off increases the probability of the solution being infeasible. The 56 percentage change in total expected cost, as shown in Table 4.2, ranges from 14.37% to 42.07% as the conservatism degree of demand parameter increases, reinforcing the sensitivity of the optimization model to adjust the conservatism degree. The impact of the conservatism degree on the uncertain demand parameter extends beyond cost reduction to facility optimization. The regularization parameter influences the robust values of uncertain demand. As this parameter decreases, there is a corresponding increase in the values assigned to demand parameters. This escalation may necessitate opening new facilities, mainly when existing facility capacities limit the ability to meet customer demands. Table 4.2 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand for DDRO Scenarios Demand Regularization Return Regularization Total Expected Cost Change % R S C D 1 (Base-Case) 0.01 0.01 1,914,377 0.00% 6 4 5 7 2 0.1 0.01 1,639,277 -14.37% 5 4 5 7 3 0.2 0.01 1,546,065 -19.24% 5 4 5 6 4 0.3 0.01 1,496,966 -21.80% 5 4 5 6 5 0.4 0.01 1,417,036 -25.98% 5 4 5 6 6 0.5 0.01 1,372,973 -28.28% 5 4 5 6 7 0.6 0.01 1,309,679 -31.59% 5 4 5 6 8 0.7 0.01 1,246,239 -34.90% 4 4 5 5 9 0.8 0.01 1,175,083 -38.62% 4 4 5 5 10 0.9 0.01 1,108,959 -42.07% 4 4 5 5 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors Like the demand parameter, the regularization value of return also influences the optimization outcomes of DDRO models. As the regularization value of the return parameter varies from 0.01 57 to 0.9, a consistent decrease in total expected costs is observed across all scenarios. This decline in expenses indicates the efficacy of regularization value of return in enhancing SC efficiency and reducing operational expenses. As shown in Table 4.3, the percentage change in total expected cost ranges from 0.88% to 3.44% with increasing conservatism degrees or regularization values, emphasizing its significant impact on cost optimization within the DDRO framework. The impact of the conservatism degree on the uncertain parameter is more significant than the return parameter, as the change in the number of facilities and increase in production would be the main factor of costs. Meanwhile, virgin plastic produces new products instead of recycled plastics that have been returned from the market. Table 4.3 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Return for DDRO Scenarios Demand Regularization Return Regularization Total Expected Cost Change % R S C D 1 (Base) 0.01 0.01 1,914,377 0.00% 6 4 5 7 2 0.01 0.1 1,897,520 -0.88% 6 4 5 7 3 0.01 0.2 1,890,436 -1.25% 6 4 4 7 4 0.01 0.3 1,882,758 -1.65% 6 4 4 7 5 0.01 0.4 1,876,862 -1.96% 6 4 4 7 6 0.01 0.5 1,872,872 -2.17% 6 4 4 7 7 0.01 0.6 1,866,805 -2.48% 6 4 4 7 8 0.01 0.7 1,862,269 -2.72% 6 4 4 7 9 0.01 0.8 1,855,047 -3.10% 6 4 4 7 10 0.01 0.9 1,848,535 -3.44% 6 4 4 7 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors The simultaneous adjustments of demand and regularization value of return parameters from 0.01 to 0.9 reveal significant changes in the total costs and overall configuration of the plastic 58 CLSN. This simultaneous tuning of regularization parameters offers insights into their collective influence on SC optimization. The interaction between demand and regularization value of return parameters is evident in the notable variations in total expected costs across all scenarios. With increasing values of both regularization parameters, there is a consistent decrease in total expected costs, indicative of enhanced SC efficiency and cost reduction. The percentage change in total expected cost, as shown in Table 4.4, ranges from 15.26% to 45.53% as demand and regularization value of return values increase, underscoring the compounded effect of simultaneous adjustments on cost optimization within the DDRO framework. Table 4.4 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand and Return for DDRO Scenarios Demand Regularization Return Regularization Total Expected Cost Change % R S C D 1 (Base-Case) 0.01 0.01 1,914,376 0.00% 6 4 5 7 2 0.1 0.1 1,622,246 -15.26% 5 4 5 7 3 0.2 0.2 15,22,247 -20.48% 5 4 5 6 4 0.3 0.3 1,465,647 -23.44% 5 4 5 6 5 0.4 0.4 1,379,802 -27.92% 5 4 5 6 6 0.5 0.5 1,331,680 -30.44% 5 4 4 6 7 0.6 0.6 1,262,187 -34.07% 5 4 4 6 8 0.7 0.7 1,194,100 -37.62% 4 4 4 5 9 0.8 0.8 1,115,598 -41.73% 4 4 4 5 10 0.9 0.9 1,042,829 -45.53% 4 4 4 5 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors 59 Figure 4.3 visualizes the status of the facilities by implementing VeRoViz (L. Peng and Murray, 2022). It reveals 5 different types of facility layouts even when some scenarios have similar numbers of facilities with the number of the scenarios at the upper left corner of each image. Figure 4.3 DDRO Deterministic Model Visualizations 4.4.4 SKL-based OC-SVM with KWA Approach This section presents a comprehensive comparative analysis between SKL-based OC-SVM and SKL-based OC-SVM with KWA, highlighting the significant enhancements introduced in the latter. The main objective of this analysis is to evaluate the impact of varying demand and regularization value of return parameters on the total expected cost across different scenarios for both approaches of OC-SVM and OC-SVM with KWA. Three distinct scenarios are considered for each model, and the total expected cost is calculated alongside the percentage change compared to the base-case scenario. Additionally, the distribution of resources across different categories is examined to understand the shifts in resource allocation patterns. The modified approach demonstrates significant advantages over the original OC-SVM in both evenly distributed (Figure 4.4) and elliptically distributed (Figure 4.5) data sets. For evenly 60 distributed data, the modified approach provides a more flexible boundary that reduces the influence of outliers and better represents the actual shape of the data distribution. In the case of elliptically distributed data, it aligns more accurately with the data’s inherent variability, capturing the distribution nuances more effectively. This flexibility ensures a more accurate fit, reducing conservatism and enhancing the robustness of the optimization model. Consequently, the modified approach leads to better decision-making in SC networks by accurately reflecting the uncertainty, which is crucial for effective optimization in real-world scenarios. Figure 4.4 Comparison of Supporting Vectors for Evenly Distributed Data Set between Original DDRO and New DDRO 61 Figure 4.5 Comparison of Supporting Vectors for Ellipse Distributed Data Set Between Original DDRO and New DDRO The SKL-based OC-SVM model presented a range of total expected costs. The numerical results for varying levels of return regularization while keeping demand regularization constant are presented in Table 4.5. Table 4.5 shows that as the return regularization increases from 0.01 to 0.9, the total expected cost decreases from 1,920,999 to 1,855,525, a reduction of approximately 3.41%. For instance, when the return regularization increases to 0.4, the total expected cost decreases to 1,884,036, marking a reduction of 1.92%. The number of recovery centers (R) and collection centers (C) also decreased, indicating an optimized and more efficient network configuration that can handle returns more effectively. 62 Table 4.5 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Return for New DDRO Scenarios Demand Regularization Return Regularization Total Expected Cost Change % # of R # of S # of C # of D 1 (Base-Case) 0.01 0.01 1,920,999 0.00% 6 4 5 7 2 0.01 0.1 1,906,975 -0.73% 6 4 5 7 3 0.01 0.2 1,898,943 -1.15% 6 4 4 7 4 0.01 0.3 1,891,306 -1.55% 6 4 4 7 5 0.01 0.4 1,884,036 -1.92% 6 4 4 7 6 0.01 0.5 1,879,685 -2.15% 6 4 4 7 7 0.01 0.6 1,870,044 -2.65% 6 4 4 7 8 0.01 0.7 1,864,938 -2.92% 6 4 4 7 9 0.01 0.8 1,859,306 -3.21% 6 4 4 7 10 0.01 0.9 1,855,525 -3.41% 6 4 4 7 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors Table 4.6 indicates that increasing the demand regularization from 0.01 to 0.9 significantly reduces the total expected costs from 1,920,999 to 1,063,459, approximately 44.64%. For example, at a demand regularization of 0.4, the cost drops to 1,437,237, representing a 25.18% decrease. This table highlights the importance of accommodating demand uncertainties within the network. The number of manufacturers (R) and distributors (D) also decreases at higher levels of demand regularization, reflecting an optimized network structure. 63 Table 4.6 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand for New DDRO Scenarios Demand Regularization Return Regularization Total Expected Cost Change % # of R # of S # of C # of D 1 (Base-Case) 0.01 0.01 1,920,999 0.00% 6 4 5 7 2 0.1 0.01 1,724,553 -10.23% 6 4 5 7 3 0.2 0.01 1,632,665 -15.01% 5 4 5 7 4 0.3 0.01 1,535,417 -20.07% 5 4 5 6 5 0.4 0.01 1,437,237 -25.18% 5 4 5 6 6 0.5 0.01 1,387,473 -27.77% 5 4 5 6 7 0.6 0.01 1,352,513 -29.59% 5 4 5 6 8 0.7 0.01 1,334,814 -30.51% 5 4 5 6 9 0.8 0.01 1,182,560 -38.44% 4 4 5 5 10 0.9 0.01 1,063,459 -44.64% 4 4 5 5 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors In contrast, the SKL-based OC-SVM with the KWA model showed a total expected cost ranging from $1,920,999 to $997,629 across the same scenarios, as shown in Table 4.7. The percentage change in the total expected cost varied from 0.00% in the base-case scenario to 48.07% in Scenario 10, which indicates that the OC-SVM with the KWA model is more conservative than the traditional OC-SVM approach due to the structure and distribution of the historical data. The OC-SVM with KWA adjustment modifies the uncertainty set boundary to minimize the total distance of all support vectors, resulting in conservative solutions. Table 4.7 demonstrates that increasing demand and return regularization parameters significantly reduces expected costs. For example, when both parameters are set to 0.9, the cost drops to $997,628, a 48.07% reduction from the base case. This substantial cost reduction highlights the combined effect of addressing demand 64 and return uncertainties. Additionally, the network configuration adjusts with fewer manufacturers (R), suppliers (S), and distributors (D), reflecting a more resilient and efficient network. Table 4.7 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand and Return for New DDRO Scenarios Demand Regularization Return Regularization Total Expected Cost Change % # of R # of S # of C # of D 1 (Base-Case) 0.01 0.01 1,920,999 0.00% 6 4 5 7 2 0.1 0.1 1,710,481 -10.96% 6 4 5 7 3 0.2 0.2 1,610,611 -16.16% 5 4 5 7 4 0.3 0.3 1,505,964 -21.61% 5 4 5 6 5 0.4 0.4 1,400,509 -27.09% 5 4 5 6 6 0.5 0.5 1,346,314 -29.92% 5 4 4 6 7 0.6 0.6 1,301,476 -32.25% 5 4 4 6 8 0.7 0.7 1,278,426 -33.45% 5 4 4 6 9 0.8 0.8 1,120,595 -41.67% 4 4 4 5 10 0.9 0.9 997,629 -48.07% 4 4 4 5 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors Table 4.8 compares the original and new DDRO approaches, showing that the new DDRO approach generally results in a marginally higher total expected cost across various scenarios. For example, in the base-case scenario, the cost for the new DDRO is 1,920,999 compared to the original DDRO’s 1,914,376. Despite this slight increase, the new DDRO approach incorporates additional considerations or constraints that lead to a more comprehensive optimization. This comparison highlights the robustness and effectiveness of the new DDRO approach in managing uncertainties within the CLSC network. 65 Table 4.8 Comparison of Nine Scenarios of Deterministic Model by Changing Regularization of Demand and Return between Original DDRO and New DDRO Scenarios Demand Regularization Return Regularization Total Expected Cost ORG Total Expected Cost NEW 1 (Base-Case) 0.01 0.01 1,914,376 1,920,999 2 0.10 0.10 1,622,246 1,710,482 3 0.20 0.20 1,522,247 1,610,611 4 0.30 0.30 1,465,647 1,505,964 5 0.40 0.40 1,379,802 1,400,509 6 0.50 0.50 1,331,680 1,346,314 7 0.60 0.60 1,262,187 1,301,476 8 0.70 0.70 1,194,100 1,278,426 9 0.80 0.80 1,115,598 1,120,595 10 0.90 0.90 1,042,829 997,629 This section provides a comprehensive comparative analysis of OC-SVM and OC-SVM with KWA approaches. We focus on the impact of altering the demand and regularization value of return parameters on the total expected cost across various scenarios. The original DDRO model produced a total expected cost ranging from 1,914,376 to 1,042,829 across ten scenarios. The percentage change in the total expected cost varied from a reduction of 4.33% in Scenario 10 to an increase of 7.06% in Scenario 8. These results indicate that the optimization level in cost reduction varied significantly when the regularization parameters were altered. On the other hand, the revised new DDRO model generated a total expected cost ranging from 1,920,999 to 997,629 across the exact scenarios. The percentage change in the total expected cost 66 in this model ranged from an increase of 5.80% in Scenario 3 to a reduction of 4.33% in Scenario 10. The revised model demonstrated a more consistent and improved performance in cost reduction across the scenarios, particularly with moderate regularization parameters. One of the significant enhancements in the revised new DDRO model is the modification of the Cm value in the SKL-learn model. This refinement has benefited datasets that scatter like an ellipse, ensuring a more accurate and efficient fit. The revised model’s improved Cm value has allowed for a better fit to the data distribution, given that the data sample is an ellipse from the lower left to the upper right. Consequently, the revised model can more effectively optimize resource allocation and minimize costs across different scenarios than the original DDRO model. Overall, the revised new DDRO model has demonstrated a more effective and consistent optimization of resource utilization and cost reduction across varying demand and regularization values of return parameters. This enhanced model offers significant decision-making and strategy formulation insights in robotic optimization applications. The model’s superior capability in optimizing resource utilization, reducing costs, and improving the overall performance of robotic operations showcases its potential to revolutionize the field. It benefits datasets that deviate from a perfect circle and exhibit an elliptical distribution. Figure 4.6 and Figure 4.7 then reveal the comparison of facility numbers and expected cots between OC-SVM and OC-SVM with KWA approaches, which indicates that the OC-SVM with KWA approaches have higher values than the OC-SVM approaches. 67 Figure 4.6 Comparison of Operating Facilities Between OC-SVM and OC-SVM with KWA Approaches Figure 4.7 Comparison of Expected Costs Between OC-SVM and OC-SVM with KWA Approaches In the following, 4 different datasets are generated to evaluate the uncertainty sets yield from the OC-SVM and OCSVM with KWA approaches. The RFIC algorithm is employed to calculate the RF index of each algorithm. Figure 4.8 shows the uncertainty sets constructed using both approaches with 80% coverage. As shown in Figure 4.8, the One-Class SVM with KWA tends to construct an uncertainty set with equal distances from each SV in different directions because the OC-SVM with KWA considers the dispersion degree of data samples along each direction when constructing data-driven uncertainty sets. 68 Dataset 1 Dataset 2 Dataset 3 Dataset 4 Figure 4.8 Comparison of the OC-SVM and OC-SVM with KWA Approaches in Constructing Uncertainty Sets with 80% Coverage Table 4.9 also shows the RF index, mean, and standard deviation of the distance of all SVs from the boundary of the uncertainty sets for both approaches with different conservatism degrees. As shown in Table 4.9, the RF index of the OC-SVM with KWA has a lower value for all datasets and conservatism degrees. It indicates that the OC-SVM with KWA provides a more balanced uncertainty set than the OC-SVM approach, as the total cumulative distance of all SVs (RF index) is lowered. 69 Table 4.9 Comparing OC-SVM and OC-SVM with KWA Approaches in Constructing Balanced Uncertainty Sets Using RF Index Data Coverage OC-SVM OC-SVM with KWA Mean Std sum Mean Std sum 70% 69 65 10,672 63 60 9,615 Dataset 1 80% 61 58 6,305 53 52 5,448 90% 57 50 2,968 51 48 2,656 70% 70 57 10,760 76 53 10,311 Dataset 2 80% 60 50 6,196 54 45 5,546 90% 44 43 2,340 37 38 1,893 70% 73 75 22,300 59 59 18,116 Dataset 3 80% 67 69 13,483 52 53 10,630 90% 56 61 5,768 38 41 4,012 70% 82 77 12,445 72 73 11,051 Dataset 4 80% 76 74 7,755 70 70 7,220 90% 74 69 4,015 71 66 3,740 4.4.5 Scenario Analysis: Disruption of Nodes and Links In this section, we analyze the impact of disruptions on the SC network, focusing on two types of disruptions: link disruptions and node disruptions. Nodes refer to facilities such as manufacturers, suppliers, recovery centers, and distribution centers that play crucial roles in the SC network. Links represent the connections between these nodes, such as transportation routes and communication channels. Various factors, including natural disasters, pandemics, geopolitical conflicts, and other unforeseen events can cause disruptions. For instance, the COVID-19 pandemic disrupted SCs worldwide by causing factory shutdowns, reducing labour availability, and interrupting transportation networks. Natural disasters like wildfires, hurricanes, and earthquakes can lead to the unavailability of specific nodes or damage the links between them, further complicating SC operations. We consider various scenarios to understand how these disruptions affect the network’s performance, measured by the total expected cost and changes in the number of recovery centers, 70 suppliers, distribution centers, and manufacturers. The numerical results provide valuable insights into the SC's robustness and resilience under different conditions. Figure 4.9 reveals the CLSCN between suppliers, manufacturers, and distributors with V = 0.3 as the base case based on OC-SVM with KWA. Figure 4.9 Optimal Configuration of the CLSN Using DDRO with 70% Coverage for Scenario 1 For disrupted links, the quantity of transportation between two facilities is set to zero. For example, suppose the road for transportation between supplier 1 and manufacturer 5 is disrupted. In that case, no raw material can be transported along this link. This disruption could be due to 71 road closures caused by unforeseen incidences like landslides, floods, earthquake infrastructure damage, or temporary maintenance. Consequently, the SC must adjust by either finding alternative routes or relying on inventory buffers, leading to potential delays and increased costs: ∑𝑄𝑆𝑅[𝑆, 𝑅, 𝐼]𝑅= 0, 𝑆 = 1 𝑎𝑛𝑑 𝑅 = 5 (4.18) For disrupted nodes, the Boolean expression for the corresponding facility is set to zero. For example, suppose a manufacturing facility is disrupted due to a wildfire. In that case, it becomes inoperative, meaning no production can occur at that site. This could be caused by unforeseen incidences, such as earthquakes or floods, or other disruptions like pandemics, which force the facility to shut down. As a result, the SC must reallocate production to other facilities, which can lead to increased production costs and logistical challenges: 𝐵𝑅[𝑅] = 0, 𝑅 = 2 (4.19) Facilities disruptions may result in unfulfilled demand, which could cause shortages of items in real life. Therefore, to reduce the impact of demand shortages, a shortage cost is added to the objective function of the optimization model. In this problem, each unit of unmet demand incurs a shortage cost of 50 units. The base-case scenario, shown in Figure 4.9 with a conservative demand variation of 0.3, serves as a benchmark. There are no disruptions or backup facilities, resulting in the lowest expected cost of $1,505,963.604. In Scenario 2, shown in Figure 4.10, a single link disruption occurs specifically between Supplier 1 and Manufacturer 5 without node disruptions or backup facilities. Despite this, our SC network demonstrates its resilience, leading to only a slight increase in cost by 0.29%. This is a testament to the robustness of our network. The increase in 72 total costs is due to rerouting products or raw materials from different facility locations, often farther apart. The rerouting increases transportation costs and may involve more complex logistics to ensure timely deliveries, thereby increasing the overall cost. By examining these scenarios, we can better understand the SC network’s resilience to various types of disruptions and the importance of having backup facilities and contingency plans to mitigate the impact of such events. Figure 4.10 Disrupted Link of the CLSN Using DDRO with 70% Coverage for Scenario 2 73 Node disruptions, analyzed in Figure 4.11, have a more significant impact, raising the total expected cost by 46.05%. In this scenario, one manufacturer (precisely Manufacturer 2) is considered for disruption. It demonstrates the critical role of individual nodes in the network. The reduction in the number of manufacturers from 5 to 4 illustrates the network’s need to adapt to the loss of a critical node. The reason behind this substantial increase in total costs is that when a critical node, such as a manufacturer, is disrupted, the remaining manufacturers must compensate for the loss, often leading to higher production costs, increased transportation distances, and potential delays, significantly increasing overall operational costs. Figure 4.11 Disrupted Node of the CLSN Using DDRO with 70% Coverage for Scenario 3 and Scenario 6 74 When both link and node disruptions co-occur in Figure 4.12, the total expected cost increases slightly further to 46.28%, highlighting the compounded effect of multiple disruptions. In this scenario, the disruption of the link between Supplier 1 and Manufacturer 5 and the disruption of Manufacturer 2 exacerbates the SC inefficiencies. The simultaneous occurrence of these disruptions forces the network to rely even more on alternative routes and facilities, further increasing transportation and operational costs. This compounded effect underscores the importance of robust contingency planning and the potential need for backup facilities to mitigate the impacts of such disruptions effectively. These scenarios underscore the network’s vulnerability to node disruptions, which have a much more substantial impact than link disruptions. The significant increase in costs and the reduction in operational facilities indicate that work needs backup plans to maintain a license under severe disruption. One of the reasons for the increase in total costs is related to the shortage cost. Figure 4.12 Disrupted Link and Node of the CLSN Using DDRO with 70% Coverage for Scenario 4 75 In Scenario 5 to 7, the optimization model incorporates backup facilities, alternative entities or suppliers within the SC network that can fill the gaps created by disruptions. Unlike Scenarios 2 to 4, where no backup facilities were considered and disruptions to links and nodes could not be compensated for, Scenarios 5 to 7 demonstrate their robustness by actively switching to these cooperating entities. The introduction of backup facilities in Figure 4.13 is particularly noteworthy as it shows no additional cost compared to Scenario 2, indicating that the network’s design is robust and cost-effective. Scenario 6, combining node disruptions with backup facilities, significantly reduces the expected cost increase to 14.44%. It highlights the effectiveness of backup facilities in enhancing network resilience against node disruptions. The reduction in the number of recovery centers from 5 to 4 in this scenario further reflects the network’s adaptation to the disrupted node, underscoring the role of backup facilities in maintaining operational efficiency. 76 Figure 4.13 Disrupted Link of the CLSN Using DDRO with 70% Coverage for Scenario 5 In the most complex scenario in Figure 4.14, which combines link and node disruptions with backup facilities, the expected cost increase is limited to 16.04%. It substantially improves the cost increases observed in scenarios without backup facilities, demonstrating the critical importance of backup facilities in maintaining network performance under severe and compounded disruption conditions. 77 Figure 4.14 Disrupted Link and Node of the CLSN Using DDRO with 70% Coverage for Scenario 7 The analysis of these scenarios provides clear insights. While the SC network is resilient to link disruptions, node disruptions profoundly impact the total expected cost and network structure. The introduction of backup facilities is a valuable strategy for mitigating the adverse effects of node disruptions and combined disruptions, enhancing the network’s overall resilience. These findings underscore the necessity for strategic planning and risk management to ensure the robustness and sustainability of SC operations in the face of various disruption scenarios. The stark contrast between scenarios without backup plans (2, 3, and 4) and those with backup plans (5, 6, and 7) 78 further emphasizes the importance of incorporating backup facilities into SC design to mitigate risks and maintain operational efficiency, as shown in Table 4.10. Table 4.10 Comparison of Seven Scenarios of the Deterministic Model with Node and Link Disruptions Scenarios Link Disruption Node Disruption Back-up Facility Total Expected Cost Shortage Cost Change % R S C D 1 (Base) - - - 1,505,964 0 0.00% 5 4 5 6 2 √ - - 1,510,295 0 0.29% 5 4 5 6 3 - √ - 2,199,462 812733 46.05% 4 4 5 6 4 √ √ - 2,202,985 812733 46.28% 4 4 5 6 5 √ - √ 1,510,295 0 0.29% 5 4 5 6 6 - √ √ 1,723,461 0 14.44% 5 4 4 6 7 √ √ √ 1,747,521 0 16.04% 5 4 5 6 R: number of manufacturers, S: number of suppliers, C: number of collection center, D: number of distributors 79 Chapter 5: Conclusion 5.1 Research Summary This thesis explores the application of mathematical modelling and data-driven approaches in designing a CLSC network for the plastic industry. In Chapter 2, a literature review is conducted on developing mathematical models for designing CLSC networks and advancing DDRO methods to address parameter uncertainty, and the research gap is identified. Chapter 3 proposes a deterministic MILP model for designing a CLSC network for the plastic industry. Additionally, a real-world case study in Vancouver, Canada, is provided to evaluate the proposed model’s effectiveness in designing a CLSC network. Following that, a sensitivity analysis is conducted to assess the impact of input parameters on the overall configuration of the CLSC network and the optimal value of the objective function. The deterministic mathematical model proposed in Chapter 3 effectively captures real-world conditions and complexities within the CLSC network, yielding an optimal configuration that minimizes total costs. However, the sensitivity analysis reveals that the deterministic model is highly susceptible to variations in key input parameters, such as demand, return rates, and disposal fractions. In other words, variations in the value of demand, return, or disposal fractions change the configuration of the CLSC network, the number of required facilities, the flow of products among each facility or stakeholder, and the total costs. Moreover, the values of these parameters are uncertain and change over time. Therefore, incorporating uncertainty in key input parameters is crucial when designing a CLSC network to ensure the resulting network is resilient to variations in uncertain parameters. 80 Chapter 4 delves into developing DDRO approaches to address parameter uncertainty in CLSC network design. At the beginning of Chapter 4, the SKL-based OC-SVM approach developed by Han et al. (2021a) is introduced. However, a significant limitation of the original DDRO model is its inability to account for the degrees of dispersion when constructing uncertainty sets. The model’s dependence on pre-set and equal weights for basis kernel matrices corresponding to varying levels of data dispersion can lead to an inaccurate representation of uncertainty. To this end, this approach proposed by Han et al. (2021a) is then extended to an SKL-based OC-SVM with KWA to incorporate fairness in constructing uncertainty sets. The concept of a fair and balanced uncertainty set is first introduced in this thesis. A balanced uncertainty set refers to a set where the distance from the uncertainty set’s boundary to the dataset’s edge is nearly uniform in all directions. In other words, the distances of SVs—historical data points not included within the uncertainty sets—from the boundary of the uncertainty set exhibit minimal variation relative to one another. The advantage of a balanced uncertainty set compared to an imbalanced uncertainty set lies in two folds. The advantage of a balanced uncertainty set over an imbalanced one is twofold. First, a balanced uncertainty set yields robust solutions with consistent levels of conservatism across different optimizations under uncertainty problems due to the uniform distance from the uncertainty set’s boundary to the edge of the dataset in all directions. Additionally, a balanced uncertainty set more accurately represents the true distribution of uncertain parameters by accounting for the degree of dispersion of data samples in different directions. As noted in Chapter 2, a critical research gap in DDRO methodologies is the need for a standardized index to evaluate and compare data-driven uncertainty sets constructed using different DDRO approaches. To address this gap, Chapter 4 introduces the RF index, designed to assess the construction of balanced and fair uncertainty sets by different DDRO methods, including 81 OC-SVM and OC-SVM with KWA. An accompanying RFIC algorithm is also developed in Chapter 4 to calculate the RF index. In Chapter 4, both the OC-SVM and OC-SVM with KWA approaches are applied to address uncertainty in optimizing a CLSC network for the plastic industry, and their results are compared. The RF index, calculated using the RFIC algorithm, is applied to the uncertainty sets generated by both DDRO approaches. The numerical results indicate that the proposed DDRO approach in this thesis outperforms the SKL-based OC-SVM approach developed by Han et al. (2021a) in constructing balanced uncertainty sets. Additionally, the SKL-based OC-SVM with KWA has produced a broader spectrum of robust solutions, ranging from more conservative to less conservative outcomes, compared to those generated by the standard SKL-based OC-SVM approach. Integrating DDRO with KWA significantly enhances robustness and fairness, making the approaches discussed in this thesis particularly valuable for real-world applications, where the accuracy and reliability of the optimization model are critical for effective decision-making. This combination represents a substantial advancement in the field, providing a comprehensive solution to SC managers' complex challenges. The numerical results indicate that the proposed DDRO approach in this research enhances the construction of fair uncertainty sets and reduces the cumulative distance of SVs from the boundary of the uncertainty set by an average of 13.6%. 5.2 Research Contributions This thesis contributes significantly to the optimization field by addressing the challenges posed by uncertainty in the design of CLSC networks. Unlike studies that aim to deliver universal CLSC design solutions, this research focuses on developing and evaluating robust optimization methods, specifically the KWA approach and the RFIC. These methodological advancements provide a 82 systematic framework for improving uncertainty modelling and assessing the robustness of optimization techniques in the context of CLSC network design for the plastic industry. One of the significant contributions of this thesis is the integration of machine learning algorithms and DDRO approaches with mathematical modelling to address uncertainties in CLSC networks. The design of CLSC networks in the plastic industry using data-driven methods has yet to be explored. This research bridges this gap by employing DDRO techniques that construct uncertainty sets aligned with historical data, thereby enhancing the feasibility and resilience of CLSC networks. These methods focus on high-probability uncertainty regions, mitigating overly conservative solutions that fail to reflect practical operational scenarios. Another significant contribution is developing the SKL-based OC-SVM framework enhanced with the KWA approach. The SKL-based OC-SVM, as Han et al. (2021a) proposed, effectively constructs compact uncertainty sets that reflect historical data while maintaining the tractability of robust counterpart models. However, this method assumes equal weighting of basis kernel matrices, disregarding variations in data dispersion along different directions. This limitation results in imbalanced uncertainty sets, which may cause inconsistencies in the conservatism of optimization models. The KWA approach adjusts kernel weights according to the degree of data dispersion, producing more balanced uncertainty sets that reduce variability in the levels of conservatism across different directions. This improvement significantly enhances the reliability of decision-making in CLSC network design. A further essential contribution of this research is the introduction of the RF Index and the development of the RFIC. The RF Index serves as a novel metric for evaluating and comparing data-driven uncertainty sets, addressing the lack of standardized metrics in the literature. The RF Index promotes fairness in uncertainty modelling by ensuring consistent levels of conservatism 83 across all directions. It evaluates how uniformly the reduction rates of uncertainty sets are distributed, ensuring that the proper distribution of uncertain parameters is adequately captured. The RFIC algorithm, developed as part of this thesis, efficiently computes the RF Index by measuring the cumulative distances of support vectors from the uncertainty set boundary. This computationally efficient algorithm enables robust comparisons between DDRO approaches, including the KWA-enhanced SKL-based OC-SVM. Rather than providing a definitive solution for CLSC network design, this thesis emphasizes the development of robust evaluation frameworks and analytical tools. Introducing the RFIC and KWA approaches allows for a systematic analysis of robustness, fairness, and balance in uncertainty modelling. These contributions provide critical insights into the trade-offs between conservatism and reliability in robust optimization models, advancing state of the art in data-driven decision-making for CLSC network design. 5.3 Research Limitations While this research presents significant advancements in optimizing CLSC utilizing DDRO approaches, such as OC-SVM with KWA, several limitations must be acknowledged. These limitations highlight areas where the current model may fall short and suggest further research and development avenues. Firstly, due to the unavailability of actual data, the model is based entirely on generated rather than real-world data. The generated data samples effectively demonstrate the capabilities of the proposed DDRO approach in this paper, highlighting its advantages over the method proposed by Han et al. (2021a). However, incorporating real-world data would further enhance the development of a resilient CLSC for the plastic industry. 84 Another limitation of this research is the simplified assumptions regarding the problem and the available data. This thesis assumes sufficient historical data is available to construct data-driven uncertainty sets. However, in cases where only limited historical data is accessible, constructing uncertainty sets from such data may not accurately represent the proper distribution of uncertain parameters. Another critical limitation is the deterministic nature of the model, which does not account for multi-objective optimization. Real-life scenarios often require balancing multiple factors, such as economic costs and environmental impacts, like carbon taxes from emissions. The current model’s focus on single-objective optimization may only partially capture the trade-offs necessary in real-world decision-making. Additionally, the DDRO approach is highly affected by the regularization parameter, which determines the conservatism degree. This research established a base scenario with a value of 0.01, covering nearly 99% of all data samples within the data-driven uncertainty set. However, the decision-maker manually enters this value rather than being determined by an automated process or algorithmic logic. This manual input introduces subjectivity and potential bias, which could affect the consistency and reliability of the optimization process. Although sensitivity analysis is conducted to observe how changes in the value impact the objectives, an objective method needs to be used to determine them. In summary, while this research advances the field of robust optimization, particularly in the context of CLSC, several limitations must be addressed. These include the reliance on generated data, low historical data availability, the absence of multi-objective optimization, and the manual determination of the regularization parameter. These limitations underscore the urgent need for further research to enhance the model’s applicability, scalability, and robustness in real-world scenarios. Addressing these limitations in future research could significantly advance the 85 development and practical implementation of decision-making models for designing CLSC networks. 5.4 Future Research Based on this thesis's findings and limitations, several promising directions for future research could further expand and enhance its contributions. While this study focused on designing a resilient CLSC network for the plastic industry using the SKL-based OC-SVM with KWA, these methods could be further developed and applied in broader contexts. Additionally, addressing the limitations and integrating novel methodologies could significantly advance robust optimization and supply chain design. A crucial avenue for future research involves extending the proposed model to incorporate multi-objective functions. The current thesis focuses on minimizing total costs as a single objective. However, real-world CLSC design often requires balancing multiple objectives, such as minimizing environmental impacts, maximizing social benefits, and enhancing operational efficiency. For example, incorporating objectives like reducing greenhouse gas emissions, promoting regional job creation, and improving social equity could lead to more holistic and sustainable CLSC designs. Developing a multi-objective optimization framework would allow decision-makers to assess trade-offs and prioritize goals based on their specific requirements. Another area for exploration is the application of the RF index as a tool for comparative analysis across various DDRO approaches. While this thesis introduced the RF index as a measure of fairness and balance in uncertainty sets, its potential can be extended by combining it with other evaluation criteria, such as the tractability of robust counterpart models, the degree of conservatism, and computational efficiency. A more comprehensive evaluation framework incorporating these 86 criteria would enable a systematic comparison of different data-driven uncertainty modelling approaches and provide deeper insights into their respective strengths and limitations. Additionally, the thesis assumes the availability of sufficient historical data to construct data-driven uncertainty sets. However, in many real-world scenarios, such data may need to be more comprehensive, posing challenges in designing balanced and accurate uncertainty sets. Future research could focus on developing methods to construct uncertainty sets with limited data, leveraging techniques such as transfer learning, synthetic data generation, or Bayesian inference. These approaches could improve the robustness of models under data-constrained conditions and expand the applicability of DDRO methods in industries where data collection is challenging. Another promising research direction is the broader applicability of the SKL-based OC-SVM with KWA. While this thesis specifically addressed CLSC network design for the plastic industry, the proposed approach could be adapted to various fields, such as healthcare logistics, financial risk management, and disaster response. For instance, the framework could optimize resource allocation under uncertainty in healthcare to improve system resilience and efficiency. Similarly, the logistics industry could leverage this framework to address transportation, inventory management, and uncertainty of dynamic demand patterns. Validation of the proposed methods using real-world data remains a critical research need. The current study relies on data generated through estimations, which may only partially capture the complexities and variability of real-world conditions. Future studies should aim to test the SKL-based OC-SVM with the KWA approach using actual datasets from relevant industries. This validation would not only demonstrate the model's practical applicability and robustness but also identify potential areas for refinement, ensuring its reliability in operational settings. 87 Lastly, to address uncertainties more comprehensively, the integration of advanced uncertainty modeling techniques with multi-objective optimization could be explored. 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Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Tosarkani, Babak Mohamadpour
Robust optimization for plastic closed-loop supply chains: advancing data-driven models with kernel weight adjustment and fairness metrics
Text