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UBC Theses and Dissertations

Optimization of forest-based biomass logistics at the operational level Malladi, Krishna Teja 2020

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  Optimization of forest-based biomass logistics at the operational level  Krishna Teja Malladi  M.Sc., Simon Fraser University, 2014 B.Tech., The LNM Institute of Information Technology, 2011   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF   Doctor of Philosophy in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Forestry)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  February, 2020  © Krishna Teja Malladi, 2020    ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Optimization of forest-based biomass logistics at the operational level  submitted by Krishna Teja Malladi in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Forestry  Examining Committee: Dr. Taraneh Sowlati, Forestry Supervisor  Dr. Julie Cool, Forestry Supervisory Committee Member  Dr. James A. Vercammen, Sauder School of Business University Examiner Dr. Philip David Evans, Forestry University Examiner             iii  Abstract Logistics cost and emissions are important factors affecting the utilization of forest-based biomass. While numerous studies optimized biomass logistics at the tactical level, those at the operational level are limited. The focus of most of the previous studies was on cost minimization, while emission reduction from biomass logistics received less attention. Few recent studies analyzed the impact of carbon pricing policies on the optimum cost and emissions of biomass logistics. However, due to the focus on specific case studies, the results obtained in these studies may not be generalized. Moreover, these studies combined the cost and the emissions into one objective function resulting in the loss of information about the trade-off between the two objectives. The overall goal of this dissertation is to optimize biomass logistics at the operational level considering biomass storage, pre-processing and transportation decisions, and to analyze the impacts of carbon pricing policies on biomass logistics optimization models independent of the underlying case study. First, optimization models are developed to minimize the total cost of biomass logistics considering all logistics operations. The models are applied to the case study of a large logistics company. The results indicated a potential to save up to 12% of the total cost compared to the actual plans implemented by the company. Next, several properties of the optimal cost and emissions of case-independent logistics models under different carbon pricing policies are proposed and proved mathematically. The properties are numerically verified using the case study of a biomass-fed district heating plant. The results indicate that the carbon tax and the carbon cap-and-trade models result in equal emissions for equal carbon prices. The carbon cap-and-trade model is more cost-effective than the carbon tax model only if the carbon price is more than the price of initial allowances. Finally, the optimization model developed for the biomass-fed district heating plant is extended to incorporate cost and emissions as two separate objectives. A new algorithm is proposed to solve bi-objective optimization models considering carbon pricing policies to overcome the computational complexity involved in solving these models. Being case-independent, the algorithm can be applied to other cases.     iv  Lay Summary High logistics cost and emissions impede the widespread utilization of biomass. While numerous studies attempted to optimize biomass logistics, those related to short-term planning are limited in the literature. The objective of this study is to optimize biomass logistics for short-term planning and to analyze the impacts of carbon pricing policies on the optimal cost and emissions of biomass logistics models. The optimization models, when applied to the case of a large logistics company, indicate a potential to save up to 12% of total logistics cost. The impact of carbon pricing policies on biomass logistics optimization models is analyzed using the case of a biomass-fed district heating plant. The results indicate that the carbon tax and the carbon trading models result in equal emissions for equal carbon prices. Furthermore, bi-objective biomass logistics optimization models are developed considering carbon pricing policies, and an algorithm is developed to solve them.    v  Preface All the work presented in this dissertation was carried out by the author, Krishna Teja Malladi, during his PhD program. The work was conducted under the supervision of Dr. Taraneh Sowlati at the Industrial Engineering Research Group of the University of British Columbia, Vancouver, Canada.  Parts of this dissertation is presented in the following publications.  • A version of Chapter 2 has been published in the following two articles.  o Malladi, K.T, and Sowlati, T. (2017) Optimization of operational level transportation planning in forestry: a review. International Journal of Forest Engineering. 28 (3): 198-210. o Malladi, K.T, and Sowlati, T. (2018) Biomass logistics: A review of important features, optimization modeling and the new trends. Renewable and Sustainable Energy reviews. 94: 587-599. I was responsible for defining the topic, searching the literature, synthesizing the content and writing the manuscript. Sowlati, T. was the supervisory author and was involved throughout the process of reviewing the literature and preparing the manuscript.  • A version of Chapter 3 has been published in the following article.  o Malladi, K.T, Quirion-Blais, O. and Sowlati, T. (2018) Development of a decision support tool for optimizing the short-term logistics of forest-based biomass. Applied Energy. 216: 662-677.  I was the main author of this article. I defined the problem, collected data from the industrial partner by visiting their site multiple times, developed and solved the mathematical models, and prepared the manuscript. Sowlati, T. was the supervisory author who provided advising support in defining the problem, identifying the industrial collaborator, collecting data, developing and solving the mathematical models, and preparing the manuscript. Quirion-Blais, O. developed the decision support tool that was delivered to the industrial collaborator and contributed to manuscript edits.  • A review article has been published which provided the basis for the optimization models developed in Chapter 3.  o Malladi, K.T, and Sowlati, T. (2018) Sustainability aspects in Inventory Routing Problem: A review of new trends in the literature. Journal of Cleaner Production. 197 (Part 1): 804-814.    vi  I was responsible for defining the topic, searching the literature, synthesizing the content and writing the manuscript. Sowlati, T. was the supervisory author and was involved throughout the process of reviewing the literature and preparing the manuscript. • A version of Chapter 4 is submitted for publication.  o Malladi, K.T, and Sowlati, T. (2019) Impact of carbon pricing policies on the cost and emissions of the biomass supply chain optimization models.  I was the main author of this article. I defined the problem, developed mathematical models, proposed several properties and proved them mathematically, developed optimization models for the case study, solved the models, and prepared the manuscript. Sowlati, T. was the supervisory author who provided advising support in defining the problem, proposing and proving the properties, developing and solving the mathematical models, and preparing the manuscript.  • The case study presented in Chapter 4 is based on an article that has been published.  o Quirion-Blais, O., Malladi, K.T., Sowlati, T., Gao, E., and Mui, C. (2019) Analysis of feedstock requirement for the expansion of a biomass-fed district heating system considering daily variations in heat demand and biomass quality. Energy Conversion and Management. 187: 554-564.  I contributed to writing the entire manuscript, collecting data, and synthesizing the results of the simulation model. The initial data was gathered, and the simulation model was developed by the main author, Quirion-Blais, O.  Sowlati, T. was the supervisory author who provided advising support in defining the problem, gathering the data, developing the models, and preparing the manuscript. Gao, E. contributed to gathering the data and synthesizing the results. Mui, C. was the industrial collaborator who was involved in defining the problem, data gathering, and synthesis of the results. • A version of Chapter 5 is accepted for publication.  o Malladi, K.T, and Sowlati, T. (2019) Bi-objective optimization of biomass supply chains considering carbon pricing policies. Applied Energy (accepted) I defined the problem, developed mathematical models and applied them to the case study, developed the algorithm, and prepared the manuscript. Sowlati, T. was the supervisory author who provided advising support in defining the problem, developing and solving the mathematical models, developing the algorithm, and preparing the manuscript.    vii  Table of Contents Abstract ....................................................................................................................................................... iii Lay Summary ............................................................................................................................................. iv Preface .......................................................................................................................................................... v Table of Contents ...................................................................................................................................... vii List of Tables ............................................................................................................................................... x List of Figures ............................................................................................................................................ xii Acknowledgements .................................................................................................................................. xiv Dedication ................................................................................................................................................. xvi Chapter 1: Introduction ............................................................................................................................. 1 1.1 Background .................................................................................................................................. 1 1.2 Research objectives ..................................................................................................................... 7 1.3 Outline of the thesis ..................................................................................................................... 7 Chapter 2: Literature review ..................................................................................................................... 9 2.1 Synopsis ....................................................................................................................................... 9 2.2 Biomass logistics operations: key features and decisions ........................................................... 9 2.2.1 Biomass collection .................................................................................................................. 9 2.2.2 Storage................................................................................................................................... 10 2.2.3 Transportation ....................................................................................................................... 10 2.2.4 Pre-processing ....................................................................................................................... 11 2.3 Biomass logistics optimization modeling .................................................................................. 12 2.3.1 Biomass logistics optimization models at the tactical level .................................................. 12 2.3.2 Biomass logistics optimization models at the operational level ............................................ 15 2.4 Impact of carbon pricing policies in biomass supply chain optimization models ..................... 17 2.5 Conclusions ............................................................................................................................... 22   viii  Chapter 3: Optimization of operational level forest-based biomass logistics considering transportation, storage, and pre-processing operations ........................................................................ 24 3.1 Synopsis ..................................................................................................................................... 24 3.2 Problem description ................................................................................................................... 24 3.3 Optimization models.................................................................................................................. 26 3.3.1 Transshipment model ............................................................................................................ 27 3.3.2 Routing model ....................................................................................................................... 32 3.4 Data............................................................................................................................................ 38 3.5 Results ....................................................................................................................................... 42 3.6 Discussion .................................................................................................................................. 48 3.7 Decision support tool ................................................................................................................. 50 3.8 Conclusions ............................................................................................................................... 52 Chapter 4: Impact of carbon pricing policies on the cost and emissions of the biomass supply chain optimization models .................................................................................................................................. 54 4.1 Synopsis ..................................................................................................................................... 54 4.2 Case-independent optimization models with carbon pricing policies ....................................... 54 4.2.1 Optimization model with the carbon tax policy .................................................................... 56 4.2.2 Optimization model with the carbon cap-and-trade policy ................................................... 56 4.2.3 Optimization model with the carbon offset policy ................................................................ 57 4.3 Total cost of optimization models with carbon pricing policies ................................................ 57 4.4 Pair-wise comparison of optimization models considering carbon pricing policies ................. 61 4.5 Case study .................................................................................................................................. 68 4.5.1 Data ....................................................................................................................................... 70 4.6 Optimization model ................................................................................................................... 73 4.7 Results ....................................................................................................................................... 77 4.7.1 Impact of carbon pricing policies on the total cost of the optimization models .................... 79   ix  4.7.2 Pair-wise comparison of the optimization models with carbon pricing policies ................... 83 4.7.3 Impact of carbon pricing policies on optimal feedstock mix at the district heating plant ..... 87 4.8 Discussion .................................................................................................................................. 89 4.9 Conclusions ............................................................................................................................... 90 Chapter 5: Multi-objective biomass supply chain optimization models considering carbon pricing policies ........................................................................................................................................................ 92 5.1 Synopsis ..................................................................................................................................... 92 5.2 Bi-objective optimization models .............................................................................................. 93 5.3 Results ....................................................................................................................................... 94 5.4 Mathematical properties of bi-objective optimization models with carbon pricing policies ..... 98 5.5 Algorithm for solving bi-objective optimization models with carbon pricing policies ........... 105 5.6 Illustration of the algorithm ..................................................................................................... 107 5.7 Discussion ................................................................................................................................ 111 5.8 Conclusions ............................................................................................................................. 113 Chapter 6: Conclusions .......................................................................................................................... 115 6.1 Summary and conclusions ....................................................................................................... 115 6.2 Strengths .................................................................................................................................. 117 6.3 Limitations ............................................................................................................................... 119 6.4 Future work ............................................................................................................................. 120 References ................................................................................................................................................ 121    x  List of Tables Table 2-1: Summary of important features of biomass logistics at the tactical level, examples of studies that incorporated them ....................................................................................................................................... 15 Table 2-2: Summary of important features of biomass logistics at the operational level, examples of studies that incorporated them ................................................................................................................................ 17 Table 2-3: Summary of studies that analyzed the impacts of carbon pricing policies on biomass supply chain optimization models .................................................................................................................................... 21 Table 3-1: Notations used in the transshipment model ............................................................................... 28 Table 3-2: Notation used in the routing model ........................................................................................... 35 Table 3-3: Characteristics of different types of truck ................................................................................. 40 Table 3-4: Biomass type, origin and destination of trucks, and compatible truck type .............................. 40 Table 3-5: Total and weekly costs obtained by summing the costs for each routes per day ...................... 43 Table 4-1: Notations used in case-independent optimization models for different carbon pricing policies55 Table 4-2: Summary of the analyses and pairwise comparisons of optimization models with carbon pricing policies from optimal emissions and cost perspectives............................................................................... 67 Table 4-3: Quality, price, emission, and transportation characteristics of wood residues, briquettes, and pellets .......................................................................................................................................................... 71 Table 4-4: Notations used in the models ..................................................................................................... 73 Table 4-5: Total cost, emission, and optimal feedstock mix per year from feedstock cost minimization and emission minimization models ................................................................................................................... 77 Table 5-1: Pareto-optimum solutions obtained for the bi-objective model without carbon pricing ........... 95 Table 5-2: Notations used in the case-independent bi-objective optimization models ............................... 98 Table 5-3: Costs of all Pareto-optimum solutions to the model without carbon pricing under different carbon pricing policies .......................................................................................................................................... 108 Table 5-4: Matrix evaluating the pairwise trade-off between total cost and emissions of the Pareto-optimum solutions of the model without carbon pricing under the carbon tax policy (tax = $50 per tonne CO2-eq.) and the carbon cap-and-trade policy (carbon price = $50 per tonne CO2-eq. and initial allowance = 2000 tonnes of CO2-eq.) .................................................................................................................................... 109   xi  Table 5-5: Matrix evaluating the pairwise trade-off between total cost and emissions of the Pareto-optimum solutions of the model without carbon pricing under the carbon offset policy (carbon price = $50 per tonne CO2-eq. and compliance target = 2000 tonnes of CO2-eq.) ...................................................................... 110    xii  List of Figures Figure 1-1: Biomass supply chain and logistics activities ............................................................................ 2 Figure 3-1: Schematic representation of operations of the forest-based biomass logistics company ......... 26 Figure 3-2: Decomposition-based solution approach to solve biomass logistics problem at the operational level ............................................................................................................................................................. 27 Figure 3-3: A sample of an auxiliary graph with 3 nodes showing the 3 types of truckloads in the routing problem ....................................................................................................................................................... 34 Figure 3-4: Sample of a daily cartage report ............................................................................................... 41 Figure 3-5: Comparison of different cost components in the company's original routes and the routes obtained from the optimization models ...................................................................................................... 44 Figure 3-6: Number of truckloads for each type of truck ........................................................................... 45 Figure 3-7: Transportation cost per week for each truck type .................................................................... 46 Figure 3-8: Number of truckloads per week of each truckload type in the supply chain ........................... 47 Figure 3-9: Average number of truckloads of each truck type and each truckload type in the supply chain .................................................................................................................................................................... 48 Figure 3-10: Snapshot of the main worksheet of the decision support tool ................................................ 51 Figure 3-11: Snapshot of the results sheet obtained from the decision support tool .................................. 52 Figure 4-1: Efficiencies of the existing and the proposed new gasification systems (provided by Nexterra Systems Corp.) ............................................................................................................................................ 72 Figure 4-2: Total emission in carbon tax (and carbon cap-and-trade) and carbon offset models for different carbon prices ............................................................................................................................................... 79 Figure 4-3: Total cost in the carbon tax model for different carbon prices ................................................ 80 Figure 4-4: Optimum costs of the four carbon cap-and-trade models for different carbon prices .............. 81 Figure 4-5: Optimum costs of carbon offset models for different carbon prices ........................................ 82 Figure 4-6: Optimal costs of the carbon tax and carbon cap-and-trade models .......................................... 84 Figure 4-7: Optimal costs in the carbon tax and carbon offset models ....................................................... 85   xiii  Figure 4-8: Optimum costs of carbon cap-and-trade and carbon offset models for different carbon prices .................................................................................................................................................................... 86 Figure 4-9: Optimal decisions of the models with carbon tax and the four carbon cap-and-trade policies (optimal decisions of the models with carbon tax and carbon cap-and-trade policies are equal for a given carbon price) ............................................................................................................................................... 87 Figure 4-10: Annual biomass and natural gas consumption prescribed by the model with carbon offset policy with a compliance target of 2000 tonnes CO2-eq for different carbon prices ................................. 88 Figure 4-11: Annual biomass and natural gas consumption prescribed by the model with carbon offset policy with a compliance target of 3000 tonnes CO2-eq for different carbon prices ................................. 88 Figure 5-1: Pareto-optimum solutions for the model without carbon pricing and the model with the carbon tax model (tax = $50 per tonne CO2-eq), the carbon cap-and-trade policy (initial allowance = 2000 tonnes and price=$50 per tonne CO2-eq), and carbon offset policy (compliance target = 2000 tonnes and price=$50 per tonne CO2-eq) ....................................................................................................................................... 96    xiv  Acknowledgements I would like to wholeheartedly thank my supervisor, Prof. Taraneh Sowlati for her incessant support during my PhD. Her knowledge, passion, patience and hard work constantly encouraged me throughout this journey. I feel very fortunate to have been supervised by her. I fall short of words to express my thanks to her for the support and encouragement she provided especially during many stressful times.  I would like to take this opportunity to thank my supervisory committee member, Dr. Julie Cool, for always being available for committee meetings and providing feedback on my research work and the dissertation. I would like to thank Dr. Bruce Larson for being in my supervisory committee until he retired and sharing his knowledge on forest operations and transportation. I would like to specially thank Prof. James Vercammen for examining the thesis and providing constructive comments to improve the overall quality of the thesis, especially the chapters on carbon pricing policies. The comments provided by Prof. Phil Evans were very helpful to improve the overall quality and highlight the scientific contribution of the work.  This work would not have been possible without the financial support I received from UBC (strategic recruitment fellowship and several internal awards) and from NSERC for several projects I worked on. I express my gratitude to these agencies.  The support I received from the industry and academic collaborators requires a special mention. I would like to thank Mr. Robby Gill and his staff at the logistics company for sharing the data and details about their operations. Special thanks to Dr. Olivier Quirion-Blais for being a part of the work in Chapter 3 and developing the decision support tool for the company. I cannot miss to appreciate Mr. Cliff Mui of Nexterra Systems Ltd., and Mr. Faisal Mirza of the City of Vancouver for their collaboration on different projects.  I made great friends at the Industrial Engineering Research Group, UBC. Working with Shaghayegh and Evelyn on different projects was a pleasure. Taking courses and doing assignments with Luke Opacic was very helpful. Many interactions and experiences with my group members, Claudia Cambero, Mahdi Mobini, Mehdi Piltan, Diana Siller and Jan Maier, added a great flavor to my time at UBC. Research-related and other informal conversations with the newer members of the group, Sahar Ahmadvand, Salar Ghotb, Rohit Arora, and Maziyar Khadivi were great fun! I thank my research group for all the great experiences. I cannot miss the chance to thank the amazing staff at the Department of Wood Science and the Faculty of Forestry at UBC.  A significant part of my experience at UBC involved UBC Residence Life for whom I worked as a Residence Advisor. This job taught me many life skills that I wouldn’t have learnt otherwise. I worked with   xv  amazing people who never failed to surprise me with their energy and enthusiasm. I thank them for all the wonderful times.  During many stressful times, music was my stress-buster. I would like to express special thanks to my music school, Pandit Jasraj School of Music Foundation, Vancouver, for being extremely supportive. I would like to thank my music teacher and a wonderful human being, Asha Aunty. The school has given me many wonderful opportunities to practice and sing in front of the audience. Serving on the board of the school helped me improve my inter-personal and management skills. I cannot thank the school enough for giving me all these opportunities. My Sargam-UBC family deserves a special mention here. This music club at UBC is the idea of my friend and roommate, Abhiram, who asked me to support as the Vice President of the club. Since my first year at UBC, the activities we had at Sargam made my free time very enjoyable. Working with enthusiastic musicians always motivated me to learn and practice more. Sargam gave me a chance to make great friends with Sneha with whom I practiced and sang several times. The long and interesting conversations I had with Sneha can never be forgotten!  I am also very thankful to my amazing roommates, Abhiram and Ankur, for being family away from India. Having good roommates is extremely important, and I think I have been very lucky in this aspect. I would also like to thank my friends Bhavin, Payel, Rohan, Shruti and Hamza for all the fun times we had together.   I need to specially thank my friends from India who have always been there for me. Our hangouts, trips and some serious conversations kept me going in my PhD. Divyanshu, Arvind, Ankita, Anubha, Sonal and BV, thanks to you all!  I want to mention my professor from my undergrad, Prof. S.K. Gupta, who inspired me to pursue research in Operations Research. His advice of asking the question “why” before doing anything in research helped me a lot, not just in research but also in my personal life. It was a privilege to be mentored by him. Thanks to my previous supervisors, Prof. Abraham Punnen and Dr. Snezana Mitrovic-Minic, for helping me grow as a researcher.  I need to specially thank my family in India who supported me in all possible ways. My wife, Varaynya, has always been very supportive during this journey. There were times when I did not have enough time to talk to her due to my work. She was always understanding. I cannot thank her enough! My parents incessantly encouraged me to pursue my PhD and supported me throughout this journey. My brother and sister-in-law, with their immense grit and determination, always inspired me to face difficult times with courage. My tata garu may not be around now, but his great words of advice always stay with me. My family, despite the physical distance, always supported me mentally! As a gesture of gratitude, I dedicate this dissertation to my family and teachers.     xvi  Dedication          To my family and teachers      1  Chapter 1: Introduction  1.1 Background Biomass is a clean and renewable source of energy that has gained importance in the recent years. Utilizing biomass as a source of energy has several advantages. It is a versatile source that can be used to generate heat, electricity, biofuels or a combination of them (Shabani and Sowlati 2013). Biomass can also be stored and be used on-demand (Rentizelas et al. 2009). Because of its local availability, biomass can increase fuel security and reduce carbon dioxide emissions (Asadullah 2014). Due to numerous advantages of using biomass, significant effort has been made in developing advanced technologies to convert it to energy and fuels.  Despite several advantages, numerous barriers impede the wide-spread utilization of biomass for energy generation. One of the main barriers is that not all technologies that convert biomass to energy and fuels are commercialized yet (Malico et al. 2019). Logistics are also realized to be important in planning bioenergy/fuel production systems (Ba et al. 2016). The logistics cost is a major component of bioenergy and biofuel costs (Rentizelas et al. 2009). In some cases it could represent as much as 90% of the total feedstock cost (Ekşioğlu et al. 2010). In addition to cost, due to the dependence on fossil fuels, logistics operations also contribute  to emissions from utilizing biomass (Cambero et al. 2015). Therefore, improvements in logistics could play a key role in enhancing biomass utilization (Gold and Seuring 2011).  Biomass logistics involve operations related to collection, storage, pre-processing and transportation of biomass (Ekşioğlu et al. 2010). Collection of biomass deals with picking up biomass from supplier locations, and it can be demand-driven or supply-driven. In demand-driven collection, biomass is picked up from suppliers to meet the demand at conversion facilities. Supply-driven collection deals with the pickup of the entire quantity of biomass available at the supply points, irrespective of the demand. Storage of biomass deals with decisions related to the location and quantity of biomass to be stored. Biomass can be stored at supply points, conversion facilities, and/or at intermediate storage facilities. Pre-processing includes activities such as comminution and drying of biomass to meet the size and quality requirements for the conversion process. Transportation relates to the movement of biomass between different locations of the network. Similar to other forest products, biomass is transported by trucks, trains and ships, while trucking is the main mode of transportation in many regions (Atashbar et al. 2018). Planning biomass logistics is complex due to the characteristics of biomass such as its seasonal availability, scattered geographical distribution, and quality variations, as well as inter-dependencies among logistics operations (Caputo et al. 2005). Figure 1-1 shows a schematic representation of biomass logistics operations.     2  Biomass supply areas• Harvest/ collection• Storage• Pre-processingAgriculture farmsForest sitesWood processing millsIntermediate storage facilities• Storage• Pre-processingBio-conversion facilities• Storage• Pre-processing• Energy productionBio-conversion facilitiesIntermediate storage facilitiesAgriculture-based biomassMill residuesAgriculture-based biomassForest-based biomassMill residuesFeedstockUpstreamStorage sites/ end usersBiofuel/BioenergyBiofuel storage sites/end users• StorageDownstreamForest-based biomassLogistics activitiesTransportation Transportation TransportationTransportationBiomass supply chain Figure 1-1: Biomass supply chain and logistics activities     3  Numerous studies developed mathematical models to optimize biomass logistics. These models can be divided into two groups based on the decision planning level: tactical and operational levels. Tactical level planning deals with medium-term planning horizon such as a year with either weekly (e.g., (van Dyken et al. 2010)) or monthly decisions (e.g., Flisberg et al. (2012) and Gunnarsson et al. (2004)). Most of the literature on biomass logistics optimization focused on the tactical level planning. All these studies included transportation decisions in their mathematical models as product flow values between different locations of the network during each period of the planning horizon. A few of these studies, such as those by Akhtari et al. (2014) and De Meyer et al. (2015), included decisions related to the storage of biomass at intermediate facilities to account for interrupted supply of biomass. Other studies including Shabani and Sowlati (2013) and Memişoğlu and Uster (2015) considered direct delivery of biomass from supply points to demand points without including an option to store biomass at intermediate facilities. Decisions related to pre-processing of biomass were included in some studies (e.g., (Akhtari et al. 2014; Gautam et al. 2017)), while other studies that did not include these decisions, and either considered the supply of biomass in its pre-processed form (e.g., Shabani and Sowlati (2013)) or assumed pre-processing to happen at the conversion facilities (e.g., Huang et al. (2014) and  Memişoğlu and Üster (2015)). While significant work has been done in optimizing biomass logistics at the tactical level, optimization of logistics decisions at the operational level has received relatively limited attention. Biomass logistics optimization at operational level involves short-term planning horizon such as a week or a day. Transportation planning at the operational level involves decisions related to biomass flow between different locations and daily routes to be taken by trucks. Storage and pre-processing decisions at the operational level must be taken along with the biomass flow and truck routing decisions. Operational level plans are crucial for efficient implementation of logistics decisions (D’Amours et al. 2010). Previous studies that considered optimization of biomass logistics at the operational level focused on either the optimal allocation of harvest equipment over biomass supply sites for collection of biomass (e.g., Aguayo et al. (2017) and Zamar et al. (2017a)) or optimization of truck routes for transporting biomass over a single-day planning horizon (e.g., Han and Murphy (2012) and Zamar et al. (2017b)). Studies that focused on transportation of biomass considered biomass in the form of chips, therefore biomass pre-processing was not included in these studies. In addition, since the models were developed for single-day planning horizon, storage decisions were not included in them. Further, the network considered in these studies included only biomass suppliers and conversion plants. Intermediate storage facilities were not included in the networks.  Moreover, transportation decisions were simplified in the models. For example, Han and Murphy (2012) considered pre-determined number of truckloads of each product to be picked up from each   4  supply point and be delivered to each demand point. Therefore, the mathematical model in their study focused on routing of trucks to satisfy the pre-determined transportation orders. The study by Zamar et al. (2017b) included decisions related to the quantity of biomass to be collected from each supplier under biomass quality uncertainty. However, their study considered delivery to a single demand point making the transportation decisions less complex. To the best of my knowledge, biomass logistics optimization models for short-term planning including supply points, intermediate storage facilities and multiple demand points, along with decisions related to biomass storage, pre-processing, flow, and truck routing using heterogeneous trucks were not developed in the literature. Most of the literature on biomass logistics optimization focused on minimizing the total cost of biomass logistics, with limited attention  given to mitigating emissions from biomass logistics operations. Few recent studies considered emissions from biomass logistics operations by including the cost of emissions in their cost objective functions (e.g., Palak et al. (2014) and Memari et al. (2018)).  The cost of emissions in these studies was defined based on the carbon pricing policies that are currently implemented in different jurisdictions in the world.  Carbon pricing policies are instruments that aim at mitigating carbon emissions by putting a price on them. Carbon pricing gives emitters a choice between halting their activities, altering their activities, or paying for the emissions (The World Bank 2019). The revenue generated from carbon pricing is utilized for green initiatives by governments. Governments around the world agree that pricing carbon emissions is the cheapest and most effective tool to meet their emission reduction targets (Environment and Climate Change Canada 2018). According to a report by the World Bank, as of April 1st, 2019, carbon pricing has been implemented or planned to be implemented in 46 national and 28 sub-national jurisdictions (World Bank Group 2019).  There are three main carbon pricing policies in practice: 1) carbon tax, 2) carbon cap-and-trade, and 3) carbon offset policies. In the carbon tax policy, emitters are charged a fee for every unit of carbon they emit. The carbon tax policy is implemented in countries/regions such as the Province of British Columbia in Canada, Finland, Sweden, and Switzerland (Haites 2018). It is believed that a high carbon tax rate motivates the replacement of fossil fuels with cleaner options by making it more economical to generate cleaner energy (Carbon Tax Center 2019b).  In the carbon cap-and-trade policy, governing authorities set an overall cap on carbon emissions of a jurisdiction and split the emission allowances among emitters (Environmental Defense Fund 2019). Emission allowances are given to emitters either for free or through an auction (Goulder et al. 2010; Waltho et al. 2019). Emitters can sell (or buy) emission credits to (or from) the carbon market when their total   5  emission is less (or more) than the allocated initial allowance. The initial allowances allocated to emitters is reduced over time which gives incentives to emitters to shift to low-carbon technologies (Environmental Defense Fund 2019). The carbon cap-and-trade system is in place in regions such as the European Union, California in the United States of America, the province of Quebec in Canada, and seven regions in China (Haites 2018). Due to the involvement of auctioning and trading carbon allowances, the implementation of carbon cap-and-trade system is complex (Wittneben 2009). In the carbon offset policy, emitters offset their emissions beyond a compliance target by purchasing emission credits from certified emission reduction sources (Zhou and Wen 2019). Emission credits are purchased from low-carbon energy projects and projects that reduce emissions (United Nations Carbon Offset Platform 2019; Carbon Tax Center 2019a). The carbon offset policy is adopted in Australia under the name of “Safeguard Mechanism” (Commonwealth of Australia 2018). According to the Australian Government, uncertainty associated with carbon policies is a key risk that can deter investment in businesses (Commonwealth of Australia 2018). The carbon offset policy, which is characterized with certainty in the compliance target and carbon price values, can mitigate the risk associated with uncertainties in carbon policies.    Recent studies that considered emissions in biomass supply chain optimization models analyzed the impacts of carbon pricing policies on the optimal decisions, cost, and emissions of the models. These studies can be categorized into those that dealt with strategic planning level and tactical planning level. Studies that focused on the strategic level considered either the carbon cap-and-trade policy (e.g., Giarola, et al. (2012) and Ortiz-Gutiérrez et al. (2013)), or the carbon tax policy (e.g., Mohamed Abdul Ghani et al. (2018)), or all the three carbon pricing policies (e.g., Marufuzzaman et al. (2014) and Marufuzzaman et al. (2014)). These studies developed optimization models to determine the location of bio-conversion facilities for converting biomass to energy. Studies that analyzed the impact of carbon pricing policies on biomass logistics optimization models at the tactical level focused on decisions related to biomass supplier selection, transportation mode selection, and inventory of biomass at conversion facilities (e.g., Palak et al. (2014) and Memari et al. (2018)). It was generally observed in these studies that increasing the carbon price and decreasing the carbon cap result in emissions reduction. Few studies mentioned that the carbon cap-and-trade policy is more effective than the carbon tax policy for emissions mitigation (e.g., Marufuzzaman et al. (2014a)). No previous study analyzed the impacts of carbon pricing policies on biomass logistics models at the operational level, to the best of my knowledge.  Although previous studies attempted to analyze the impacts of carbon pricing policies on biomass supply chain optimization models, the analyses conducted in these studies were restricted to the considered case   6  studies. Therefore, the observations in those studies were specific to the data used in the case studies. As mentioned by Zakeri et al. (2015), insights derived from such case-specific studies may not be applicable to other case studies and industries.  While different case studies and optimization models were considered in previous studies, the modelling framework for incorporating carbon pricing policies in these studies is the same. First, these studies developed economic optimization models without including the cost of emissions. Then, the cost-only optimization models were extended to incorporate carbon pricing policies. The carbon tax policy was considered by adding the cost of emissions to the objective function, and no new constraints were added to the cost-only optimization models. For incorporating the carbon cap-and-trade policy, the deviation of total emissions from the initial allowance is determined, and the resultant cost/revenue was included in the objective function of the cost-only optimization model. To include the carbon offset policy, the deviation of total emissions from the compliance target was determined, and the cost of purchasing carbon offsets was added to the objective function of the model. Because of this common modeling framework, the impact of carbon pricing policies on optimal cost and emissions of biomass supply chain models can be analyzed using case-independent optimization models. Conducting such analysis using case-independent optimization models is important because insights derived from these studies can be generalized and be applied to all optimization models irrespective of the underlying case study. No previous study in the literature analyzed the impacts of carbon pricing policies on optimal solutions of case-independent optimization models, to the best of my knowledge.   Previous studies that analyzed the impact of carbon pricing policies on biomass supply chain optimization models combined cost and emissions into a single-objective function by adding the cost of emissions to the cost objective function. With this type of modelling, the information about the trade-off between cost and emissions of supply chains would be lost (Wu et al. 2010). However, decision makers prefer having a set of solutions that capture the information about the trade-off between different objectives to choose an alternative based on their preferences (Konak et al. 2006; Deb and Deb 2014). Therefore, multi-objective optimization models should be developed for biomass logistics considering carbon pricing policies. Solving multi-objective optimization models to obtain the set of trade-off solutions could involve solving several single-objective optimization models separately (Miettinen 1998). This process could increase the computational effort, especially if the single-objective models are large and complex to solve. Furthermore, in order to analyze the impact of carbon pricing policies on the optimum solutions of the models with carbon pricing policies, bi-objective optimization models with different carbon policies have to be solved separately. This increases the number of single-objective models that should be solved. In addition, to assess   7  the impact of varying carbon prices and initial allowances/compliance targets, bi-objective models for a considered policy must be solved several times for different carbon prices and initial allowances/compliance targets. This increases the number of single-objective optimization models to be solved significantly. Developing efficient strategies to determine the optimal solutions of multi-objective biomass supply chain optimization models considering carbon pricing policies could reduce the computational effort by reducing the number of single-objective models to be solved.  1.2 Research objectives The overall goal of this work is to optimize forest-based biomass logistics cost at the operational level considering transportation, storage and pre-processing decisions, and to analyze the impacts of carbon pricing policies on optimal cost and emissions biomass logistics optimization models independent of the underlying case study. This goal is achieved through the three following objectives. 1. Develop mathematical models to optimize forest-based biomass logistics at the operational level considering biomass flow, storage, pre-processing and truck routing operations. These models will consider multiple types of biomass and multiple types of trucks, where each type of truck can carry specific types of biomass. The models will include biomass suppliers, intermediate storage facilities and bio-conversion facilities, where each location can be visited by specific types of trucks. The models will be applied to a real case study.  2. Analyze the impacts of carbon pricing policies on optimal cost and emissions of case-independent biomass logistics optimization models. The impact of different carbon prices and initial allowances/compliance targets on the optimal cost and emissions of the models will be analyzed. For this purpose, several propositions that describe the dependence of optimal cost and emissions of case-independent models on carbon pricing policies will be proposed and proved mathematically. The propositions will be numerically verified using a case study.  3. Develop bi-objective biomass logistics optimization models considering carbon pricing policies. The models will have two objective functions: 1) minimizing total cost including the cost of emissions defined by carbon pricing policies, and 2) minimizing total emissions. A new algorithm will be developed to reduce the computational effort involved in solving bi-objective optimization models considering carbon pricing policies. The applicability of the algorithm will be demonstrated using a case study.  1.3 Outline of the thesis In addition to the current chapter, this dissertation consists of the following chapters.    8  The studies on biomass logistics optimization models and those that analyzed the impacts of carbon pricing policies are reviewed in Chapter 2. In Chapter 3, mathematical models are developed for optimizing biomass logistics at the operational level considering transportation, storage and pre-processing decisions. The models are applied to the case study of a large biomass logistics company.  The impact of carbon pricing policies on optimal cost and emissions of case-independent biomass logistics is described in Chapter 4. Several properties of optimum cost and emissions of models under different carbon pricing policies are proposed and proved mathematically. The case of a biomass-fed district heating plant is described, and an optimization model is developed to determine the optimal feedstock mix at the plant to meet its heat demand. The impact of different carbon pricing policies on the optimization model developed for the district heating plant is analyzed to verify the propositions.  In chapter 5, bi-objective optimization models are developed for the case of the biomass-fed district heating plant described in Chapter 4. The two objectives considered are minimizing total emissions and minimizing total cost, which includes the cost of emissions. A new algorithm is developed to solve bi-objective optimization models considering carbon pricing policies.  Finally, Chapter 6 concludes the dissertation with a note on strengths and limitations of this work and the direction for future work.     9  Chapter 2: Literature review  2.1 Synopsis Biomass logistics involve inter-dependent operations related to harvesting and collection, storage, transportation, and pre-processing of biomass. Each of these operations has many practical features that have to be taken into account for planning at tactical and operational levels. This chapter describes the important features of each biomass logistics operation and reviews previous studies that developed mathematical models to optimize forest-based and agricultural-based biomass logistics. The studies are categorized and reviewed based on the planning level considered in the models. Furthermore, studies that analyzed the impacts of carbon pricing policies on the optimal cost and emissions of biomass supply chain optimization models are reviewed. Most of the studies in the literature focused on economic optimization at the tactical planning level, while the literature on operational planning level is limited. No previous study on operational level planning considered biomass flow, storage, pre-processing and truck routing decisions simultaneously. Studies that analyzed the impacts of carbon pricing policies on biomass supply chain models focused on specific case studies. Whether the results reported in these studies are generalizable is not clear. Moreover, these studies combined cost and emissions into a single objective function due to which the information about the trade-off between the two objectives is lost.  2.2 Biomass logistics operations: key features and decisions Biomass logistics involve four main operations, namely, biomass collection, storage, transportation, and pre-processing. Key features and decisions involved in each of these operations for forest-based and agriculture-based biomass are briefly described in this section.  2.2.1 Biomass collection Biomass collection deals with procuring the required quantities of biomass from supply areas. It may include harvest operations when biomass, such as energy wood, is not readily available and must be harvested before collection (Gunnarsson et al. 2004). Planning the collection of forest residues may not include harvest planning as harvesting for logs is usually planned prior to biomass collection.  Interrupted availability characterizes forest-based biomass supply. Inaccessibility to forest areas during some months results in interrupted supply of forest residues in countries such as Canada (Akhtari et al. 2014; Gautam et al. 2017), Sweden (Gunnarsson et al. 2004), Austria (Kanzian et al. 2013), and the United States of America  (Ekşioğlu et al. 2009; Marufuzzaman and Ekşioğlu 2017). Due to the restricted supply season, the collection period of several supply areas may overlap making the collection process equipment-   10  and labor-intensive. The harvest equipment may have to be scheduled among several suppliers under tight time windows making the collection-scheduling complex. Biomass collection includes decisions related to the selection of suppliers and determination of the quantity of biomass to harvest/collect from each supplier. At the operational level, biomass collection includes decisions related to scheduling of harvest and collection equipment to harvest/collect biomass from each supplier (Aguayo et al. 2017). 2.2.2 Storage The main driver for storing biomass is to match its supply and demand during the entire planning period (Gold and Seuring 2011). Interrupted supply of biomass, which is resulted from seasonalities, and uncertainties present in biomass supply chain makes storage a crucial logistics operation. Storage decisions include the quantity of biomass to be stored and the type of storage system to be used at different locations of the supply chain. Biomass storage decisions are generally taken along with other logistics decisions in the optimization models. Incorporating storage decisions requires models with multiple planning periods where the quantity of biomass stored at the end of each period is considered as a decision variable.  Biomass can be stored at different locations of the network including supply sites, intermediate storage facilities and conversion facilities (Sowlati 2016). Storing biomass at different locations may have different logistical implications. For example, storing residues at wood processing mills could be time-constrained due to limited storage space in over-head bins used to store the residues (Nadimi 2015). Therefore, mill residues are typically not stored at their respective supply areas for a long time.  On the other hand, forest residues may be stored at the forest sites for several months after harvest to reduce the high moisture content through open-air drying (Sowlati 2016). Storage of biomass at conversion facilities may also be constrained by the limited storage capacity (Gronalt and Rauch 2007).  While storage of biomass is essential to maintain a consistent supply of feedstock, biomass dry matter losses caused due to quality deterioration is a risk associated with storing biomass for a long duration (He et al. 2014). Biomass stored for long periods of time may decompose, and as a result, it may not be useful in the conversion process (Memişoğlu and Üster 2015). In addition, Storing biomass for long duration may pose a risk of fire due to the internal heat generated as a result of respiration of living cells in biomass (Fuller 1985). 2.2.3 Transportation Transportation deals with the movement of biomass between different locations of the network. High transportation cost due to long transportation distances is observed as one of the main contributors to the   11  high biomass logistics cost. Moreover, due to its low energy density, large quantities of biomass is required to meet the demand, and as a result the total cost is increased  (Rentizelas et al. 2009; Sosa et al. 2015). Different modes of transportation such as truck, rail and barge are used to transport biomass. Trucks, which are used widely for biomass transportation, are found to be economical only when the transportation distances are short (Roni et al. 2017). Rail and barge are considered cost-effective for long distance and high volume transportation of biomass (Marufuzzaman and Ekşioğlu 2017; Andersen et al. 2017). However, the use of these modes may be restricted due to the limited access of biomass supply and demand locations to these modes of transportation. Transportation decisions are made at both tactical and operational levels. For tactical level planning, the decisions include the biomass flow quantity between different locations per period, which is typically a month or a week (e.g., Ekşioğlu et al. (2009) and Akhtari et al. (2014)). When multiple modes of transportation are used, the decisions also include the selection of the mode of transportation (e.g., Palak et al. (2014)). Transportation of biomass at the operational level deals with biomass flow decisions over a short-term horizon such as a week or a day, and decisions on daily truck routes and schedules to carry out biomass pickup and delivery operations (e.g., Zamar et al. (2017b) and Han and Murphy (2012)). 2.2.4 Pre-processing Biomass pre-processing includes operations such as sorting, grinding/chipping, drying and densification (Humboldt 2012). Pre-processing is done in order to increase the transportation efficiency and improve the feedstock quality (Gold and Seuring 2011; Humboldt 2012). Biomass may or may not require additional pre-processing operations depending on its type and the harvesting method (Humboldt 2012). Logging residues such as non-merchantable logs, tops and branches which are larger in size require comminution before they are used in the conversion process (e.g., Akhtari et al. (2014)). On the other hand, mill residues including sawdust and shavings do not need comminution as they are in a usable form (e.g., Shabani and Sowlati (2013)).  Pre-processing of biomass can take place at different locations including the supply sites, intermediate facilities, or conversion facilities (Sowlati 2016; De Meyer et al. 2015). Transportation of pre-processed biomass is considered more efficient due to the increase in biomass density (Humboldt 2012; Flisberg et al. 2015). Thus, pre-processing of biomass at forest sites and transporting them to storage or conversion facilities would be the most efficient strategy for reducing transportation-related costs. However, this requires equipment such as mobile chippers and grinders which have higher cost and lower efficiency compared to stationary equipment at storage sites or conversion facilities (Flisberg et al. 2015). As a result, pre-processing of biomass is more efficient at intermediate facilities or conversion facilities. Depending on   12  the distances, type of equipment used for pre-processing, and transportation mode and capacity, one of these alternatives may be preferable.  Biomass pre-processing includes decisions related to the location and the type of pre-processing, the quantity of biomass to be pre-processed, and scheduling the pre-processing operations. Pre-processing of biomass has mostly been considered along with transportation and storage decisions in the optimization models. While biomass pre-processing includes drying and densification as well, most of the models only looked at grinding and chipping decisions. 2.3 Biomass logistics optimization modeling This section provides a brief review of the studies that developed optimization models for both forest-based and agriculture-based biomass logistics at the tactical and operational levels. Detailed review of studies that developed optimization models for biomass logistics is published in three articles that are listed in the Preface of this dissertation. To avoid a lengthy literature review chapter, examples of studies that incorporated different features that are relevant to the topics presented in this dissertation are reviewed in this section.  2.3.1 Biomass logistics optimization models at the tactical level The studies that developed optimization models at the tactical level considered medium-term planning horizons such as one year with either weekly (e.g., van Dyken et al. (2010)) or monthly decisions  (e.g., Flisberg et al. (2012) and  Gunnarsson et al. (2004)).  To handle interrupted biomass supply, a group of studies in this category included decisions related to the storage of biomass at intermediate facilities in their optimization models (e.g., Akhtari et al. (2014), Flisberg et al. (2012) and De Meyer et al. (2015)). In these models, biomass from supply areas could directly be sent to conversion facilities, or to intermediate facilities for storage. Another group of studies such as those by Shabani and Sowlati (2013) and Memişoğlu and Üster (2015) considered direct delivery of biomass from supply points to demand points without including an option to store biomass at intermediate facilities. Storing biomass at intermediate storage facilities enables handling and storing large volumes of biomass for long durations (Zhang et al. 2016). However, storage at intermediate facilities requires additional transportation of biomass, from the supply sites to storage facilities and from the storage sites to conversion facilities (Rentizelas et al. 2009). Due to the additional costs for transportation, loading and unloading operations, storing biomass at intermediate storage facilities might increase the total logistics cost (Akhtari et al. 2014; Kanzian et al. 2009). A general observation in the literature was that the direct delivery of biomass from supply sites to conversion facilities, whenever possible, was more economical than storing   13  biomass at intermediate facilities to avoid the additional transportation and handling costs at the storage sites.  On the contrary, few studies such as Gunnarsson et al. ( 2004) and Gautam et al. ( 2017) reported a decrease in the total logistics cost when biomass was stored at intermediate storage facilities. Gautam et al. (2017) highlighted that cost reduction was possible due to the improvement in biomass quality when it was stored at intermediate facilities. Pre-processing of forest residues at supply sites, intermediate facilities, and conversion facilities was considered in few models in the literature (e.g., Akhtari et al. (2014) and Kanzian et al. (2009)). Pre-processing costs were assumed to be different at different locations. Although pre-processing of biomass at intermediate facilities is considered more efficient, these models suggested that pre-processing of biomass at forest sites was the most economical option as pre-processing and storing biomass at intermediate facilities required additional handling and transportation operations. All the studies that developed biomass logistics optimization models at the tactical level included transportation decisions as biomass flow values between different locations of the network during each period of the planning horizon. Most of the studies assumed transportation cost parameters such as transportation and loading/unloading costs per unit flow of biomass (e.g., Zhang et al. (2016) and Roni et al. (2017)). Transportation cost was included in the objective function in the models by multiplying the total flow of biomass with the cost parameters.  As highlighted in the literature, trucks are considered to be economical for transporting biomass over short distances (Roni et al. 2017). Most of the studies in the literature considered trucks for biomass transportation (e.g., Shabani and Sowlati (2013) and Akhtari et al. (2014)). Several recent studies considered the transportation of biomass over long distances using rail and barge (e.g., Marufuzzaman and Ekşioğlu (2017), Roni et al. (2017) and Andersen et al. (2017)). These studies assumed that trucks were used to transport biomass from supply areas to multi-modal facilities, which had access to rail or barge. Biomass was transported over long distances using rail or barge because of their larger capacity compared to trucks. Andersen et al. (2017) reported 8% increase in the profit margin when the barge-truck combination was used compared to using only trucks.  While most of the studies focused only on the economic optimization of biomass logistics, the environmental objective of minimizing the total emissions due to logistics operations was considered in few studies (e.g., Ng and Maravelias (2016) and Roni et al. (2017)). Ng and Maravelias ( 2016) included emissions due to biomass harvesting, pre-processing, conversion process and transportation in their model, which dealt with determining the locations of the conversion facilities and intermediate depots for pre-processing of biomass. They concluded that setting up a single conversion facility was the most economical   14  solution, while setting up two facilities was shown to have lower emissions due to shorter transportation distances. However, they did not consider total emissions due to setting up the two facilities while the economic objective function included cost due to facility setup. A similar observation was made in the study by Roni et al. (2017) whose single-objective model with cost minimization suggested to set up a single facility with large capacity, while their multi-objective model with economic, environmental and social objectives suggested multiple facilities of smaller capacities. Moreover, their cost minimization model suggested increased use of trucks to avoid capital costs for setting up rail hubs, whereas their multi-objective model suggested increased usage of rail for transporting biomass as it created more jobs and reduced total emissions. They included the total emissions from biomass transportation, biofuel production and setting up transportation hubs in their environmental objective function. The studies reviewed in this section are summarized in Table 2-1.      15  Table 2-1: Summary of important features of biomass logistics at the tactical level, examples of studies that incorporated them Important logistics features Examples of studies Modeling aspects/important findings Interrupted availability of biomass Kanzian et al. (2013) Akhtari et al. (2014)  Biomass supply was assumed to be inconsistent due to seasonal availability.   Storage of biomass at intermediate facilities Akhtari et al. (2014)  Kanzian et al. (2009) Biomass storage at intermediate facilities was considered. Storage at intermediate facilities increase the cost due to the extra transportation and handling operations. Gunnarsson et al. (2004)  Gautam et al. (2017) Biomass storage at intermediate facilities was included. Storage at intermediate facilities would result in cost savings due to biomass quality improvement. Pre-processing of forest residues at intermediate sites Kanzian et al. (2013) Akhtari et al. (2014)  Pre-processing of biomass was considered at forest areas, intermediate sites and conversion facilities. Results suggested that pre-processing at forest sites was the most economical option.  Transportation cost using cost per unit volume or cost per vehicle load Akhtari et al. (2014)  Zhang et al. (2016) Roni et al. (2017) Transportation decisions were modeled as the flow of biomass between different locations. Cost parameters for unit flow of biomass were considered to calculate the total cost. Roni et al. (2017) considered the flow values were defined as number of vehicle loads. Transportation cost was associated with each vehicle load of biomass Inter-modal distribution for long distance transportation of biomass  Marufuzzaman and Ekşioğlu (2017)  Roni et al. (2017)  Andersen et al. (2017) A combination of trucks and rail was used to transport biomass over long distances. Andersen et al. (2017) reported 8% increase in total profit when barge and truck were used compared to using only trucks. Emissions from logistics operations Ng and Maravelias (2016)  Roni et al. (2017) Models for minimizing total emissions suggested to install multiple conversion facilities, while models for minimizing the total cost suggested setting up a single large facility.   2.3.2 Biomass logistics optimization models at the operational level While significant work has been done in optimizing biomass logistics at the tactical level, its short-term planning (operational level) has received limited attention. Biomass logistics optimization at operational level includes decisions related to collection, transportation, storage and pre-processing of biomass over a short-term planning horizon such as a week or a day. Previous studies in this category developed optimization models considering single-day planning horizon, and focused either on the allocation of   16  harvest equipment among biomass collection facilities (e.g., Zamar et al. (2017a) and Aguayo et al. (2017)) or the transportation of biomass (e.g., Han and Murphy (2012) and Zamar et al. (2017b)). None of the studies considered decisions related to storage of biomass. This section provides a brief review of these studies.  Studies related to optimizing the collection of biomass focused on allocating the biomass collection equipment over either a single supply site (e.g., Zamar et al. (2017a)), or multiple supply areas (e.g., Aguayo et al. (2017)). Zamar et al. (2017a) addressed the issue of dispersed availability of biomass in several small piles within a supply area by developing two optimization models. The first model was used to cluster biomass into several piles in the supply site, and the second model was used to optimize the collection of biomass piles using the collection equipment. Aguayo et al. (2017) studied the collection of biomass from several supply sites which had overlapping harvesting periods using limited equipment. They assumed that one conversion facility was responsible for managing the biomass harvest and collection operations. They developed an optimization model which included decisions related to allocation of different types of collection equipment to the biomass supply sites and the routing of equipment between them. Transportation of biomass at the operational level includes decisions related to the flow of biomass between different locations and the routing of biomass trucks. Previous studies that focused on biomass transportation at the operational level focused only on either biomass flow decisions (e.g., Zamora-Cristales et al. (2015)) or truck routing decisions (e.g., Han and Murphy (2012), Gracia et al. (2014) and Zamar et al. (2017b)). None of the previous studies considered both biomass flow and truck routing decisions in their models. Zamora-Cristales et al. (2015) addressed the complexity related to the inaccessibility of large chip trucks that pick up wood chips from forest sites. They addressed this issue in their model by considering moving biomass from harvest areas to concentration yards using smaller trucks, where the concentration yards are within the harvest area but more accessible to the road. Biomass was pre-processed at the concentration yards and transported to conversion facilities using large chip vans.  Studies that focused on optimizing the truck routes dealt with the transportation of mill residues (e.g., Han and Murphy (2012) and Zamar et al. (2017b)). The network considered in these studies included biomass supply sites and either a single destination (e.g., Zamar et al. (2017b)) or multiple destinations (e.g., Han and Murphy (2012)). The study by Han and Murphy (2012) considered the number of truckloads of biomass to be picked up from each supply point to be pre-determined. Therefore, the model in their study focused on routing of trucks such that the pre-determined transportation orders were satisfied. On the contrary, instead of assuming pre-determined transportation orders, Zamar et al. (2017b) included decisions related to the quantity of biomass to be collected from each supplier under biomass quality uncertainty. In their   17  model, trucks could pick up residues from multiple sawmills to accumulate full truckloads of biomass, and the full-truckload was delivered to the destination. However, their study considered delivery to a single demand point using a homogeneous fleet of trucks making the transportation decisions less complex. To the best of my knowledge, biomass logistics optimization models at the operational level including supply points, intermediate storage facilities and demand points, with decisions related to biomass storage, pre-processing, biomass flow and truck routing have not been developed in the literature yet.  The studies reviewed in this section are summarized in Table 2-2. Table 2-2: Summary of important features of biomass logistics at the operational level, examples of studies that incorporated them Important logistics features Examples of studies Modeling aspects/important findings Optimizing biomass collection Zamar et al. (2017a) Aguayo et al. (2017) Biomass piles were partitioned into several clusters, and the collection equipment was routed in each cluster to collect biomass in the study by Zamar et al. (2017a). Biomass collection equipment was allocated to different suppliers and routed between supply areas in Aguayo et al. (2017).  Routing of biomass trucks Han and Murphy (2012) Zamar et al. (2017b)  Han and Murphy (2012) assumed pre-determined transportation orders, and developed routing model to satisfy the transportation orders. Zamar et al. (2017b) did not include biomass decisions as their network had a single demand point.   Biomass flow and pre-processing Zamora-Cristales et al. (2015) Biomass was transported from forest sites to concentration yards using small trucks. Biomass was pre-processed at concentration yards and sent to conversion facilities using large trucks.  2.4 Impact of carbon pricing policies in biomass supply chain optimization models This sections reviews the studies that analyzed the impacts of carbon pricing policies on optimal cost and emissions of biomass supply chain optimization models.  Environmental concerns related to emissions from logistics operations have been considered in few recent studies by considering carbon pricing in their models (e.g., Palak et al. (2014) and Memari et al. (2018)). Studies that considered carbon pricing policies in biomass supply chain optimization models can be categorized into those that dealt with strategic and tactical level planning.  Studies that focused on the strategic level planning considered either the carbon cap-and-trade policy (e.g., Giarola et al. (2012) and Ortiz-Gutiérrez et al. (2013)), or the carbon tax policy (e.g., Mohamed Abdul Ghani et al. (2018)), or all the three carbon pricing policies (e.g., Marufuzzaman et al. (2014a) and   18  Marufuzzaman et al. (2014b)). These studies developed optimization models to determine decisions related to the location of bio-conversion facilities for converting biomass to energy. Giarola et al. (2012) concluded that optimization models with carbon cap-and-trade policy prescribed investment in bioethanol production because a part of the total revenue was from selling emission credits in the carbon market. Ortiz-Gutiérrez et al. (2013) reported that low price of carbon did not have any impact on the optimal supply chain design, while higher prices resulted in different designs. Studies by Marufuzzaman et al. (2014a) and Marufuzzaman et al. (2014b) concluded that in order to reduce emissions, having a strict cap on total emissions was more effective than pricing emissions. Recently, Mohamed Abdul Ghani et al. (2018) studied the impact of pricing carbon emissions and incentives for utilizing corn stover for biofuel production instead of burning them in the fields. They observed that both biomass conversion incentives and emissions pricing resulted in increased biomass utilization and decreased emissions compared with burning biomass in open fields. However, biomass conversion incentives resulted in increased profits while emissions pricing resulted in decreased profits compared with burning corn stover in open fields.  Studies that analyzed the impact of carbon pricing policies on biomass supply chain optimization models at the tactical level planning focused on decisions related to biomass supplier selection, transportation mode selection, and inventory of biomass at conversion facilities (e.g.,, Palak et al. (2014) and Memari et al. (2018)). Palak et al. (2014) concluded that the optimization models were less sensitive to changes in carbon price compared to changes in the initial allowance. In addition, they reported that as the carbon price increases and/or the initial allowance decreases, the model suggests procuring more biomass from local suppliers as it is a greener option as compared to procuring biomass from non-local suppliers. Memari et al. (2018) concluded that with increasing carbon prices in the carbon tax model, the total cost of the model increases and total emissions decrease.  Although studies reviewed in this section analyzed the impacts of carbon pricing policies on biomass supply chain optimization models, the models were developed specific to the considered case studies. The policy insights derived in these studies were restricted to the considered case studies. Whether these results are applicable to other case studies is not clear. As mentioned by Zakeri et al. (2015), insights derived from such case-specific studies may not be applicable to other case studies and industries. While different case studies and optimization models were considered in previous studies, the underlying idea in all these studies was similar. In all these studies, a cost-only optimization model specific to the considered case study was developed. The developed cost-only optimization model was extended to incorporate carbon pricing policies by modifying the objective function of the model. The cost-only optimization model was extended to consider the carbon tax policy by adding an extra term representing   19  the cost of emissions to the objective function of the model. The carbon cap-and-trade policy was considered in the optimization model by determining the deviation of total emissions from the initial allowance, and by including the cost or revenue from buying and selling additional allowances in the objective function. The cost-only optimization model was modified to consider the carbon offset policy by determining the number of carbon offsets purchased, and adding the cost of purchasing carbon offsets to the objective function of the model. With this standard process to consider carbon pricing policies in optimization models, general and case-independent supply chain optimization models can be extended to consider carbon pricing policies. Using mathematical properties of optimization models, analysis on how carbon pricing policies impact the optimal solutions of the case-independent models can be conducted. Insights derived from case-independent models can be generalized and be applied to all optimization models irrespective of the underlying case study. No previous study in the literature on biomass supply chain optimization models analyzed the impacts of carbon pricing policies on optimal solutions of case-independent optimization models, to the best of my knowledge.   Several studies analyzed the impact of carbon cap-and-trade policy on optimal solutions of optimization models (e.g., Ortiz-Gutiérrez et al. (2013) and Palak et al. (2014)). However, these studies assumed that initial emission allowance allocated to emitters to be given free of cost. Initial allowance is given free of cost in the carbon cap-and-trade system to mitigate adverse economic impacts associated with pricing carbon emissions (Haites 2018). However, in practice, the proportion of initial allowance given free of cost would reduce over time. For example, in the European Union, the proportion of allowance given free of cost was 60% in 2013, while it is planned to decrease it to 50% during the period 2013-2020 (European Commission 2016). In the European Union, the manufacturing industry received 80% of its emission allowance free of cost in 2013, while this would decrease to 30% in 2020 (European Commission 2016).  With the increase in the proportion of auctioned allowance, it is important to analyze the impact of the price of initial allowance on optimal cost and emissions of optimization models considering the carbon cap-and-trade policy. The impact of the price of initial allowance on optimization models has not been discussed in the literature to the best of my knowledge.  Previous studies that analyzed the impact of carbon pricing on biomass supply chain optimization models focused only on strategic and tactical planning levels. The impact of carbon pricing on the operational level planning of biomass supply chains was not studied earlier to the best of my knowledge.  All studies that analyzed the impacts of carbon pricing policies on biomass supply chain models combined cost and emissions into one objective function. These single-objective optimization models resulted in one optimum solution that minimized the total cost of supply chain models for a considered carbon pricing   20  policy. Although these models determined the least-cost alternative, by combining the cost and emissions into one objective,  the trade-off between them is not obtainable  (Wu et al. 2010). Multi-objective optimization models that result in a set of Pareto-optimum solutions provide information about the trade-off between different objectives (Savic 2002; Carrillo and Taboada 2012). A solution is called Pareto-optimal if one of its objectives cannot be improved without sacrificing at least one other objective function. Therefore, Pareto-optimum solutions capture the information about the trade-off between different objectives of the model. Decision makers prefer having a set of trade-off solutions over having a single “best” solution in most real-life applications (Konak et al. 2006). This is because decision makers base their choices on their preferences which can often be qualitative, non-technical and experience-driven (Deb 2014). In such cases, providing a set of trade-off solutions can be helpful to evaluate the pros and cons of each candidate solution based on the decision makers’ preferences (Deb 2014). Other considerations such as budget and emission reduction targets can also be incorporated in the decision making process when a set of trade-off solutions is available to decision makers. In addition, the literature highlights that carbon pricing policies may impact the trade-off between these objectives of supply chain optimization models (Jin et al. 2014). Therefore, it is important to consider total cost and emissions as separate objectives in multi-objective optimization models under carbon pricing policies. However, to the best of the author’s knowledge, no previous study developed multi-objective optimization models for biomass supply chains considering carbon pricing policies.  There are three approaches for solving multi-objective optimization models depending on the point of time at which decision makers’ preferences are considered. These include a priori, a posteriori and interactive methods (Miettinen 1998). A priori and interactive methods require decision makers’ preferences to be known before and during the decision making process, respectively (Branke et al. 2008). However, obtaining the information about decision makers’ preferences without the knowledge of the trade-off solutions is a difficult task (Deb 2014). In contrast, the set of Pareto-optimum solutions is first generated, then the most preferred solution is chosen by decision makers in a posteriori method. An advantage of a posteriori method is that the decision makers’ preferences are not needed beforehand (Branke et al. 2008). However, solving multi-objective models using a posteriori method involves solving several single-objective optimization models separately (Miettinen 1998). This process could involve significant computational effort, especially if the single-objective models are large and complex to solve. Therefore, developing efficient methodologies to determine the trade-off solutions of multi-objective supply chain optimization models considering carbon pricing policies could mitigate the computational effort involved in solving these models.    21  Studies reviewed in this section are summarized in Table 2-3.  Table 2-3: Summary of studies that analyzed the impacts of carbon pricing policies on biomass supply chain optimization models  Study Considered problem Main decisions Considered carbon policies Planning horizon Case study Strategic level models Giarola et al. (2012) Biomass supply chain design for producing bioethanol  Technology selection and biomass flow.  Carbon cap-and-trade 20 years Bioethanol infrastructure in Northern Italy Ortiz-Gutiérrez et al. (2013) Biomass supply chain design for producing ethanol and biogas Technology and location selection, biomass flow, fuel flow Carbon cap-and-trade 15 years  Bioethanol production in Northern Italy Marufuzzaman et al. (2014a) Biodiesel supply chain design Facility location, transportation mode selection, biomass flow  Carbon cap, carbon tax, carbon cap-and-trade, carbon offset 10 years  Data from the state of Mississippi, USA  Marufuzzaman et al. (2014b) Biodiesel supply chain design Biocrude plant location, transportation mode selection, biomass flow  Carbon cap, carbon tax, carbon cap-and-trade, carbon offset Single period model Data from the state of Mississippi, USA Mohamed Abdul Ghani et al. (2018) Biodiesel supply chain design  Biomass harvest, biomass burnt, transportation of biomass,  Carbon tax Single period Corn farms in North Dakota, USA Tactical level models Palak et al. (2014) Biomass supplier and transportation mode selection  Supplier and transportation mode selection, biomass flow, inventory of biomass  Carbon cap, carbon tax, carbon cap-and-trade, carbon offset 1 year  Hypothetical case study using randomly generated data Memari et al. (2018) Biomass flow and biomass inventory  Transportation mode selection, inventory of biomass, backlog of biomass at different plants Carbon tax and carbon cap-and-trade 1 year  Oil-palm fruit bunches used at CHP plants in Malaysia   22  2.5 Conclusions Overall, the literature on biomass logistics optimization is vast. Most of the previous studies focused on tactical level planning with yearly planning horizons (e.g., Akhtari et al. (2014) and Ekşioğlu et al. (2009)). Studies on tactical level planning of biomass logistics incorporated several practical aspects like interrupted supply of biomass (e.g., Kanzian et al. (2009)), pre-processing of biomass at several locations (e.g., Akhtari et al. (2014)), and long-distance transportation of biomass using multiple modes of transportation (e.g, Andersen et al. (2017)). The focus of most of these studies has been on minimizing the total cost of biomass logistics, while minimizing emissions from biomass logistics operations has received limited attention in the literature (e.g., Ng and Maravelias (2016) and Roni et al. (2017)).  The literature on operational level optimization of biomass logistics is limited. Previous studies in this category focused either on optimizing the allocation of equipment for harvesting/collecting biomass (e.g., Aguayo et al. (2017) and  Zamar et al. (2017a)), or optimizing truck routes for transporting biomass for one-day planning horizon (e.g., Han and Murphy (2012) and Zamar et al. (2017b)). Studies that considered routing decisions in their models either assumed biomass flow decisions to be pre-determined (Han and Murphy 2012), or did not include biomass flow decisions (Zamar et al. (2017b)). None of these studies included biomass storage and pre-processing decisions in their models. Moreover, the network considered in these studies included biomass supply points and demand points. Intermediate storage facilities, which are used to store and pre-process biomass, were not considered in previous studies. Overall, no previous study considered biomass flow, storage, pre-processing and truck routing decisions simultaneously using multiple types of trucks in their models. Few recent studies considered emissions in biomass logistics optimization models by including the cost of emissions in the cost objective function of the models (e.g., Ortiz-Gutiérrez et al. (2013) and Palak et al. (2014)). In these studies, the cost of emissions was calculated using carbon pricing policies, namely, the carbon tax, the carbon cap-and-trade, and the carbon offset policies. These studies either considered the decisions related to biomass supply chain design and technology selection at the strategic level (e.g., Ortiz-Gutiérrez et al. (2013) and Marufuzzaman et al. (2014a)), or transportation mode selection and biomass flow decisions at the tactical level (e.g., Palak et al. (2014) and Memari et al. (2018)). These studies focused on specific case studies and reported results that were pertinent to the specific data. Whether these results are generalizable to other case studies is not clear. Moreover, no previous study analyzed the impacts of carbon pricing policies on biomass supply chain models at the operational level.  Studies that focused on analyzing the impacts of carbon pricing policies on biomass supply chain models combined the cost of the supply chain and the cost of emissions into a single objective function. These   23  single objective models prescribed one cost-minimization solution for each carbon pricing policy. However, with this type of modelling, the information about the trade-off between total cost and emissions of supply chains would be lost (Wu et al. 2010). Decision makers prefer to have a set of trade-off solutions instead of a single cost-minimization alternative to incorporate their preferences in the decision making process. Therefore, it is important to develop multi-objective biomass supply chain optimization models considering carbon pricing policies. Since solving multi-objective optimization models to obtain a set of trade-off solutions could involve solving several single-objective optimization models separately. This process could involve significant computational if the single-objective models are large and complex to solve. Furthermore, to assess the impact of carbon pricing policies on the multi-objective optimization models, several multi-objective models with different carbon pricing policies should be solved separately. This could further add to the computational effort. Therefore, developing strategies to determine the optimum solutions of the multi-objective models considering carbon pricing policies without solving several single-objective models could reduce the computational effort required to solve the models.      24  Chapter 3: Optimization of operational level forest-based biomass logistics considering transportation, storage, and pre-processing operations  3.1 Synopsis Logistics operations can contribute to up to 40% of total supply chain cost (Haridass et al. 2014), and in some cases they can contribute up to 90% of the total feedstock cost (Ekşioğlu et al. 2010). Due to this large proportion, even a small reduction in the logistics cost can lead to substantial savings (Palmgren et al. 2004). Biomass logistics decisions are taken at both tactical and operational levels. While the literature on biomass logistics at the tactical level is vast, that at the operational level is limited.  Previous studies on biomass logistics optimization at the operational level focused mainly on optimizing truck routes without including decisions related to storage and pre-processing of biomass, which are essential steps in utilizing biomass for energy generation. Moreover, the networks considered in previous studies did not consider intermediate storage sites. In this chapter, mathematical models are developed for optimizing forest-based biomass logistics at the operational level considering biomass flow, storage, pre-processing, and truck routing decisions simultaneously. The considered network includes biomass supply sites, intermediate storage sites, and biomass demand points. The models are applied for the short-term planning of a large biomass logistics company located in the Lower Mainland region of British Columbia, Canada. The company deals with the collection, storage, pre-processing and transportation of biomass. Multiple types of biomass are transported using different types of trucks. Several operational constraints related to truck-location compatibilities and truck-biomass compatibilities arising from heterogeneity of trucks and biomass types which further complicate the logistics planning are incorporated in the models. First, a transshipment model is developed and solved using a mixed integer formulation to determine pre-processing schedules and the number of truckloads of each biomass type to be transported each day using each type of truck. Then, a routing model, which uses the results of the transshipment model, is developed to determine the optimal routing for the available trucks. Experiments were conducted on real data from the company over a span of four weeks. The results indicate a potential to reduce 12% of total average cost compared to the actual routes implemented by the company. It is suggested that savings could be obtained by using larger trucks for longer distance transportation and smaller trucks for shorter distances. Direct delivery of biomass from suppliers to customers, bypassing the yard, could result in cost savings.  3.2 Problem description  The optimization models developed in this chapter are applied to the case of a large biomass logistics company located in the Lower Mainland, British Columbia, Canada. The company deals with collecting   25  biomass from several industrial sites and delivering feedstock to customers. Biomass types collected from suppliers include sawdust, shavings, clean wood and unclean wood. While sawdust and shavings are delivered to customers without pre-processing, clean and unclean wood must be comminuted into chips and hog fuel, respectively, before they are delivered to customers. The company owns a central yard where biomass collected from suppliers can be stored, and clean and unclean wood can be comminuted using chippers and grinders. Each supplier can supply multiple types of biomass and each customer can demand multiple types of feedstock. Transportation of biomass is carried out using a heterogeneous fleet of trucks, and restrictions related to truck-location compatibilities and truck-product compatibilities further complicate the problem. Each supplier and customer could be visited by more than one truck type depending on the truck-location compatibilities, and each truck type can carry more than one biomass type.  Each week, the company receives information about supply and demand quantities from each supplier and each customer for the following week. Depending on the information received, the company makes decisions related to storage, pre-processing and transportation of biomass. Pre-processing decisions prescribe the quantity of each type of biomass to be comminuted at the yard each day. Transportation decisions include the quantities of biomass to be transported to the yard, to be sent directly from suppliers to customers, and quantities of feedstock to be sent from the yard to customers. Transportation decisions also include the truck type to be sent to each location and the resultant routes to be taken by each truck. Currently, the logistics decisions are made by the managers of the company. Figure 3-1 shows a schematic of the logistics operations of the company.   26  123S123DSuppliers CustomersYardStorage and/orcomminution .. ..SawdustShavingsClean woodClean wood chipsUnclean woodHog fuel Figure 3-1: Schematic representation of operations of the forest-based biomass logistics company 3.3 Optimization models In this section, the mathematical models developed for the optimization of short-term logistics of forest-based biomass are presented. The problem is solved using a decomposition-based approach, where the problem is divided into two sub-problems. Two optimization models, one for each sub-problem, are developed. The first sub-problem is the transshipment problem used to determine the flow of biomass using different truck types along with the storage and pre-processing decisions at the storage yard. The results of transshipment model are used as inputs of the truck routing problem. The routing model is used to determine the daily routes for each truck. Similar decomposition-based approach has been used in previous studies for solving transportation and routing problems (e.g., (Flisberg et al. 2009; Cordeau et al. 2015)). In these studies, the problem was decomposed into a transportation problem to determine product flow quantities, and a Vehicle Routing Problem which used the results of the transportation problem to determine the daily truck routes. However, these studies considered networks involving only suppliers and customers, and did   27  not include the option of storage and pre-processing of products at intermediate facilities. A schematic of the solution framework is shown in Figure 3-2. Transshipment ModelDaily truck availabilityDaily biomass supply at each supplierDaily biomass demand at each customerInput DataDaily truckloads for each truck typeDaily Truck Routing ModelOptimal daily truck routesOutput of the transshipment modelOutput of the routing modelOptimal biomass flow quantity in the network Biomass comminution schedules at the yard Biomass storage quantity at the yard  Figure 3-2: Decomposition-based solution approach to solve biomass logistics problem at the operational level 3.3.1 Transshipment model A transshipment model is developed to determine the number of truckloads of each biomass type to be sent from each source to each destination using each truck type. Other decisions in the transshipment model include the quantity of biomass to be comminuted and stored at the yard during each period of the planning horizon. Constraints of the model include those related to supply, demand, storage and comminution of biomass on a daily basis. Constraints related to maximum operating hours per truck and the number of trucks available for each truck type are also included in the transshipment model to prevent infeasibility in the routing model. This requires an estimation of the total operation time per truck type which can be used in the objective function and constraints of the transshipment model.  The notations used in the transshipment model are shown in Table 3-1.     28  Table 3-1: Notations used in the transshipment model Sets Definition 𝑃 Set of all biomass types 𝑝 ∈ {0, 1, … , P} including biomass before and after comminution. 𝑃𝑛𝑐 Subset of biomass that do not need comminution, 𝑃𝑛𝑐 ⊂ 𝑃 𝑃𝑏𝑐 Subset of biomass that need comminution, before comminution, 𝑃𝑏𝑐 ⊂ 𝑃 𝑃𝑎𝑐 Subset of biomass that need comminution, after comminution, 𝑃𝑎𝑐 ⊂ 𝑃 𝑆 Set of all suppliers, 𝑖 ∈ {0,1, … , S} where {0} is the yard and 𝑆′ = 𝑆\{0} are the other suppliers 𝐷 Set of all customers 𝑘 ∈ {0,1, … , D} where {0} is the yard and 𝐷′ = 𝐷\{0} are the other customers 𝑇 Set of all time periods 𝑡 ∈ {1, … , T} in the planning horizon 𝐶 Set of vehicle types c ∈ {1, … , C} 𝑉𝑐 Set of vehicles 𝑣 ∈ {1, … , Vc} of type 𝑐 ∈ 𝐶 Decision variables Definition 𝑞𝑖𝑘𝑝𝑐𝑡 Volume of biomass 𝑝 ∈ 𝑃 to be transported from supplier 𝑖 ∈ 𝑆 to customer 𝑘 ∈ 𝐷 using vehicle type 𝑐 ∈ 𝐶 at period 𝑡 ∈ 𝑇 𝑛𝑖𝑘𝑝𝑐𝑡 Number of truckloads of biomass 𝑝 ∈ 𝑃 from supplier 𝑖 ∈ 𝑆 to customer 𝑘 ∈ 𝐷 using vehicle type 𝑐 ∈ 𝐶 at period 𝑡 ∈ 𝑇 𝐼𝑝𝑡 Inventory of biomass 𝑝 ∈ 𝑃 to be stored during period 𝑡 ∈ 𝑇 𝑐𝑝𝑡 Volume of biomass 𝑝 ∈ 𝑃𝑏𝑐 to be comminuted at period 𝑡 ∈ 𝑇 Parameters Definition 𝑠𝑖𝑝𝑡 Volume of biomass type 𝑝 ∈ 𝑃 available from supplier 𝑖 ∈ 𝑆′ at period 𝑡 ∈ 𝑇 𝑑𝑘𝑝𝑡 Volume of biomass type 𝑝 ∈ 𝑃 required by customer 𝑘 ∈ 𝐷′ at period 𝑡 ∈ 𝑇 𝐼𝑝,0 Initial inventory of biomass type 𝑝 ∈ 𝑃 at the yard 𝛽𝑖𝑐 Binary parameter equal to 1 if vehicle type 𝑐 ∈ 𝐶 can pick up/deliver biomass at location 𝑖 ∈ 𝑆 ∪ 𝐷 𝛽𝑝𝑐 Binary parameter equal to 1 if truck type 𝑐 ∈ 𝐶 can pick up/deliver biomass type 𝑝 ∈ 𝑃  𝛿𝑝 Quantity of biomass type 𝑝 ∈ 𝑃𝑏𝑐 that can be comminuted per minute at the central yard 𝛾𝑝 Maximum quantity of comminuted biomass type 𝑝 ∈ 𝑃𝑎𝑐   that can be produced during each day 𝜆𝑝 Conversion factor for comminuting biomass type 𝑝 ∈ 𝑃𝑏𝑐 (volume of comminuted biomass obtained from unit volume of uncomminuted biomass) 𝑄𝑐 Capacities of vehicle type 𝑐 ∈ 𝐶 𝑇𝑇𝑖𝑘𝑐 Total estimated travel time to transport a truckload from supply point 𝑖 ∈ 𝑆 to delivery point 𝑘 ∈ 𝐷 using truck type 𝑐 ∈ 𝐶 (includes back-and-forth travel of trucks from supply point 𝑖 to delivery point 𝑘, and truck loading and unloading times) 𝑇𝑇𝑚𝑎𝑥 Maximum time of travel permitted per truck per day 𝑇𝐶𝑐 Transportation cost per minute for vehicle type 𝑐 ∈ 𝐶 𝐶𝐶𝑝 Cost per minute for comminuting biomass type 𝑝 ∈ 𝑃𝑏𝑐 𝐿𝐶 Cost of loading per minute   29  𝜖 Factor to limit the empty truck volume in each truckload Objective function The objective function of the model is to minimize the sum of transportation, loading, unloading and comminution costs as shown in Expression (3.1).  Minimize ∑ ∑ ∑ ∑ 𝑛𝑖0𝑝𝑐𝑡 ∗ 𝑇𝐶𝑐 ∗ 𝑇𝑇𝑖0𝑐𝑡∈𝑇𝑐∈𝐶𝑝∈𝑃𝑖∈𝑆′+ ∑ ∑ ∑ ∑ 𝑛0𝑘𝑝𝑐𝑡 ∗ 𝑇𝐶𝑐 ∗ 𝑇𝑇0𝑘𝑐𝑡∈𝑇𝑐∈𝐶𝑝∈𝑃𝑘∈𝐷′+  ∑ ∑ ∑ ∑ ∑ 𝑛𝑖𝑘𝑝𝑐𝑡 ∗ 𝑇𝐶𝑐 ∗ 𝑇𝑇𝑖𝑘𝑐𝑡∈𝑇𝑐∈𝐶𝑝∈𝑃𝑘∈𝐷′𝑖∈𝑆′+ ∑ ∑ 𝐶𝐶𝑝 ∗ 𝑐𝑝𝑡/𝛿𝑝𝑝∈𝑃𝑏𝑐𝑡∈𝑇 (3.1) The transportation cost is obtained by multiplying the estimated total transportation time with the unit transportation cost for each type of truck, where the total transportation time includes the full and empty travels of the truck, as well as its loading and unloading time. The first, second and third terms in the objective function represent the transportation, loading and unloading costs of truckloads from suppliers to yard, yard to customers, and suppliers to customers, respectively. The final term of the objective function represents the total comminution cost.  Constraints Constraints related to the supply and demand of biomass are shown in Expressions (3.2) and (3.3).  ∑ ∑ 𝑞𝑖𝑘𝑝𝑐𝑡𝑐∈𝐶𝑘∈𝐷= 𝑠𝑖𝑝𝑡  ∀  𝑖 ∈ 𝑆′, 𝑝 ∈ 𝑃, 𝑡 ∈ 𝑇 (3.2) ∑ ∑ 𝑞𝑖𝑘𝑝𝑐𝑡𝑐∈𝐶𝑖∈𝑆= 𝑑𝑘𝑝𝑡 ∀ 𝑘 ∈ 𝐷′, 𝑝 ∈ 𝑃, 𝑡 ∈ 𝑇 (3.3) Constraint set (3.2) implies that the whole quantity of biomass supplied at each supplier is picked up daily. In these constraints, the daily quantity of biomass transported from a supplier to the yard and to all customers is equal to the biomass supply quantity. Similarly, constraint set (3.3) makes sure that the demand is met on a daily basis for each customer. In these constraints, the total biomass received by a customer from the yard and from all suppliers is equal to its biomass demand. Certain biomass types must be comminuted at the yard before the corresponding feedstock is delivered to customers. This is ensured by constraint set (3.4), which prevents the biomass types that require comminution from being sent directly from suppliers to customers without transiting through the yard.   30  ∑ ∑ ∑ 𝑞𝑖𝑘𝑝𝑐𝑡𝑐∈𝐶𝑘∈𝐷′𝑖∈𝑆′= 0, ∀ 𝑝 ∈ (𝑃𝑏𝑐 ∪ 𝑃𝑎𝑐), 𝑡 ∈ 𝑇 (3.4) Trucks that carry biomass from one location to another must be compatible with the biomass type and the pickup and delivery locations. Constraint set (3.5) ensures the truck compatibility with biomass type, the supplier and customer locations. If the truck type is not compatible with either the biomass type or the pickup or delivery location, this constraint set makes the biomass flow quantity equal to zero.  𝑞𝑖𝑘𝑝𝑐𝑡 ≤ 𝑀 ∗ 𝛽𝑖𝑐 ∗ 𝛽𝑘𝑐 ∗ 𝛽𝑝𝑐  ∀ 𝑖 ∈ 𝑆, 𝑘 ∈ 𝐷, 𝑝 ∈ 𝑃, 𝑐 ∈ 𝐶, 𝑡 ∈ 𝑇 (3.5) Constraint set (3.6) imposes that the number of truckloads of a given truck type determined by the model is sufficient to move the given quantity of biomass. For a given quantity of biomass transported between two locations using a particular truck type, these constraints estimate the number of truckloads required by dividing the biomass quantity with the capacity of the truck.  ( 𝑛𝑖𝑘𝑝𝑐𝑡 − 1) ∗ 𝑄𝑐 ≤ 𝑞𝑖𝑘𝑝𝑐𝑡 ≤ 𝑛𝑖𝑘𝑝𝑐𝑡 ∗ 𝑄𝑐 , ∀ 𝑖 ∈ 𝑆, 𝑘 ∈ 𝐷, 𝑝 ∈ 𝑃, 𝑐 ∈ 𝐶, 𝑡 ∈ 𝑇 (3.6) Constraint set (3.7) is added for practical reasons to prevent the trucks from travelling when the load is less than a certain percentage of the capacity of the vehicle. 𝑛𝑖𝑘𝑝𝑐𝑡 ∗ 𝑄𝑐 − 𝑞𝑖𝑘𝑝𝑐𝑡 ≥ 𝜖 ∗ 𝑄𝑐 , ∀ 𝑖 ∈ 𝑆, 𝑘 ∈ 𝐷, 𝑝 ∈ 𝑃, 𝑐 ∈ 𝐶, 𝑡 ∈ 𝑇 (3.7) Constraints (3.8) and (3.9) are related to the inventory balance for biomass types that require comminution, and constraint set (3.10) represents the inventory balance for biomass types that do not require comminution. The inventory balance constraints are included to make sure that the inventory of all biomass types at the yard is maintained from one period to another. 𝐼𝑝𝑡 = 𝐼𝑝,𝑡−1 − 𝑐𝑝𝑡 + ∑ ∑ 𝑞𝑖0𝑝𝑐𝑡𝑐∈𝐶𝑖∈𝑆′, ∀  𝑝 ∈ 𝑃𝑏𝑐 , 𝑡 ∈ 𝑇 (3.8) 𝐼𝑝𝑡 = 𝐼𝑝,𝑡−1 + 𝜆𝑝 ∗ 𝑐𝑝𝑡 − ∑ ∑ 𝑞0𝑘𝑝𝑐𝑡𝑐∈𝐶𝑘∈𝐷, ∀ 𝑝 ∈ 𝑃𝑎𝑐 , 𝑡 ∈ 𝑇 (3.9) 𝐼𝑝𝑡 = 𝐼𝑝,𝑡−1 + ∑ ∑ 𝑞𝑖0𝑝𝑐𝑡𝑐∈𝐶𝑖∈𝑆′− ∑ ∑ 𝑞0𝑘𝑝𝑐𝑡𝑐∈𝐶𝑘∈𝐷′, ∀  𝑝 ∈ 𝑃𝑛𝑐 , 𝑡 ∈ 𝑇 (3.10) Constraint set (3.11) ensures that the number of trucks available for each type are sufficient considering the maximum shift time per day. In these constraints, the total daily estimated travel time for each truck type is limited by the number of available trucks of that type multiplied by the maximum allowed operation time for each truck. Due to these constraints, the solution obtained from the transshipment model can be used in the routing model without any infeasibilities.   31  ∑ ∑ 𝑛𝑖0𝑝𝑐𝑡 ∗ 𝑇𝑇𝑖0𝑐𝑝∈𝑃𝑖∈𝑆′+ ∑ ∑ 𝑛0𝑘𝑝𝑐𝑡 ∗ 𝑇𝑇0𝑘𝑐𝑝∈𝑃𝑘∈𝐷′+ ∑ ∑ ∑ 𝑛𝑖𝑘𝑝𝑐𝑡 ∗ 𝑇𝑇𝑖𝑘𝑐𝑝∈𝑃𝑘∈𝐷′𝑖∈𝑆′ ≤  𝑇𝑇𝑚𝑎𝑥 ∗ |𝑉𝑐|, ∀ 𝑐 ∈ 𝐶, 𝑡 ∈ 𝑇 (3.11) Constraint set (3.12) limits the total volume of each biomass type that is comminuted at the yard on each day to the maximum comminution capacity. 𝜆𝑝 ∗ 𝑐𝑝𝑡 ≤ 𝛾𝑝, ∀  𝑝 ∈ 𝑃𝑏𝑐 , 𝑡 ∈ 𝑇 (3.12) Constraint sets (3.13), (3.14), (3.15) and (3.16) are used to define the decision variables of the model. 𝑛𝑖𝑘𝑝𝑐𝑡 ∈ 𝕫, ∀ 𝑖 ∈ 𝑆, 𝑘 ∈ 𝐷, 𝑝 ∈ 𝑃, 𝑐 ∈ 𝐶, 𝑡 ∈ 𝑇 (3.13) 𝑞𝑖𝑘𝑝𝑐𝑡 ≥ 0, ∀ 𝑖 ∈ 𝑆, 𝑘 ∈ 𝐷, 𝑝 ∈ 𝑃, 𝑐 ∈ 𝐶, 𝑡 ∈ 𝑇 (3.14) 𝐼𝑝𝑡 ≥ 0, ∀  𝑝 ∈ 𝑃, 𝑡 ∈ 𝑇 (3.15) 𝑐𝑝𝑡 ≥ 0, ∀  𝑝 ∈ 𝑃, 𝑡 ∈ 𝑇 (3.16) Previous studies on the transshipment problem mostly considered transportation cost to be dependent only on the quantity of product that is transported, and the total transportation cost was calculated using the cost per unit flow of product (Guastaroba et al. 2016). However, the actual cost structure could be more complicated and may depend on several distribution attributes such as the number of truckloads required and the duration of empty truck travel.  Few studies on the transportation problem considered different cost structures to incorporate empty truck travel cost in their models. For example, Flisberg et al. (2009) who studied the transportation of logs from several suppliers to multiple customers assumed trucks always traveled back-and-forth between the log pickup and delivery locations. This way, they incorporated both truck loaded and empty travel costs in their objective function. This cost structure which assumes back-and-forth travel of trucks between pickup and delivery locations could be appropriate when the total supply and demand volumes exceed the volume of one truckload, and multiple truckload pickups and deliveries are required.  However, the cost structure used by (Flisberg 2009) may not be applicable when the supply and demand quantities are less than a truckload, and trucks visit multiple suppliers and customers during each trip. Cordeau et al. ( 2015), who studied the Inventory Routing Problem for transporting a product from a supplier to multiple customers with less-than-truckload demand, mentioned that assuming back-and-forth travel of trucks between pickup and delivery locations could over-estimate the total cost. Instead, they mentioned that the transportation cost could be approximated by the cost of traveling from the supplier to the customer, without considering the cost of traveling back empty to the supply point. The transportation cost estimation used by Cordeau et al. (2015) is a better approximation than the back-and-forth travel   32  assumption when a truck begins its route from the supply point and visits several customers to meet their demand before returning back to the supply point.  Neither of the two cost estimation approaches described above can be used directly in our problem which includes multiple suppliers, multiple customers, a transshipment center which is also the tucks depot, and full truck-loads delivery. The cost of delivering biomass from a supply point to a demand point in our case depends on the type of truckload that is transported. There are three types of truckloads of biomass: (1) truckload from a supplier to the yard, (2) truckload from the yard to a customer, and (3) truckload directly from a supplier to a customer. According to the data we received from the company, most of the truckloads of types 1 and 2 require back-and-forth travel of trucks between the yard and suppliers, and the yard and customers, respectively. Therefore, for each truckload of types 1 and 2, the transportation time coefficients 𝑇𝑇𝑖0𝑐 and 𝑇𝑇0𝑘𝑐 include the truck travel time to perform a round trip between the yard and suppliers, and the yard and customers, respectively. On the other hand, trucks that deliver biomass directly from suppliers to customers (i.e., truckload of type 3) mostly do not go back to the yard until the end of the day. These trucks start at the yard, perform a series of pickups and deliveries between suppliers and customers before returning to the yard. Moreover, the quantities of biomass picked up and delivered are often more than a truckload. This situation is similar to that of Flisberg et al. (2009) who considered log transportation between multiple suppliers and customers without including an intermediate storage yard. Following their cost estimation, the transportation time coefficient 𝑇𝑇𝑖𝑘𝑐 for transporting a truckload of biomass from a supplier 𝑖 ∈ 𝑆′ to a customer 𝑘 ∈ 𝐷′ includes truck loaded travel time from 𝑖 to 𝑘 and its empty travel time from 𝑘 to 𝑖. All transportation time coefficients also include the time required for loading and unloading the trucks at pickup and delivery points, respectively. 3.3.2 Routing model The transshipment model described in Section 3.3.1 provides the details of full-truckloads of biomass to be transported each day. Each truckload is defined by the pickup point, delivery point, biomass type, truck type and the day of transportation. Since the pickup and delivery point of each truckload are known, the time required for transporting each truckload is known. The objective of the routing model is to determine the best route for each truck such that all the truckload deliveries are completed with a minimum cost. A truck route is defined by a sequence of full-truckloads of biomass to be transported by the truck. Since each truckload has a pickup and delivery location, a truck route consists of a sequence of locations to be visited by the truck to pick up and deliver biomass. Truck routes must be determined such that all the truckload deliveries determined in the transshipment model are performed using appropriate truck types and each truck is limited by the maximum number of operation hours per day.    33  The routing model is developed on an auxiliary graph described as follows. Let 𝐿 be the set of all truckloads obtained from the transshipment model and let 𝑁 = {0} ∪ 𝑁′ be the set of nodes and 𝐴 be the set of arcs of the auxiliary graph. {0} represents the yard and 𝑁′is the set of nodes where each node 𝑗 ∈ 𝑁′ represents a truckload 𝑙 ∈ 𝐿. For every node 𝑗 ∈ 𝑁′, let 𝑙𝑗 be the corresponding truckload in 𝐿, 𝑠𝑗 and 𝑑𝑗 be the pickup and delivery locations of the truckload 𝑙𝑗, respectively.  Every pair of nodes in 𝑁 has an arc in the network. Every arc has an associated time required to travel over it. Let 𝑇𝑇𝐴𝑖𝑗 be the time required to travel on arcs (𝑖, 𝑗) ∈ 𝐴 in the auxiliary graph. Since each node 𝑗 ∈ 𝑁′ represents a truckload delivery of biomass, there is a service time 𝑆𝑇𝐴𝑗, which includes truck travel, loading and unloading times. The travel times on arcs and the service times of the nodes are defined according to the following steps:  Step 1. For every node 𝑗 ∈ 𝑁′, set the travel time from the yard to the node 𝑗 equal to the travel time from the yard to the pickup location of truckload 𝑙𝑗. Mathematically, set 𝑇𝑇𝐴0𝑗 =  𝑇𝑇0,𝑠𝑗. For a truckload 𝑙𝑗 of type 2 which starts at the yard, 𝑇𝑇𝐴0𝑗 =  0. Step 2. For every node  𝑗 ∈ 𝑁′, set the travel time from 𝑗 to the yard equal to the travel time from the delivery location of the truckload 𝑙𝑗 to the yard. Mathematically, set  𝑇𝑇𝐴𝑗0 =  𝑇𝑇𝑑𝑗,0. For a truckload 𝑙𝑗 of type 1 whose destination is the yard,  𝑇𝑇𝐴𝑗0 = 0. Step 3. For every pair of nodes {𝑖, 𝑗|𝑖 < 𝑗} ∈ 𝑁′, set the travel time from 𝑖 to 𝑗 equal to the travel time between the delivery location of truckload 𝑙𝑖 and the pickup location of truckload 𝑙𝑗. Similarly, set the travel time from 𝑗 to 𝑖 equal to the travel time between the delivery location of truckload 𝑙𝑗 and the pickup location of truckload 𝑙𝑖. Mathematically, set 𝑇𝑇𝐴𝑖𝑗 = 𝑇𝑇𝑑𝑖,𝑠𝑗 and 𝑇𝑇𝐴𝑗𝑖 = 𝑇𝑇𝑑𝑗,𝑠𝑖. Step 4. For each node 𝑗 ∈ 𝑁′, set the service time equal to the sum of travel time between the pickup and delivery locations of the truckload 𝑙𝑗, truck loading time at the pickup location of the truckload 𝑙𝑗 and the truck unloading time at the delivery location of the truckload 𝑙𝑗. Mathematically,  𝑆𝑇𝐴𝑗 =𝑇𝑇𝑠𝑗,𝑑𝑗 + 𝐿𝑇𝑐 + 𝑈𝑇𝑐. An example of the auxiliary graph for delivering three truckloads of biomass is shown in Figure 3-3. For clarity, each of the three truckloads represent one truckload type. The travel times between each node of the auxiliary network are shown on the arcs connecting them.    34  d2s3TT0,s3TTd3,0TTd2,s1 TT0,0=0TTd2,0TT0,0=0TT0,0=0TT0,s1TTd3,s1TT0,s3TTd3,s2TTd2,s3Yard Truckload Type 1 Truckload Type 3Truckload Type 20d30s10 Figure 3-3: A sample of an auxiliary graph with 3 nodes showing the 3 types of truckloads in the routing problem The routing problem deals with defining daily routes for each truck located at the yard such that every node 𝑗 ∈ 𝑁′ in the auxiliary graph, which represents one truckload, is visited exactly once. The routing problem is a variant of the Multiple Traveling Salesmen Problem where multiple salesmen located at a central depot are routed such that all the nodes in the network are visited exactly once by the salesmen (Bektas 2006). Since we deal with routing of a heterogeneous fleet of trucks, the routing problem in biomass logistics is a variant of Multiple Traveling Salesmen Problem with heterogeneous trucks. Since the truckloads are defined daily, solving the routing model for all the days together is equivalent to solving the routing model for each day separately. Moreover, our experiments suggested that solving the routing model for all days together is computationally intractable using commercial MIP solvers due to the size of the model. Therefore, the routing model is run on the auxiliary network for each day of the week separately.  The notation used in the routing model is shown in Table 3-2.      35  Table 3-2: Notation used in the routing model Sets Definition 𝑁 Set of all nodes in the auxiliary graph for the routing formulation with 𝑗 ∈ 𝑁 ={0,1, … , 𝑛}, where {0} is the yard and 𝑁′ = 𝑁\{0}, the subset of all truckloads obtained from the transshipment model 𝑁𝑐 Set of truckloads for vehicle type 𝑐 ∈ 𝐶 excluding the depot and 𝑁𝑐′ = 𝑁𝑐\{0} 𝐴 Set of all arcs (𝑖, 𝑗) ∈ {0,1, … , |𝐴|} in the auxiliary graph for the routing formulation Decision variables Definition 𝑥𝑖𝑗𝑐𝑣 Binary variable equal to 1 if vehicle 𝑣 ∈ 𝑉𝑐 traverses from 𝑖 ∈ 𝑁 to 𝑗 ∈ 𝑁 and equal to 0 otherwise 𝑦𝑖𝑐𝑣 Binary variable equal to 1 if vehicle 𝑣 ∈ 𝑉𝑐 visits node 𝑖 ∈ 𝑁 and equal to 0 otherwise 𝑤𝑖 Integer variable representing the node potential of  𝑖 ∈ 𝑁′. These are used to eliminate sub-tours 𝑧𝑐𝑣 Binary variable equal to 1 if vehicle 𝑣 ∈ 𝑉𝑐 is used and 0 otherwise Parameters Definition 𝑇𝑇𝑚𝑎𝑥 Maximum time of travel permitted per vehicle per day 𝑇𝑇𝑚𝑖𝑛 Minimum travel time required per vehicle per day 𝑇𝑇𝐴𝑖𝑗 Travel time between nodes 𝑖, 𝑗 ∈ 𝑁 in the auxiliary graph 𝑆𝑇𝐴𝑖 Time required to service node 𝑖 ∈ 𝑁′ in the auxiliary graph 𝑇𝐶𝑐 Transportation cost per minute for vehicle type 𝑐 ∈ 𝐶 𝑐𝑖 Truck type of node 𝑖 ∈ 𝑁′ 𝑡𝑖 Day of the planning horizon to which the truckload 𝑖 ∈ 𝑁′ belongs to Objective function Expression (3.17) represents the objective function of minimizing the routing cost of trucks in the auxiliary network. The routing model uses the truckloads resulted from the transshipment model, thus the travel cost of loaded trucks between the locations in the supply chain is a constant in the routing model. Consequently, the model minimizes the empty travel cost of trucks in order to minimize the routing cost. 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 ∑ ∑ ∑ ∑ 𝑥𝑖𝑗𝑐𝑣 × 𝑇𝑇𝐴𝑖𝑗 × 𝑇𝐶𝑐𝑣∈𝑉𝑐𝑐∈𝐶𝑗∈𝑁𝑖∈𝑁 (3.17) Constraints Constraint set (3.18) ensures every node in the auxiliary graph is visited once by any vehicle. This means that each truckload of biomass is picked up and delivered exactly once, and no truckloads are left out.  ∑ ∑ ∑ 𝑥𝑖𝑗𝑐𝑣𝑣∈𝑉𝑐𝑐∈𝐶𝑖∈𝑁= 1, ∀ 𝑗 ∈ 𝑁′ (3.18)   36  Vehicle flow balance constraints in (3.19) ensure that each vehicle that visits a node 𝑗 ∈ 𝑁′ also leaves that node. These constraints imply that a truck would move on to deliver another truckload or go back to the yard at the end of the day after a truckload of biomass is delivered. ∑ 𝑥𝑖𝑗𝑐𝑣𝑖∈𝑁=  ∑ 𝑥𝑗𝑘𝑐𝑣𝑘∈𝑁, ∀ 𝑗 ∈ 𝑁′, 𝑐 ∈ 𝐶, 𝑣 ∈ 𝑉𝑐 (3.19) Constraint set (3.20) limits the number of vehicles of each type that leave the yard on each day to the number of vehicles available. ∑ ∑ 𝑥0,𝑗𝑐𝑣𝑣∈𝑉𝑐𝑗∈𝑁′≤ |𝑉𝑐|, ∀ 𝑐 ∈ 𝐶 (3.20) Constraint set (3.21) is used to define the variables 𝑧𝑐𝑣 indicating if a vehicle is used or not. The variable 𝑧𝑐𝑣 is equal to ∑ 𝑥0,𝑗𝑣𝑐𝑗∈𝑁′  if a vehicle 𝑣 of type 𝑐 travels from the yard to a node 𝑗 ∈ 𝑁′, and 0 if it does not leave the yard.  ∑ 𝑥0,𝑗𝑣𝑐𝑗∈𝑁′− 𝑧𝑐𝑣 = 0, ∀ 𝑐 ∈ 𝐶, 𝑣 ∈ 𝑉𝑐 (3.21) The yard is the trucks depot, this means that the trucks start their trip from the yard and return to the yard at the end of their trip. Constraints that impose that every truck that leaves the yard returns to the yard are shown in Expression (3.22). ∑ 𝑥0,𝑗𝑐𝑣𝑗∈𝑁′=  ∑ 𝑥𝑗,0,𝑐𝑣𝑗∈𝑁′, ∀ 𝑐 ∈ 𝐶, 𝑣 ∈ 𝑉𝑐 (3.22) Constraint set (3.23) ensures that if a node is visited by a truck, the truck comes from one of the arcs to that node. It means if 𝑦𝑗𝑐𝑣 = 1, i.e. a node 𝑗 is traversed by a given vehicle 𝑣 of class 𝑐, then there must be one 𝑥𝑖𝑗𝑐𝑣 = 1, i.e. the same vehicle 𝑣 of class 𝑐, must transit through an arc coming from node 𝑖, to reach node 𝑗. 𝑦𝑗𝑐𝑣 =  ∑ 𝑥𝑖𝑗𝑐𝑣𝑖∈𝑁, ∀ 𝑗 ∈ 𝑁′, 𝑐 ∈ 𝐶, 𝑣 ∈ 𝑉𝑐 (3.23) Sub-tour elimination is ensured using constraint set (3.24). A sub-tour is a tour within the set of nodes 𝑁′ where a vehicle traverses in a cycle. A cycle results when a vehicle visits the node that was visited before. For example, consider nodes 𝑖, 𝑗, 𝑘, 𝑙 ∈ 𝑁′. Assume that the same vehicle visits all the four nodes, and traverses in the following order: 𝑖 → 𝑗 → 𝑘 → 𝑙 → 𝑖. This is a sub-tour as the vehicle visits the node 𝑖 twice forming a cycle. On the other hand, if the truck travels to another node 𝑚 after visiting node 𝑙 or goes back   37  to the yard, then it would not form a sub-tour. Therefore, sub-tours can be avoided by constraining a vehicle not to visit a node which was previously visited. This can be achieved by defining non-negative numbers, called node potentials, to each node that is visited by each vehicle. For two nodes 𝑖, 𝑗 ∈ 𝑁′ which are visited by the same vehicle, node potentials 𝑤𝑖 and 𝑤𝑗 are defined such that 𝑤𝑗 > 𝑤𝑖 if the vehicle travels from 𝑖 to 𝑗. Since the node potentials of the nodes visited by a vehicle are always in the ascending order, the vehicle would not re-visit a node which was already visited. For the example explained above, since the vehicle visits node 𝑖 before node 𝑙, then  𝑤𝑙 > 𝑤𝑖. Now, the vehicle cannot go back to node 𝑖 from node 𝑙 as this traversal would contradict the definition of node potentials. Constraint set (3.24) assigns non-negative node potentials in an ascending order to nodes along the routes traversed by each truck. As a result, sub-tours are eliminated in the optimal routes. 𝑤𝑖 − 𝑤𝑗 + 𝑛 ∑ ∑ 𝑥𝑖𝑗𝑐𝑣𝑣∈𝑉𝑐𝑐∈𝐶≤ 𝑛 − 1, ∀ (𝑖, 𝑗) ∈ 𝐴 (3.24) Constraint set (3.25) limits the maximum travel time per truck per day to the maximum allowable working hours per driver. The travel time of each truck includes the routing time in the auxiliary graph, loaded travel time, and loading and unloading times obtained from the transshipment model. ∑ ∑(𝑥𝑖𝑗𝑐𝑣 × 𝑇𝑇𝐴𝑖𝑗)𝑗∈𝑁𝑖∈𝑁+ ∑ (𝑦𝑖𝑐𝑣 × 𝑆𝑇𝐴𝑖)𝑖∈𝑁′≤ 𝑇𝑇𝑚𝑎𝑥, ∀ 𝑣 ∈ 𝑉𝑐 , 𝑐 ∈ 𝐶 (3.25) Symmetry-breaking constraints that ensure that a truck 𝑣 ∈ 𝑉𝑐 is used only after the truck 𝑣 − 1 ∈ 𝑉𝑐 is used are represented by constraint set (3.26). 𝑧𝑐,𝑣−1 ≥ 𝑧𝑐𝑣 , ∀ 𝑐 ∈ 𝐶, 𝑣 ∈ 𝑉𝑐 (3.26) Each node in the auxiliary graph represents one truckload of a truck type. Therefore, each node in the auxiliary graph is associated with a truck type. Constraint set (3.27) ensures that a node is visited by a truck type defined by the truckload.  𝑦𝑖𝑐𝑣 = 0, ∀ 𝑐 ≠ 𝑐𝑗 (3.27) The decision variables of the model are defined in constraint sets (3.28), (3.29) and (3.30).  𝑥𝑖𝑗𝑐𝑣 ∈ {0,1}, ∀ 𝑖, 𝑗 ∈ 𝑁, 𝑐 ∈ 𝐶, 𝑣 ∈ 𝑉𝑐 (3.28) 𝑦𝑖𝑐𝑣 ∈ {0,1}, ∀ 𝑖, 𝑗 ∈ 𝑁, 𝑐 ∈ 𝐶, 𝑣 ∈ 𝑉𝑐 (3.29) 𝑤𝑖 ∈ 𝕫, ∀ 𝑖 ∈ 𝑁 (3.30)   38  Solving the model provides the optimal routing on each day for the trucks available considering the given truckloads. The following procedure can be used to obtain the routes from the variables: let 𝑁𝑐𝑣𝑡 be the set of nodes visited by truck 𝑣 ∈ 𝑉𝑐 , 𝑐 ∈ 𝐶 on day 𝑡 ∈ 𝑇. For each 𝑖 ∈ 𝑁𝑐𝑣𝑡, the value of 𝑤𝑖 can be found from the optimal solution of the model. By the definition of node potentials, 𝑤𝑗 > 𝑤𝑖 if the truck 𝑣 ∈ 𝑉𝑐 , 𝑐 ∈ 𝐶 visits node 𝑖 ∈ 𝑁𝑐𝑣𝑡 before visiting 𝑗 ∈ 𝑁𝑐𝑣𝑡. Therefore, by sorting the nodes in 𝑁𝑐𝑣𝑡 according to their node potentials, we can construct the route for the truck 𝑣 ∈ 𝑉𝑐 , 𝑐 ∈ 𝐶. The same procedure is to be followed for all trucks.  3.4 Data The developed optimization models are applied to a large biomass logistics company in Lower Mainland, British Columbia, Canada. The company collects biomass from 26 major suppliers and delivers feedstock to 9 major customers, which represents more than 80% of the business. The company is a third-party logistics provider for these suppliers and customers and has long-term contracts with them. It owns a large yard where biomass can be stored and comminuted. It also owns a fleet of trucks for biomass pickup and delivery operations.  Biomass types collected from suppliers include sawdust, shavings, clean wood, and unclean wood. Each supplier can supply more than one type of biomass. Four types of feedstock are delivered to customers, namely, sawdust, shavings, clean wood chips and hog fuel. Clean wood chips and hog fuel are produced by chipping and grinding clean wood and unclean wood, respectively, at the yard.  Each customer can demand more than one type of feedstock.  Depending on whether the biomass requires comminution, biomass can either be delivered directly to customers or be brought to the yard. Clean and unclean wood require comminution, while sawdust and shavings do not.  Since comminution happens at the yard, clean and unclean wood are always transported from the suppliers to the yard. After comminution, clean wood chips and hog fuel are delivered to respective customers from the yard. On the other hand, sawdust and shavings can either be sent to the yard for storage or be delivered directly from suppliers to customers, by-passing the yard.  Biomass is comminuted at the yard using a chipper and a grinder owned by the company. The chipper is used to comminute clean wood into clean wood chips. The chipper can produce wood chips of uniform size which is a requirement of the customers. On the other hand, the stationary electric grinder is used to grind unclean wood into hog fuel. The functioning of the grinder is similar to that of a hammer, due to which unclean wood is ground into non-uniform sizes. Both the chipper and grinder incur an operation cost per unit time, and the cost of the grinder is higher than the cost of the chipper. The chipper and the grinder can   39  produce a fixed amount of comminuted biomass per unit time, and the throughput (volume of biomass comminuted per unit time) of the grinder is greater than that of the chipper.  The company maintains inventories of sawdust, shavings, clean wood, unclean wood and clean wood chips at the yard. Capacity for storing these biomass and feedstock types is not a restrictive factor due to the large size of the yard compared to the volume of stored biomass. A minimum inventory of clean wood chips is always maintained at the yard. On the other hand, the company does not maintain inventory of hog fuel, and the quantity of unclean wood ground on a given day is equal to the total demand of hog fuel for that day.  Biomass is picked up from suppliers on a daily basis where the total quantity of biomass supplied on a given day must be picked up completely on the same day. Therefore, the company does not have control over inventory of biomass at suppliers. Similarly, feedstock is delivered to customers on a daily basis such that the demand of a customer on a given day is met on the same day. Inventory of feedstock at customer sites is not controlled by the company. Four different types of trucks are used for pickup and delivery of biomass. They are sawdust truck, long trailer truck, roll off truck, and end dump truck. Table 3-3 shows different properties of each type of truck. Different types of biomass can be carried using specific types of truck. The origin and destination of truckloads carrying different biomass types and the compatibilities between biomass types and truck types are shown in Table 3-4. Similar to truck type-biomass type compatibility, trucks also have restrictions related to whether they can visit different suppliers and customers. These restrictions are due to space constraints at the supplier and customer sites. Few suppliers and customers may be visited by multiple types of trucks.      40  Table 3-3: Characteristics of different types of truck Truck Type Number of trucks Capacity (m3) Loading time (minutes) Unloading time (minutes) Cost (CDN$/hour) Sawdust truck 4 53.56 7 5 90 Long trailer and truck 7 107.57 15 20 125 Roll off truck 2 30.60 10 10 90 End dump truck 5 68.86 10 10 120  Table 3-4: Biomass type, origin and destination of trucks, and compatible truck type Biomass type Origin Destination Truck type Sawdust, shavings Suppliers Yard, customers All truck types Yard Customers All truck types Clean and unclean wood Suppliers Yard Roll off and end dump trucks Clean wood chips and hog fuel Yard customers All truck types The pickup and delivery of the biomass at supplier and customer sites, respectively, do not require additional loading and unloading equipment. Sawdust and shavings are collected directly from the over-head bins located at supplier sites. Clean and unclean wood are deposited in metal bins or end dump trailers located at supplier sites and are collected using roll off trucks and end dump trucks, respectively. Similarly, unloading of biomass at customer sites does not require additional equipment as all truck types have unloading capability.  In contrast, loading biomass onto trucks at the yard requires loaders. The company rents at least two loaders for the loading operation, and there is a cost associated with loading each truck based on the time it takes to fill the truck. According to the logistics manager of the company, loader’s availability is not a bottleneck for their operations. Each truck that is to be loaded at the yard spends ten minutes on average waiting for the loader. The average truck waiting time at the yard, and the associated loader costs for each truckload of biomass from the yard to customers are included in the objective function of the transshipment model and in the constraints related to the maximum operation time of each truck per day.  The trucks are located at the yard, and drivers start and end their routes at the yard each day. Each driver typically drives the same type of truck and visits the same supplier and customer locations as much as possible. Each driver is limited to work no more than 13 hours each day.    41  At the end of each day, drivers fill a report called the “driver cartage report” which contains information concerning the pickup and delivery operations carried out by each driver on that day. Each entry in the cartage report depicts a truckload delivery which includes details about the product that was transported and the pickup and delivery locations for that truckload. The company keeps record of the cartage reports to determine the total cost of transportation and the total driver working time.  Figure 3-4 shows a sample of the daily cartage report filled by one driver. The report shows the transportation of four truckloads of biomass. The route adopted by the driver as per the report is (yard, supplier 1, customer 1, supplier 1, customer 1, supplier 1, yard, supplier 2, yard).  Location Name Biomass TypeFrom: Supplier 1 SawdustTo: Customer 1From: Supplier 1 ShavingsDaily Cartage ReportName of the driver: XYZ Date: May 28, 2017To: Customer 1From: Supplier 1 Clean woodTo: YardFrom: Supplier 2 Unclean woodTo: Yard Figure 3-4: Sample of a daily cartage report Each week, the company receives information about biomass supply and demand for the following week from their suppliers and customers. Based on this information, the weekly comminution schedules and daily truck routes are determined manually by the company. Comminution decisions and transportation decisions are taken separately by the yard manager and the logistics manager, respectively.  Data from the company were obtained in the form of driver cartage reports for a span of four weeks. Few drivers did not mention the biomass type that was transported in each truckload in their reports. In such cases, details regarding biomass types available at each supplier and feedstock types demanded by each   42  customer were gathered from the logistics manager of the company. This information was used to assess the product type in each truckload of the cartage reports.  The type of truck driven by each driver is not mentioned in the cartage report. This information is necessary to calculate the total quantity biomass picked up from each supplier and delivered to each customer. However, since drivers usually drive the same truck type, details about the truck type driven by each driver were obtained from the logistics manager of the company. Combining the information from the cartage reports and the logistics manager, the daily supply and demand quantities of biomass were calculated for four weeks. The compatibilities between product type, locations and truck type were retrieved from the cartage reports. The exact routes taken by each driver on each day were also retrieved from the cartage reports.  3.5 Results  In this section, the results obtained from solving the mathematical models are presented using the data obtained from the company. The models are run for a planning horizon of one week where the transshipment model is run to determine the weekly transportation decisions and comminution schedules, and the routing model is solved for each day using the results from the transshipment model. Since data were received for a span of four weeks, the models are tested on four problem instances, one for each week.  The models are assessed by determining the total cost of transportation. This cost includes truck routing cost, cost when truck waits for the loader at the yard, cost incurred when trucks wait during loading and unloading operations, and the cost associated with the loader usage. Cost of comminution are disregarded in this comparison since information about the quantity of biomass comminuted at the yard was not available from the cartage reports. Previous studies which developed transportation optimization models evaluated their results by comparing the cost of the routes obtained from their models with routes which assume that trucks always travel back-and-forth between the supply and demand points (Kim et al. 2009). The cost of routes where trucks always travel back-and-forth between supply and demand points is an upper bound to the optimal cost. However, this may not be a tight bound since these routes do not incorporate backhauling opportunities at all. For the data obtained from the company, it was observed that the cost of the actual truck routes obtained from the cartage reports was on average 7.7% better compared to the cost with truck’s back-and-forth movements, indicating that backhauling was already incorporated in the manually-developed routes. We compare the results of our models with the routes implemented by the company as retrieved from the cartage reports.   43  In Table 3-5, the total cost from the models is compared with that from the original routes. The total cost includes loaded and empty truck travel cost, trucking cost incurred during loading and unloading operations,  cost for trucks when waiting at the yard for the loader, and the loader cost. Loaded travel cost is the cost incurred when a truck travels loaded with biomass, and empty travel cost refers to the cost when the truck travels empty.  The second column of Table 3-5 shows the cost of the actual routes taken by the trucks. The third column shows the results of running the routing model using truckloads that were originally transported by the drivers. Instead of running the transshipment model to determine the number of truckloads of each truck type and each biomass type, we use the number of truckloads of each truck type and biomass type that were retrieved from driver cartage reports. The fourth column shows the improvement in total cost using the routing model for the original truckloads. The results show an average improvement of 6.6% over the original cost. This means that the routing model can increase the efficiency of the routes by exploiting backhauling opportunities in the pickup and delivery of biomass even when the truckloads were those used by the company.  The fifth column of Table 3-5 shows the results of the models where truckloads obtained from the transshipment model are used in the routing model. The sixth column shows the reduction in cost from the models as compared to the company’s original cost. An average improvement of 12% can be observed in the total cost. This can be attributed to better full-truckload transportation decisions between different locations of the network and the resultant routes which incorporate backhauling as much as possible.  Table 3-5: Total and weekly costs obtained by summing the costs for each routes per day Week Cost from driver’s reports (CDN$) Truckloads from driver’s reports & routes from the routing model Truckloads from the transshipment model & routes from the routing model Cost (CDN$) % improvement Cost (CDN$) % improvement Week 1 55895.00 51646.40 7.6 48242.33 13.7 Week 2 47467.00 44306.50 6.7 41992.57 11.5 Week 3 70085.70 66103.10 5.7 62620.25 10.7 Week 4 73063.10 68288.40 6.5 64045.30 12.3 Total 246510.30 230344.40 4.6 217105.90 12.0 Average 61627.70 57586.10 6.6 54276.50 12.0  Trucks and loaders consume fuel during operation. The cost parameters used in the models for trucks and loaders include the cost of fuel. Therefore, the cost optimization models also contribute to a reduction in the total fuel consumption. Using the truck fuel consumption rates from the literature (Dehkordi 2015), the   44  total fuel consumption during each week is calculated. The results suggest that the improvement in fuel consumption during each week relates to the improvement in total cost. The average reduction in total fuel consumption using the optimization models for four weeks was 11.7%.  Figure 3-5 compares different cost components of our solutions and the driver’s original routes. It can be observed that the cost of each component has reduced using the optimization models as compared to the company’s original routes. It can also be observed that the empty travel cost is relatively lower than loaded travel cost in the optimal solutions while it is the opposite in the original routes. In the drivers’ original routes, loaded travel cost was on average 34% of the total cost while empty travel cost was 36% of the total cost. In the routes obtained from our solutions, loaded travel cost was 36% and empty travel cost was 34% of the total cost.   Figure 3-5: Comparison of different cost components in the company's original routes and the routes obtained from the optimization models The total number of truckloads transported during each week and the distribution of the number of truckloads with respect to the truck type from the company’s original routes and the solution from our models can be seen in Figure 3-6. Except for week 2, the total number of truckloads in both the solutions is more or less the same. In addition, expect for week 2, the number of truckloads of the sawdust truck increased in the optimal solutions. The number of truckloads of long trailer decreased in the optimal 05,00010,00015,00020,00025,00030,000Week 1Week 2Week 3Week 4Week 1Week 2Week 3Week 4Week 1Week 2Week 3Week 4Week 1Week 2Week 3Week 4Week 1Week 2Week 3Week 4Loaded travel cost Empty travel cost Loading & unloadingcostWaiting cost Loader costCost (CDN$)Company's solution This study  45  solutions. The number of truckloads of roll off trucks and end dump trucks are more or less the same for all four weeks.   Figure 3-6: Number of truckloads for each type of truck Figure 3-7 shows the average cost incurred by each truck type from the company’s original routes and the solutions obtained from the optimization models. Even though Figure 6 shows that the number of truckloads of sawdust trucks is greater than that of long trailer trucks, the average total cost of long trailer trucks is much greater than that for sawdust trucks. This trend of higher cost for long trailer trucks and lower cost for sawdust trucks can also be observed in the company’s original routes. 37332444545411412015416210182129128 142113148148169 14 163733064494441401091971937878 94 98139110144137169 14 16050100150200250300350400450500Week 1Week 2Week 3Week 4Week 1Week 2Week 3Week 4Week 1Week 2Week 3Week 4Week 1Week 2Week 3Week 4Week 1Week 2Week 3Week 4Total Sawdust Long Trailer Roll Off End Dump# TruckloadsCompany's solution This study  46   Figure 3-7: Transportation cost per week for each truck type There are three types of loaded truckloads: truckload from a supplier to the yard (type 1), truckload from the yard to a customer (type 2), and truckload directly from a supplier to a customer (type 3). The total number of truckloads of each type is shown in Figure 3-8. It can be observed that the optimization models suggest transporting a smaller number of truckloads of the type 1 and more number of truckloads of type 3. Except for week 3, the number of truckloads of type 2 also decreased using the optimization models.  05,00010,00015,00020,00025,00030,00035,00040,00045,000Week1Week2Week3Week4Week1Week2Week3Week4Week1Week2Week3Week4Week1Week2Week3Week4Sawdust Long Trailer Roll Off End DumpTransportation cost (CDN$)Company's solution This study  47   Figure 3-8: Number of truckloads per week of each truckload type in the supply chain The average number of truckloads of each truck type and each truckload type are shown in Figure 3-9. Similar to the observations made in Figure 3-8, Figure 3-9 suggests that the average number of truckloads of type 1 reduced using our models for all truck types, while the number remains the same for end dump truck type. Similarly, the number of truckloads of type 3 increased on average for all truck types. With respect to the use of sawdust trucks, the models suggest decreased usage for type 1 loads and increased usage for type 2 loads from the yard to customers.  219172200 21577 81156 1677771897220115018018670 711601601028510998050100150200250Week 1 Week 2 Week 3 Week 4 Week 1 Week 2 Week 3 Week 4 Week 1 Week 2 Week 3 Week 4Type 1: Supplier-Yard Type 2: Yard-Customer Type 3: Supplier-Customer# TruckloadsCompany's solution This study  48    Figure 3-9: Average number of truckloads of each truck type and each truckload type in the supply chain The durations of chipping and grinding of clean wood and unclean wood, respectively, are determined by the transshipment model. Since the quantity of unclean wood ground daily is equal to the quantity of hog fuel demanded by the customers that day, there is no flexibility in scheduling the grinding operations. On the other hand, inventory of clean wood chips is maintained at the yard, therefore, chipping schedules can be flexible in terms of the day it takes place. According to the optimal solution obtained from the transshipment model, chipping takes place every Monday and Tuesday such that the total quantity of wood chips produced is equal to their demand over the week. The inventory of clean wood chips at the end of each week is equal to the safety stock maintained by the company. No chipping happens during the weeks when clean wood chips are not demanded by customers.  3.6 Discussion The results presented in Section 3.5 indicate that an average reduction of 12% in total cost could be achieved using the optimization models developed in this study compared to the routes taken by the drivers. This reduction in total cost can be attributed to the decisions related to the truck type and the truckload type 47.516.25128.259.573.5420.5 4.2516.551.759037.259.51239.591.7518.50.754.2530.75598.750020406080100120140Sawdust LongTrailerRoll Off EndDumpSawdust LongTrailerRoll Off EndDumpSawdust LongTrailerRoll Off EndDumpType 1: Supplier-Yard Type 2: Yard-Customer Type 3: Supplier-CustomerAverage # TruckloadsCompany's solution This study  49  transported between different locations of the network, and the resultant routes of each truck incorporating backhauling as much as possible. This can be observed from the difference in the number of truckloads of each truck type and each truckload type, and the reduction in cost of empty truck travel compared to that of loaded truck travel in the optimal solutions.  The optimization models also resulted in a reduction of 11.7% in total fuel consumption on average. This reduction in fuel consumption corresponds to the reduction in total cost as the cost parameters used in the model include fuel consumption cost. This observation is in accordance with the literature (Lin et al. 2014) which indicate that cost minimization in transportation models also contribute to a reduction in fuel consumption.  The total number of truckloads transported in the driver’s original routes is similar to that suggested by the optimization models. Roll-off and end dump trucks are mainly used to pick up clean and unclean wood from suppliers which have little flexibility in the type of truck that can visit them. Therefore, the number of truckloads of roll-off and end dump trucks is almost the same in the optimal solution of the models and the driver’s original routes. On the other hand, several suppliers and customers dealing with sawdust, shavings, wood chips and hog fuel have flexibility in the type of truck that can visit them. Therefore, the number of truckloads of sawdust and long trailer trucks which carry these types of biomass are different using the optimization models compared to the company’s solution. The models suggested transporting more truckloads of the sawdust truck and fewer truckloads of the long trailer truck. Due to the increased number of sawdust truckloads, the total cost of using sawdust trucks is greater in the optimal solutions of the models. Similarly, due to the decrease in the number of long trailer trucks, the total cost of using long trailer trucks is lower in the optimum solutions suggested by the models compared to the costs in driver’s original routes. Although fewer number of long trailer truckloads is transported compared to that of sawdust trucks, the total cost of using long trailer trucks is much greater than that of sawdust trucks in the optimum solutions of the models. This indicates that more truckloads of sawdust trucks are used for shorter distance transportation, and long trailer trucks are used for longer distances. Although sawdust trucks have higher cost per unit volume of biomass (1.68 $/hour∙m3) compared to that of long trailer trucks (1.16 $/hour∙m3), they have shorter loading and unloading times. Due to this reason, depending on the quantity of biomass and the distance travelled, the model selects sawdust trucks for short distances and long trailer trucks for long distances. This trend of higher cost for long trailer trucks and lower cost for sawdust trucks can also be observed in the company’s original routes.   50  Since clean and unclean wood must be comminuted at the yard before delivering the respective feedstock to customers, the truckloads carrying clean and unclean wood, and clean wood chips and hog fuel cannot bypass the yard. All the truckloads carrying clean and unclean wood are of truckload type 1 (from suppliers to the yard, shown in Figure 3-3), and those carrying clean wood chips and hog fuel are of type 2 (from the yard to customers, shown in Figure 3-3). Depending on the quantities of biomass supplied and demanded, truckloads carrying sawdust and shavings can bypass the yard by performing a direct delivery. A reduction in the number of truckloads from the supplier to the yard (truckload type 1, shown in Figure 3-3), which is compensated by the increase in the number of direct deliveries from the suppliers to the customers (truckload type 3), can be observed in the optimum solutions. Since direct deliveries, which avoid multiple loading and unloading operations, are cost efficient, the optimization models selects more truckloads of type 3 compared to those in the company’s solution. 3.7 Decision support tool A decision support tool is developed for the company using the Solver Studio (Mason 2013) add-in in Microsoft Excel® to make their weekly transportation and comminution decisions. The model is written in the PuLP language. The tool is made as simple as possible, and only the inputs that vary on a regular basis are to be modified by the user. These inputs include the date of the beginning of the planning horizon, the inventory of biomass at the yard at the beginning of the planning horizon, the number of vehicles available and the quantities of biomass to be picked up and delivered. Once all the required data have been entered, the user presses the optimize button located on the spreadsheet to run the tool. The button executes the model and writes back the results in the workbook. Figures of the user interface and the workbook within the tool to display the results are shown in the Figures 3-10 and 3-11.    51   Figure 3-10: Snapshot of the main worksheet of the decision support tool Figure 3-11 displays the main results given by the tool. It contains the quantity of biomass to be comminuted on each day, the level of inventory of biomass for each day and a list of orders indicating all the information required for each truckload. To facilitate the interpretation of the results, the tool displays maps showing the relative quantities of biomass to be picked up and delivered at each supplier and customer locations.    52   Figure 3-11: Snapshot of the results sheet obtained from the decision support tool 3.8 Conclusions The current literature on biomass logistics optimization at the operational level is limited and focuses on optimizing daily truck routes without including decisions related to storage and pre-processing of biomass, which are important steps in the conversion of biomass to energy and fuels. Moreover, previous studies did not include intermediate storage sites in their networks. This chapter presented the optimization models for forest-based biomass logistics at the operational level considering biomass flow, storage, pre-processing, and truck routing decisions. The network included biomass supply sites, demand points, and intermediate storage sites. The problem considered the transportation of multiple types of biomass using multiple types of trucks that differ in capacity and cost. The problem was solved using a decomposition-based solution approach by solving the transshipment and the routing models sequentially. The transshipment model was solved to determine weekly storage, pre-processing and biomass flow decisions. The output of the transshipment model was used within the routing model to determine daily truck routes. The models were applied to a large biomass logistics company based in Lower Mainland region, British Columbia, Canada.     53  The results suggested that the decomposition-based solution approach improved the total cost and fuel consumption compared to the routes implemented by the company. For the considered case study, the results indicated a potential to improve up to 12% of total logistics cost and 11.7% of total fuel consumption of trucks and loaders compared to the routes implemented by the company. The models, which were validated by the company, suggested direct delivery of biomass from suppliers to customers where possible, and larger trucks were used more for the direct delivery of biomass. The optimum solutions prescribed using large trucks for long distances and smaller trucks for shorter distances. The results of the routing model indicated a reduction in relative cost of empty truck travel compared to the relative cost of loaded truck travel, where as it was the opposite in the company’s original routes. The decomposition-based solution approach can be applied to other cases where the transportation is in full truckloads. However, the models may require additional modifications to incorporate the case-specific constraints.  Due to the reliance on fossil fuels, biomass logistics activities emit carbon emissions. Carbon pricing policies, which aim at reducing carbon emissions, impact biomass logistics operations.  Although previous studies analyzed the impact of carbon pricing policies on the optimum solutions of biomass supply chain models, the analysis was pertinent to the specific case studies. Therefore, whether these results could be generalized is not clear. In the next chapter, the impact of carbon pricing policies on optimal cost and emissions of biomass logistics optimization models is analyzed using case-independent optimization models.     54  Chapter 4: Impact of carbon pricing policies on the cost and emissions of the biomass supply chain optimization models  4.1 Synopsis Climate change is one of the major challenges faced by the world (The World Bank 2019). The concentration of carbon dioxide in the atmosphere is considered as the most important factor contributing to climate change (NASA 2019). Several jurisdictions around the world introduced carbon pricing policies to mitigate carbon emissions. The three carbon pricing policies that are practiced in the world are the carbon tax, the carbon cap-and-trade, and the carbon offset policies. Previous studies analyzed the impacts of carbon pricing policies on optimum cost and emissions of biomass supply chain optimization models. However, these studies focused on specific case studies. Therefore, the results reported in previous studies may not be generalized and applied to other cases. However, the modelling framework used in previous studies to incorporate carbon pricing policies in biomass supply chain models was similar. Adopting a similar modelling framework, the impacts of carbon pricing policies on the optimal cost, emissions, and decision variables of case-independent biomass logistics models are analyzed in this chapter. For this purpose, a case-independent optimization model is developed without considering carbon pricing policies. The model is extended to incorporate carbon pricing policies by determining the deviation of total emissions from the initial allowance/compliance target and adding the cost of emissions to the objective function. Several propositions, which describe the impact of varying carbon prices and initial allowance/compliance target on the optimal cost and emissions of the case-independent models, are made. These propositions are proved using the mathematical properties of the models. Optimum cost and emissions of the models with different carbon pricing policies are compared pairwise. The propositions are numerically verified using the case of a biomass-fed district heating plant. An optimization model is developed for the selection of biomass feedstock at the biomass-fed district heating plant, and the model is extended to incorporate carbon pricing policies. The impact of carbon pricing policies on biomass feedstock selection, and the resultant cost and emissions at the district heating plant is analyzed.  4.2 Case-independent optimization models with carbon pricing policies Previous studies that analyzed the impacts of carbon pricing policies on optimization models first developed cost-only optimization models specific to the considered case studies. The cost-only optimization models were extended to include carbon pricing policies by determining the deviation of total emissions from the initial allowance/compliance target and adding the cost of emissions to the objective function. Following a   55  similar approach, a case-independent and cost-only optimization model is extended it to include carbon pricing policies in this section. Consider a cost-only optimization model that minimizes a cost function 𝑓(𝑥). The cost function 𝑓(𝑥) represents the cost incurred by a firm by making decisions related to its operations denoted by 𝑥. The set of all possible alternatives of 𝑥 is denoted by 𝑋. Therefore, 𝑋 denotes the feasible region of the optimization model defined by all constraints of the model. The operations denoted by 𝑥 ∈ 𝑋 have associated emissions 𝑒(𝑥). The objective function 𝑓(𝑥) of the cost-only optimization model does not include the cost of emissions. Using the notation shown in Table 4-1, the cost-only optimization model is extended to incorporate the three carbon pricing policies as shown below. It is assumed that there are no other restrictions on total emissions other than the constraints used to define the carbon policy.  Table 4-1: Notations used in case-independent optimization models for different carbon pricing policies Notation Definition 𝑓(𝑥) Cost objective function without considering the cost of carbon emission 𝑒(𝑥) Total emission  𝑋 Set of all feasible solutions of the optimization model 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 Initial emission allowance (tonnes of CO2-eq.) allocated in the carbon cap-and-trade policy 𝑒𝐶&𝑇+  Positive deviation of total emissions from the initial allowance. This is the total emission allowances purchased from the carbon market in the carbon cap-and-trade policy. 𝑒𝐶&𝑇+ = 𝑒(𝑥) − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 if 𝑒(𝑥) ≥ 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 𝑒𝐶&𝑇−  Negative deviation of total emissions from the initial allowance. This is the total emission allowances sold in the carbon market in the carbon cap-and-trade policy. 𝑒𝐶&𝑇− = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 − 𝑒(𝑥) if 𝑒(𝑥) < 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 Emissions compliance target (tonnes of CO2-eq.) assigned in the carbon offset policy 𝑒𝐶𝑂+  Positive deviation of total emissions from the compliance target. This is the number of carbon offsets to be purchased. 𝑒𝐶𝑂+ = 𝑒(𝑥) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 if 𝑒(𝑥) ≥ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡  𝑒𝐶𝑂−  Negative deviation of total emissions from the compliance target.  𝑒𝐶𝑂− = 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 − 𝑒(𝑥) if 𝑒(𝑥) < 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡  𝑝 Carbon price ($/tonne of CO2-eq) in the carbon pricing policies 𝑝𝑖𝑛𝑖𝑡 Price of initial carbon allowance ($/tonne of CO2-eq) in the carbon cap-and-trade policy 𝑍𝑇 Objective function value of the carbon tax model 𝑍𝑇 = 𝑓(𝑥) + 𝑝 ∗ 𝑒(𝑥) 𝑍𝐶&𝑇 Objective function value of the carbon cap-and-trade model. 𝑍𝐶&𝑇 = 𝑓(𝑥) + 𝑝 ∗ (𝑒𝐶&𝑇+ − 𝑒𝐶&𝑇− ) + 𝑝𝑖𝑛𝑖𝑡 ∗ 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒   56  𝑍𝐶𝑂 Objective function value of the carbon offset model.  𝑍𝐶𝑂 = 𝑓(𝑥) + 𝑝 ∗ (𝑒𝐶𝑂+ ) 𝑥∗ Optimum solution of the model with carbon pricing policies. 𝑥∗ ∈ 𝑋 4.2.1 Optimization model with the carbon tax policy The formulation 𝐶𝑇 shows a case-independent optimization model with the carbon tax policy.     𝐶𝑇: Minimize 𝑍𝑇 = 𝑓(𝑥) + 𝑝 ∗ 𝑒(𝑥) (4.1)  Subject to 𝑥 ∈ 𝑋 (4.2) The cost-only optimization model is extended to include the carbon tax policy by adding the cost of emission to the objective function as shown in Eq. (4.1).  The constraint set of the model with the carbon tax policy remains the same as that of the cost-only optimization model.  4.2.2 Optimization model with the carbon cap-and-trade policy The formulation 𝐶&𝑇 shows the optimization model with carbon cap-and-trade policy.  𝐶&𝑇: Minimize 𝑍𝐶&𝑇 = 𝑓(𝑥) + 𝑝 ∗ (𝑒𝐶&𝑇+ − 𝑒𝐶&𝑇− ) + 𝑝𝑖𝑛𝑖𝑡 ∗ 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒  (4.3)  Subject to 𝑒(𝑥) − 𝑒𝐶&𝑇+ + 𝑒𝐶&𝑇− = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 (4.4)   𝑥 ∈ 𝑋 (4.5)   𝑒𝐶&𝑇+ , 𝑒𝐶&𝑇− ≥ 0 (4.6) The cost-only optimization model is extended to include the carbon cap-and-trade policy by determining the deviation of total emissions from the initial allowance (Eq. (4.4)) and adding the cost of carbon to the objective function. As shown in the expression (4.3), the objective function of the cap-and-trade optimization model includes the cost of initial emission allowances and the cost of selling or purchasing additional emission allowances. Decisions related to purchasing and selling emission allowances are related to the total emissions and the initial allowance using the constraint shown in Eq. (4.4).  According to Eq. (4.4), 𝑒𝐶&𝑇+ − 𝑒𝐶&𝑇− = 𝑒(𝑥) − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒. The term (𝑒𝐶&𝑇+ − 𝑒𝐶&𝑇− ) in the objective function shown in Eq. (4.3) can be replaced with 𝑒(𝑥) − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒. Therefore, the objective function of the model with the carbon cap-and-trade model can be re-written as  𝑍𝐶&𝑇 = 𝑓(𝑥) + 𝑝 ∗ 𝑒(𝑥) − 𝑝 ∗ 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 + 𝑝𝑖𝑛𝑖𝑡 ∗ 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒. With this expression of the objective function, the decision variables 𝑒𝐶&𝑇+  and 𝑒𝐶&𝑇− , and constraint shown in Eq. (4.4) can be eliminated from the model.    57  4.2.3 Optimization model with the carbon offset policy The mathematical formulation 𝐶𝑂 shows the optimization model with the carbon offset policy.  𝐶𝑂: Minimize 𝑍𝐶𝑂 = 𝑓(𝑥) + 𝑝 ∗ (𝑒𝐶𝑂+ ) (4.7)  Subject to 𝑒(𝑥) − 𝑒𝐶𝑂+ + 𝑒𝐶𝑂− = 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 (4.8)   𝑥 ∈ 𝑋 (4.9)   𝑒𝐶𝑂+ , 𝑒𝐶𝑂− ≥ 0 (4.10) The constraint represented by Eq. (4.8) determines the number of carbon offsets purchased for emitting more than the compliance target. The associated cost of purchasing carbon offsets is added to the objective function as shown in the expression (4.7).  4.3 Total cost of optimization models with carbon pricing policies In this section, the impact of carbon price and initial allowance/compliance target on the total cost of the optimization models with carbon pricing policies is analyzed. In all these propositions, carbon price refers to carbon tax rate in the carbon tax model, carbon trading price in the carbon cap-and-trade model, and carbon offset purchase price in the carbon offset model.  Impact of carbon price on the optimal cost of the model with carbon tax policy When the total emission is nil, the cost of emission is zero. In this case, increasing the carbon price does not have any effect on the optimum cost of the model. However, when emissions are present, increasing the carbon price results in non-decreasing the total cost. These observations are stated in Proposition 4.1.  Proposition 4.1: For increasing values of carbon price, optimal cost of the model with carbon tax policy is non-decreasing.  Proof of proposition 4.1:  The optimal objective function value of the carbon tax model is 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗). The differential of the objective function of the carbon tax model with respect to carbon tax rate, 𝑝, is 𝑑𝑍𝑇𝑑𝑝= 𝑒(𝑥∗). Since 𝑒(𝑥∗) ≥0 for all 𝑥∗ ∈ 𝑋, the optimal objective function value of the carbon tax model is non-decreasing with increasing 𝑝.  Studies such as those by Memari et al. (2018) and Palak et al. (2014) reported that the total cost of the optimization model with the carbon tax policy increases with increasing carbon price. Both studies reported that the relationship between carbon price and the total cost is almost linear. In addition, Palak et al. (2014)   58  observed that considering higher fuel-efficient technologies in the models would reduce the increase rate in the total cost under the carbon tax policy.  Impact of carbon price on optimal cost of the model with carbon cap-and-trade policy When the total emission of the carbon cap-and-trade model is less than the initial allowance, the model prescribes to sell the remaining allowances in the carbon market resulting in an additional revenue. Therefore, the total cost of the carbon cap-and-trade model decreases with increasing carbon price when total emission is less than the initial allowance. However, when the total emission is greater than the initial allowance, the model prescribes to purchase additional allowances from the carbon market. Therefore, when total emission is greater than the initial allowance, the total cost of the carbon cap-and-trade model increases with increasing carbon price. These observations are stated in Proposition 4.2.  Proposition 4.2: If the total emission is greater than the initial allowance, the optimal cost of the optimization model with cap-and-trade policy increases for increasing carbon price. When the total emission is less than the initial allowance, the optimal cost of the model decreases with increasing carbon price.  Proof of proposition 4.2:  Let 𝑝′ be the carbon price at which optimum emissions is equal to 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 in the carbon cap-and-trade model. The optimal objective function of cap-and-trade model is 𝑍𝐶&𝑇 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) +𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝). The differential of 𝑍𝐶&𝑇 with 𝑝 is 𝑑𝑧𝑇∗𝑑𝑝= 𝑒(𝑥∗) − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒. The critical point of the objective function is when 𝑒(𝑥∗) = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒. Since 𝑒(𝑥∗) =𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 when 𝑝 = 𝑝′ and 𝑒(𝑥∗) is non-increasing with 𝑝, 𝑒(𝑥∗) − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 ≥ 0 when 𝑝 ≤ 𝑝′  and 𝑒(𝑥∗) − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 ≤ 0 when 𝑝 ≥ 𝑝′. Therefore, the objective function value 𝑧𝐶&𝑇 increases when 𝑝 < 𝑝′ and decreases when 𝑝 ≥ 𝑝′.   Proposition 4.2 suggests that the total cost of the model with the carbon cap-and-trade policy depends on both the carbon price and the initial allowance. Similar observation about the non-monotonicity of optimal cost of the carbon cap-and-trade model was made in studies such as (e.g., Li et al. (2017) and Zhang and Xu (2013)). Zhang and Xu (2013) developed a model for planning the production of multiple products using a common capacity under the carbon cap-and-trade policy. Instead of minimizing the total cost, their model dealt with maximizing the total profit of the model. They observed that the total profit of the model increased when carbon allowances were sold in the carbon market, and the profit decreased when carbon allowances were purchased from the carbon trading market.  Impact of the initial allowance on optimal cost of the model with carbon cap-and-trade policy   59  Proposition 4.3 describes that the impact of the initial allowance on the optimal cost of the model with carbon cap-and-trade policy depends on the carbon trading and initial allowance prices.     Proposition 4.3: When the carbon trading price is greater than the price of initial allowances, the optimum cost of the carbon cap-and-trade model decreases with increasing initial allowance. However, when the carbon trading price is less than the price of initial allowances, optimum cost of the carbon cap-and-trade model increases with increasing initial allowance.    Proof of proposition 4.3:  Let 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒1 and 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒2 be two values of initial allowances in the carbon cap-and-trade model such that 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒1 > 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒2. Let 𝑍𝐶&𝑇1 and 𝑍𝐶&𝑇2 be the costs of carbon cap-and-trade models with initial allowance of  𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒1 and 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒2, respectively. The optimal objective functions of the two cap-and-trade models with different initial allowances are 𝑍𝐶&𝑇1 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) + 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒1 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝) and 𝑍𝐶&𝑇2 = 𝑓(𝑥∗) + 𝑝 ∗𝑒(𝑥∗) + 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒2 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝). Since 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒1 > 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒2, 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒1 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝) > 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒2 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝) if 𝑝 < 𝑝𝑖𝑛𝑖𝑡. On the contrary, if 𝑝 >𝑝𝑖𝑛𝑖𝑡,  then 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒1 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝) < 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒2 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝). Therefore, 𝑍𝐶&𝑇1 >𝑍𝐶&𝑇2 if 𝑝 < 𝑝𝑖𝑛𝑖𝑡 and 𝑍𝐶&𝑇1 < 𝑍𝐶&𝑇2 if 𝑝 > 𝑝𝑖𝑛𝑖𝑡.  Previous studies that considered the carbon cap-and-trade model assumed initial allowances to be free of cost (i.e., price of initial allowance was zero). Therefore, the carbon trading price was greater than the price of initial allowances. In this case, having a larger initial allowance results in lower optimum cost compared to the carbon cap-and-trade model with smaller initial allowance. This observation was reported in the literature (e.g.,  Palak et al. (2014)). Du et al. (2013) also assumed that initial allowances were free and reported that the total profit of the emitter increases with increasing initial allowance. However, no previous study analyzed the impact of the initial allowance when the price of initial allowance is positive.  Impact of carbon price on the optimal cost of the model with carbon offset policy In the carbon offset model, carbon emissions are priced only when the total emission is above the compliance target. Therefore, with increasing carbon price, the total cost of the model increases and the total emission decreases when total emissions is greater than the compliance target. However, when total emission is equal to the compliance target, the model does not prescribe purchasing carbon offsets and the resultant cost of carbon emissions is zero. Therefore, when total emission is equal to the compliance target, any further increase in the carbon price does not affect the optimum solution and the optimal cost and   60  emissions remain constant. This observation on the impact of carbon price on the optimum cost of the carbon offset model is stated in Proposition 4.4. Proposition 4.4: When the total emission is more than the compliance target, the optimal cost of the model with carbon offset policy increases with increasing carbon price. When the total emission is equal to the compliance target, the optimal cost of the model remains constant with increasing carbon price.   Proof of proposition 4.4:  Let 𝑒(𝑥) = 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 when carbon offset purchase price is 𝑝′ in the carbon offset model. Objective function of carbon offset model is 𝑍𝐶𝑂 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒𝐶𝑂+ . Differential of 𝑍𝐶𝑂 with respect to carbon price is 𝑑𝑍𝐶𝑂𝑑𝑝= 𝑒𝐶𝑂+ . When 𝑝 < 𝑝′, then 𝑒(𝑥∗) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 and 𝑒𝐶𝑂+ > 0. Therefore, total cost of carbon offset model increases when 𝑝 < 𝑝′.  When 𝑝 > 𝑝′, the objective function of carbon offset model 𝑍𝐶𝑂 = 𝑓(𝑥∗). Derivative of 𝑍𝐶𝑂 with respect to carbon price is 𝑑𝑍𝐶𝑂𝑑𝑝= 0. Therefore, the total cost of carbon offset model remains constant for carbon prices greater than 𝑝′.  The optimal cost of the carbon offset model depends on the carbon price and the compliance target. However, unlike the carbon cap-and-trade model, the carbon offset model does not result in the optimal emissions below the compliance target as reducing emissions below the compliance target does not have any economic incentives. Although previous studies attempted to study the impact of carbon offset policy on the solutions of optimization models, none of them reported this observation regarding the impact of carbon price on the total cost of the carbon offset model.   Impact of the compliance target on optimal cost of the model with carbon offset policy When the compliance target is larger, the emitter has more flexibility to alter their operations and keep their total emission within the target. However, when the compliance target is smaller, the emitter has limited flexibility to alter their operations and may have to purchase carbon offsets if their total emission exceeds the target. As a result, lower compliance target results in higher optimal cost compared to having a higher compliance target in the carbon offset model. This observation is stated in Proposition 4.5.   Proposition 4.5: The optimal cost of the model considering the carbon offset policy decreases with increasing compliance target.  Proof of proposition 4.5:  Let 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1 and 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2  be the two compliance targets of the two carbon offset models such that 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1 > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2. Consider the three cases: 1)   61  𝑒(𝑥∗) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1,2) 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1 ≥ 𝑒(𝑥∗) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2, and  3) 𝑒(𝑥∗) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2. Let 𝑝1′  be the carbon price at which 𝑒(𝑥∗) = 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1 and 𝑝2′  be the carbon price at which 𝑒(𝑥∗) = 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2. Since 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1 >𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2, 𝑝1′ < 𝑝2′ . Case 1: Since 𝑒(𝑥∗) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1, total cost of both the models are 𝑍𝐶𝑂1 = 𝑓(𝑥∗) + 𝑝 ∗𝑒(𝑥∗) − 𝑝 ∗ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1 and 𝑍𝐶𝑂2 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) − 𝑝 ∗ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2. Therefore, 𝑍𝐶𝑂1 < 𝑍𝐶𝑂2. Case 2: Since 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1 ≥ 𝑒(𝑥∗) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2, 𝑍𝐶𝑂1 = 𝑓(𝑥∗) and 𝑍𝐶𝑂2 =𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) − 𝑝 ∗ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2. Therefore, 𝑍𝐶𝑂1 < 𝑍𝐶𝑂2. Case 3: 𝑒(𝑥∗) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡2 < 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡1, 𝑍𝐶𝑂1 = 𝑓(𝑥∗) when carbon price is 𝑝1′  and 𝑍𝐶𝑂2 = 𝑓(𝑥∗) when carbon price is 𝑝2′ . Since 𝑝1′ < 𝑝2′ , 𝑍𝐶𝑂1 < 𝑍𝐶𝑂2. In all three cases, it is proven that 𝑍𝐶𝑂1 < 𝑍𝐶𝑂2. Previous studies also reported that increasing the compliance target results in decreasing total cost in optimization models with the carbon offset policy (e.g., Palak et al. (2014) and Marufuzzaman et al. (2014a)). Palak et al. (2014) observed that for smaller compliance targets, the model resulted in the purchase of more carbon offsets than when the model used larger targets. Moreover, they observed that the models were more sensitive to changes in carbon offset price for smaller targets compared to larger targets. This is because the model resulted in purchase of more carbon offsets when smaller targets are used in the model.   4.4 Pair-wise comparison of optimization models considering carbon pricing policies This section discusses the pair-wise comparison of optimization models with different carbon pricing policies from optimal emission and cost perspectives. For a given pair of carbon pricing policies, a comparison is drawn between optimal emission and cost of the models with the two considered policies. Since there are three carbon pricing policies, there are three pairs of policies.  Comparing optimal emissions of models with carbon tax and carbon cap-and-trade policies Proposition 4.6: For equal carbon tax rate and carbon trading price, optimization models with carbon tax and carbon cap-and-trade policies result in equal optimal decisions and emissions.  Proof of proposition 4.6: Since 𝑒𝐶&𝑇+ − 𝑒𝐶&𝑇− = 𝑒(𝑥∗) − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 from the constraints of the carbon cap-and-trade model (see Eq. (4.4)), objective function of the carbon cap-and-trade model can be re-written as the following:   62  𝑓(𝑥∗) + 𝑝 ∗ (𝑒(𝑥∗) − 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒) + 𝑝𝑖𝑛𝑖𝑡 ∗ 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒= 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) + 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝) Since 𝑝, 𝑝𝑖𝑛𝑖𝑡 and 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 are constants, the presence of the term 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 ∗ (𝑝𝑖𝑛𝑖𝑡 −𝑝)  does not influence the optimal decisions of the model. The only term in the objective function that influences the optimal solution is 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) making the optimal decisions of the carbon cap-and-trade to be equal for equal carbon prices. In addition, the difference in optimal objective value between the carbon cap-and-trade model and carbon tax model is 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝).  From a cost-optimization perspective, both carbon tax and carbon cap-and-trade policies result in equal optimal emission when the carbon prices used in these models are equal. This observation about the equal emissions in carbon tax and carbon cap-and-trade models was also reported by Zhang and Xu (2013), who studied the production planning problem at a manufacturing plant with limited capacity. Goulder and Schein (2013) also highlighted this observation and stated that firms that aim at reducing costs would reduce emissions to a point where the marginal emission abatement cost is equal to carbon price. Proposition 4.6 implies that optimal decisions and emissions of the model with carbon cap-and-trade policy depend only on the carbon price used in the model. Values of the initial allowance and the price of initial allowance do not influence optimal decisions and emissions of the carbon cap-and-trade model.  Comparing optimal cost of models with carbon tax and carbon cap-and-trade policies In this comparison, the carbon tax rate used in the carbon tax model and the carbon trading price used in the carbon cap-and-trade model are considered equal. According to Proposition 4.6, when the carbon prices used in these models are equal, they result in equal optimum emission. Although the total emissions are equal, the total cost of emissions in these models could be different. Total emission cost in the carbon tax model is equal to the product of total emission and the tax rate. Cost of emissions in the carbon cap-and-trade model is the sum of the cost of purchasing initial allowances, and the cost of purchasing allowances when emissions are above the initial allowance, or the revenue from selling allowances in the carbon market when emissions are below the initial allowance.  For equal carbon trading and initial allowance prices, the total cost of emissions in the carbon cap-and-trade model is equal to the product of total emission and the carbon trading price. This cost expression is like the cost expression of the carbon tax model. Therefore, when the carbon tax rate, the carbon trading price, and the price of initial allowances are equal, the carbon tax and carbon cap-and-trade models have equal optimum costs. This observation is described in Proposition 4.7.     63  Proposition 4.7: For equal carbon tax rate, carbon trading price, and price of initial allowances, optimal costs of the models considering carbon tax and carbon cap-and-trade policies are equal. When carbon prices in carbon tax and carbon cap-and-trade models are equal and greater than price of the initial allowance, the optimal cost of the model with carbon tax is greater than that of the carbon cap-and-trade model, and vice versa.  Proof of proposition 4.7:  From Proposition 4.6, optimal decisions of the carbon cap-and-trade and carbon tax models are equal for equal carbon prices. The objective function value of carbon tax model is 𝑍𝑇 = 𝑓(𝑥∗) + 𝑝𝑇𝑒(𝑥∗), and that of carbon cap-and-trade model is 𝑍𝐶&𝑇 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) + 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 ∗ (𝑝𝑖𝑛𝑖𝑡 − 𝑝). From these expressions, 𝑍𝐶&𝑇 = 𝑍𝑇 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) if 𝑝𝑖𝑛𝑖𝑡 = 𝑝. If 𝑝𝑖𝑛𝑖𝑡 < 𝑝 then 𝑍𝐶&𝑇 < 𝑍𝑇, and if  𝑝𝑖𝑛𝑖𝑡 > 𝑝 then 𝑍𝐶&𝑇 > 𝑍𝑇.  Previous studies that considered the carbon cap-and-trade policy assumed that the initial allowance was always given free of cost making the price of initial allowance equal to zero (e.g., Marufuzzaman et al. (2014a) and Palak et al. (2014)). Therefore, it was reported that the optimal cost of models with carbon tax policy was greater than that of the models with carbon cap-and-trade policy. Since the price of the initial allowance was considered zero, the comparison between total costs of carbon tax and carbon cap-and-trade models for different carbon prices and the price of the initial allowance was not drawn.  Comparing optimal emissions of models with carbon tax and carbon offset policies For comparing the optimum emissions of the carbon tax and carbon offset models, the carbon tax rate and the carbon offset price are considered equal. When the total emission is greater than the compliance target of the carbon offset model, the difference in total costs of the carbon tax and the carbon offset models is the product of the compliance target and the carbon price. Since both the compliance target and the carbon price are constants, optimum solutions of the carbon tax and carbon offset models are equal from a cost-minimization perspective. Therefore, when the total emission is greater than the compliance target and when the carbon tax rate is equal to the carbon offset price, the carbon tax and the carbon offset models result in equal emissions.  According to Proposition 4.4, for increasing carbon price, the optimum solution of the carbon offset model remains constant when total emission is equal to the compliance target. At this stage, any increase in the carbon offset price does not alter the total emission of the carbon offset model. On the other hand, since emissions are always priced in the carbon tax model, total emissions decrease with increasing carbon price in the model with the carbon tax model. Therefore, when the total emission of the carbon offset model is   64  equal to the compliance target, total emission of the carbon tax model is less than that of the carbon offset model. These results are described in Proposition 4.8.    Proposition 4.8: When total emissions exceed the compliance target, optimal emissions of the models with carbon tax and carbon offset policies are equal for equal carbon prices.  When total emission of the carbon offset model is equal to the compliance target, the carbon tax model results in lower emissions with increasing carbon price.  Proof of proposition 4.8:  Let 𝑒(𝑥) = 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 when carbon offset purchase price is 𝑝′ in the carbon offset model. Since total emission decreases with carbon price, 𝑒(𝑥∗) ≥ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 when 𝑝 ≤ 𝑝′. Therefore, 𝑒𝐶𝑂+ >0 when 𝑝 < 𝑝′. Therefore, the objective function value of the carbon offset model when 𝑝 ≤ 𝑝′ is  𝑓(𝑥∗) +𝑝 ∗ 𝑒𝐶𝑂+ = 𝑓(𝑥∗) + 𝑝 ∗ (𝑒(𝑥∗) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡) = 𝑍𝑇 − 𝑝 ∗ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡, where 𝑍𝑇 =𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) is the objective function value of the carbon tax model. Since 𝑝 ∗ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 is a constant value, minimizing 𝑍𝐶𝑂 is equivalent to minimizing 𝑍𝑇. Therefore, carbon offset and carbon tax models have the same optimal solution when carbon prices are equal and 𝑝 ≤ 𝑝′.  When 𝑝 > 𝑝′, then 𝑒(𝑥∗) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡. In this case, 𝑒𝐶𝑂+ = 0. Therefore, the objective function of the carbon offset model is 𝑍𝐶𝑂 = 𝑓(𝑥∗) which is different from the objective function of the carbon tax model. Therefore, optimal solution of the two models may differ when 𝑝 > 𝑝′.  Models with carbon tax and carbon offset policies were developed in previous studies (e.g., Marufuzzaman et al. (2014a) and Palak et al. (2014)). However, optimal emissions from these models were not compared.  Comparing optimal cost of models with carbon tax and carbon offset policies In the carbon offset model, only the emissions that are above the compliance target are priced and the emissions that are below the target are not priced. Moreover, there is no cost associated with acquiring the compliance target in the carbon offset model. On the other hand, all emissions are priced in the carbon tax model.  Therefore, optimal cost of the model with carbon tax policy is always greater than that of the model with carbon offset policy.  Proposition 4.9: Optimal cost of the model with carbon offset policy is always less than that of the model with carbon tax policy. Proof of proposition 4.9:  The objective function of carbon tax model is 𝑍𝑇 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒(𝑥∗) and that of the carbon offset model is 𝑍𝐶𝑂 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒𝐶𝑂+  where 𝑒𝐶𝑂+ = 𝑒(𝑥∗) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 when 𝑒(𝑥∗) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡   65  and 𝑒𝐶𝑂+ = 0 when 𝑒(𝑥∗) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡. Since 𝑒𝐶𝑂+ < 𝑒(𝑥∗) for all carbon prices, the total cost of carbon offset model is always less than that of carbon tax model.  Although previous studies studied developed models with the carbon tax and the carbon offset policies, optimum costs from these models were not compared.  Comparing optimal emissions of models with carbon cap-and-trade and carbon offset policies Since total emissions from models with carbon tax and carbon cap-and-trade policies are equal for equal carbon prices (see Proposition 4.6), the comparison of total emissions from carbon offset and carbon cap-and-trade models is similar to the comparison of total emissions from carbon offset and carbon tax models.   Comparing optimal cost of models with carbon cap-and-trade and carbon offset policies In both the carbon offset model and the carbon cap-and-trade model with free initial allowance, the initial allowance (or the compliance target) is allotted free of cost. Any emissions above the initial allowance are priced in both of these models. Therefore, when the total emission is above the initial allowance/compliance target and carbon prices are equal, total costs of the carbon offset and carbon cap-and-trade models are equal. However, when total emission goes below the initial allowance, total cost of the carbon cap-and-trade model starts decreasing (see, Proposition 4.2), and the total cost of the carbon offset model remains constant once total emission is equal to the compliance target (see, Proposition 4.4). Therefore, when total emission is less than the initial allowance, the total cost of the carbon cap-and-trade model is less than that of the carbon offset model. These observations are stated in Proposition 4.10.  Proposition 4.10: When total emissions exceed the initial allowance, the optimum costs of the carbon offset model and the carbon cap-and-trade model with free initial allowance are equal for equal carbon prices. When total emissions of the carbon cap-and-trade model is less than the initial allowance, total cost of the carbon offset model is greater than that of the carbon cap-and-trade model with free initial allowance. Proof of proposition 4.10:  Let the optimum emissions be equal to 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 when carbon price is equal to 𝑝′ in the carbon offset model. When 𝑝 < 𝑝′, 𝑒(𝑥∗) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡. Therefore, the objective function of carbon offset model is 𝑍𝐶𝑂 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒𝐶𝑂+  and that of carbon cap-and-trade model with free initial allowances is 𝑍𝐶&𝑇 = 𝑓(𝑥∗) + 𝑝 ∗ 𝑒𝐶&𝑇+ . For equal carbon prices, both the model have equal optimum objective function values if 𝑝 < 𝑝′. However, when 𝑝 ≥ 𝑝′, the objective function of carbon offset model is 𝑍𝐶𝑂 = 𝑓(𝑥∗) and that of carbon cap-and-trade model with free initial allowance is 𝑍𝐶&𝑇 = 𝑓(𝑥∗) − 𝑝 ∗ 𝑒𝐶&𝑇− . Therefore, the objective   66  function value of carbon offset model is greater than that of carbon cap-and-trade model with free allowances when 𝑝 ≥ 𝑝′  Although previous studies that considered carbon cap-and-trade and carbon offset models did not compare the optimal cost of these models, they highlighted that the carbon cap-and-trade model is more cost-effective than the carbon offset model (e.g., Marufuzzaman et al. (2014a) and Marufuzzaman et al. (2014b)). This was because of the additional revenue made from selling the unutilized emission allowances in the carbon market in the carbon cap-and-trade model. On the other hand, there was no additional economic incentive for decreasing emissions below the carbon cap in the carbon offset model. Table 4-2 summarizes the results of analyzing and comparing optimal emissions and cost of optimization models with carbon tax, carbon cap-and-trade, and carbon offset policies.      67  Table 4-2: Summary of the analyses and pairwise comparisons of optimization models with carbon pricing policies from optimal emissions and cost perspectives  Model with carbon tax policy Model with carbon cap-and-trade policy Model with carbon offset policy Model with carbon tax policy • Optimum cost increases with increasing carbon price (Proposition 4.1)  • Equal carbon price results in equal total emissions for the two models (Proposition 4.6) • For the same carbon price, costs from the two models are equal if the price of the initial allowance is equal to carbon price (Proposition 4.7) • For the same carbon price and the price of initial allowance greater than the carbon price, the optimal cost of the model with carbon cap-and-trade policy is greater than that from the model with carbon tax policy, and vice versa (Proposition 4.7) • When total emission is more than the compliance target of the carbon offset model, equal carbon price results in equal emissions in both the models (Proposition 4.8) • For equal carbon prices, optimal cost of the model with carbon tax policy is always higher than that of the model with carbon offset policy (Proposition 4.9)  Model with carbon cap-and-trade policy - • When total emission is greater than the initial allowance, total cost of the model increases with increasing carbon price (Proposition 4.2) • When total emission is less than the initial allowance, total cost of the model decreases with increasing carbon price (Proposition 4.2) • When carbon trading price is greater than the price of initial allowance, the model with higher initial allowance has lower cost compared to the model with lower initial allowance (Proposition 4.3)  • When total emission is more than the compliance target, for equal carbon cap and carbon price, total emissions from the models are equal (Propositions 4.6 and 4.8) • When total emission is more than the compliance target, for equal carbon price and compliance target, and free initial allowances, optimal cost of both models are equal (Proposition 4.10) • When total emission is not more than the compliance target, for equal carbon price and compliance target, and free initial allowance, optimal cost of the model with carbon offset policy is greater than that of the carbon cap-and-trade model (Proposition 4.10) Model with carbon offset policy - - • When total emission is more than the compliance target, total cost of the model increases with increasing carbon offset price (Proposition 4.4) • When total emission is equal to the compliance target, total cost of the model remains constant with increasing carbon offset price (Proposition 4.4) • The model with larger compliance target has lower optimal cost compared to the model with smaller compliance target (Proposition 4.5)    68  4.5 Case study  The University of British Columbia (UBC) is home to around 25,000 students, faculty, staff and other residents who live on campus (UBC Sustainability 2019).  Consumption of natural gas for heat and hot water demand of campus is identified as the single biggest source of GHG emissions at UBC (Simpson et al. 2017). UBC installed a biomass gasification facility called the Bioenergy Research and Demonstration Facility (BRDF) with a 6 MW thermal capacity in 2012 as one of the measures to achieve a 33% reduction in total GHG emissions in 2015 compared to the 2007 levels by replacing natural gas with biomass (UBC Sustainability 2011). The BRDF acts as the base-load system and currently provides over 25% of UBC’s thermal energy demand. Any heat demand of the campus that cannot be met by BRDF due to its limited capacity is met by burning natural gas. A third-party logistics company provides feedstock to BRDF. The company collects, stores and pre-processes wood residues to appropriate size and quality and delivers them to the plant using trucks. Residues are derived from sawmills, furniture manufacturers wastes, and municipal tree trimmings. The logistics company owns a large storage yard, approximately 60 km away from UBC, where feedstock is stored before delivering it to BRDF.  Although BRDF receives consistent quality of wood residues, daily variations in moisture content, heating value and bulk density of feedstock exist (Dehkordi 2015; Oveisi et al. 2018). These variations influence not only the total quantity of biomass received per truckload, but also the daily number of truckloads required by the plant. Currently with a 6 MW capacity, the plant receives 1-3 truckloads of chipped wood residues with an average moisture content of around 33% (Dehkordi 2015). The average heating value and density of wood residues received by the plant are 19.3 GJ/kg and 180 kg/m3, respectively. The volume of each truckload of wood residues is 110 m3.  BRDF has a storage area with a capacity equal to two truckloads of biomass (Wauthy , UBC Energy & Water 2018). The storage area is divided into two bays, and the capacity of each bay is 110 m3. Since the capacity of trucks used to deliver wood residues is equal to the capacity of a storage bay, at most two truckloads of biomass can be stored in the storage area.  Currently on a given day, up to ~144 MWh (=6MW * 24 hours) or 518.4 GJ of heat can be generated at BRDF. Any demand beyond 518.4 GJ of heat is transferred to a natural gas-burning heat generation facility on campus called Campus Energy Center (CEC). The natural gas boiler at CEC has an efficiency of 87% based on HHV (Wauthy, UBC Energy & Water 2018). One GJ of heat generated from burning natural gas emits 50 kg of GHG emissions, whereas wood-based biomass emits around 8 kg of GHG emissions to   69  generate the same amount of heat (Ministry of Environment and Climate Change Strategy, British Columbia 2017). As a result, consumption of natural gas is identified as a major contributor of UBC’s GHG emissions.  Burning natural gas increases not only the total emissions, but also the total cost of heat generation due to the carbon tax and carbon offsets paid by UBC to the Province of British Columbia. When this study was conducted, the Province of British Columbia charged $35 for every tonne of emitted carbon dioxide according to the Carbon Tax Act (Ministry of Environment, British Columbia 2018). In addition, UBC pays $25 per tonne of carbon dioxide to offset the emissions from burning natural gas according to the Carbon Neutral Government Program (Wauthy , UBC Energy & Water 2018). The gas commodity price paid by UBC for generating one MWh of heat, including delivery and taxes, has averaged around $30/MWh over the past 5 years, but it is forecasted to rise to $40-43/MWh over the next few years. According to its climate action plan, UBC has set an ambitious target to reduce its GHG emissions to 67% and 100% below the 2007 levels in 2020 and 2050, respectively (UBC Sustainability, Simpson, and White 2019). Reduction in natural gas consumption could be crucial in achieving this target. In order to sustainably meet the anticipated increase in the heat demand of UBC and to achieve the goal of 67% reduction in its emission levels in 2020 compared to the 2007 levels, expansion of BRDF is identified as the best option from both financial and environmental perspectives (Simpson et al. 2017). Installing a new 12 MW biomass boiler to increase the total heat generation capacity of BRDF to 18 MW was put forward as a suitable option (Simpson et al. 2017).  For a heating capacity of 18 MW, total annual heat demand allocated to BRDF based on the heat demand of UBC is 117,171 MWh (421,815.6 GJ). Out of this heat demand, around 52,272 MWh (188,179.2 GJ) is generated using the current capacity of 6 MW and the remaining is generated using natural gas. As a result, over 95% of current total CO2-eq emission is from burning natural gas. Therefore, the expansion of BRDF to 18 MW capacity can mitigate around 12095 tonnes of CO2-eq emission by replacing natural gas with biomass assuming that burning biomass results in zero CO2-eq emission.   While the proposed expansion has a potential to decrease total CO2-eq emissions of UBC, the increase in daily number of biomass trucks on campus is identified to have some impact on the campus and neighboring communities (Simpson et al. 2017). The community impact could be due to increased traffic load on campus and possible disruptions to adjacent student residences due to increased number of biomass trucks on campus. Considering the average daily heat demand of BRDF with 18 MW capacity and average quality characteristics of wood residues received over one year (2012-2013), the average number of truckloads required by the plant per day is 5.4.  Rounding this number up, the plant would require around six truckloads of wood residues per day on average after the expansion. This calculation is in accordance with Olsson et   70  al. (2016) who reported that a biomass-fed district heating plant with a capacity of 22 MW would accept 2-7 truckloads of biomass per day when there was sufficient storage of biomass.  Although the average number of truckloads of wood residues received by BRDF is six, this number could vary due to variations in feedstock quality characteristics. On days with higher heat demand and/or lower quality of wood residues, feedstock requirement could exceed the average of six truckloads. While the increase in daily truckloads negatively impacts the neighboring communities, it may also result in biomass shortage if the third-party logistics provider suppling feedstock cannot arrange the necessary transportation infrastructure. To combat potential feedstock shortage due to trucking limitation, this study considers other energy-dense feedstock options, namely, pellets and briquettes along wood residues for the expanded capacity of the plant.    4.5.1 Data  Quality of wood residues received by BRDF during the year 2016-17 are considered in this study. See Dehkordi (2015) and Oveisi et al. ( 2018) for more details on the quality of wood residues currently received by BRDF. Wood pellets are assumed to have 5% moisture content, 18.95 MJ/kg heat content and 675 kg/m3 bulk density (Pinnacle Renewable Energy Inc. 2018), and briquettes are assumed to have moisture content between 8%-12%, 18.95 MJ/kg heat content and 550 kg/m3 bulk density (Winkler 2017). Wood residues for the expanded capacity of BRDF are assumed to be delivered using trucks with a capacity of 110 m3. Pellets and briquettes are delivered using trucks that are limited by weight instead of volume due to the high density of these biomass types. Each truckload of pellets is limited to 40 tonnes (Hoque et al. 2006), and briquettes are assumed to be transported using trucks which are used to carry pellets (TecnoAmbiental et al. 2016). Feedstock is priced based on the dry tonnes of biomass received at the heating plant. The dry tonnes of biomass in one truckload depends on the moisture content and bulk density of biomass, and the truck capacity. The price of one dry tonne of wood pellets is approximately 85% more than that of dry wood residues, and the price of one dry tonne of briquettes is approximately 50% more than that of wood residues. Wood residues and briquettes are stored in the storage bays, while pellets should be stored in silos. Therefore, the storage capacities for different biomass types are different. Quality, price, emission and transportation characteristics of the three types of feedstock used in this study are summarized in Table 4-3.       71  Table 4-3: Quality, price, emission, and transportation characteristics of wood residues, briquettes, and pellets Parameter Wood residues Pellets Briquettes Average moisture content (% weight) 33 a  5 d  8-12 f  Average heating value (MJ/kg) 19 a 19 d 19 f  Average bulk density (kg/m3) 148 a 675 d 550 f  Price ($/odt) 78 b  141 d 120 f  Emission ratio (kg-CO2/GJ) 2 c  2 c 2 c  Truck capacity 110 m3 b 40 tonnes e  40 tonnes g  a: (Dehkordi 2015) b: (Wauthy, UBC Energy & Water 2018) c: (Ministry of Environment and Climate Change Strategy, British Columbia 2017) d: (Pinnacle Renewable Energy Inc. 2018) e: (Hoque et al. 2006) f: (Winkler 2017) g: (TecnoAmbiental et al. 2016) Two technologies are considered for the new system with 12 MW capacity, namely, biomass gasification and combustion. The efficiency of gasification is more than that of combustion. Nexterra Systems Corp., the company that installed the current gasification system at BRDF, provided efficiency of gasification technology (shown in Figure 4-1) as a function of biomass moisture content for the existing gasifier (6MW capacity) and the state-of-the-art gasifier which could be installed in the new system. These efficiencies are calculated according to the mass and energy protocols given in Kitto and Stultz (2005). It is mentioned that the efficiency of biomass combustion technology would be similar to that of the existing gasification system based on the same operating conditions mandated by fuel type (Mui 2017).    72    Figure 4-1: Efficiencies of the existing and the proposed new gasification systems (provided by Nexterra Systems Corp.) The provided efficiencies of the existing and new gasification systems were used to develop Equations (4.11) and (4.12) using regression. These equations are used in the optimization model described in Section 4.6.     𝑒𝑒𝑥𝑖𝑠𝑡𝑖𝑛𝑔 =  (−6 ∗ 10−5 ∗ 𝑚𝑐3 + 0.0013 ∗ 𝑚𝑐2 − 0.1145 ∗ 𝑚𝑐 + 77.341)100 (4.11)  𝑒𝑛𝑒𝑤 =(−7 ∗ 10−5 ∗ 𝑚𝑐3 + 0.0018 ∗ 𝑚𝑐2 − 0.1948 ∗ 𝑚𝑐 + 82.936)100  (4.12)   556065707580855 10 15 20 25 30 35 40 45 50 55 60% Efficiency% Moisture contentGasification in the new system Gasification in the old system  73  4.6 Optimization model The propositions shown in Sections 4.3 and 4.4 are numerically verified using the case study of the biomass-fed district heating plant described in Section 4.5.  An optimization model that optimizes the mix of feedstock and natural gas to meet a given heat demand is shown below. Since the heat generated at the district heating plant is consumed within the UBC campus,  no revenue is made by selling the heat in the market. Therefore, the objective function of the model is to minimize the total cost of biomass feedstock and natural gas to meet the heat demand.  The model assumes that the new 12 MW system installed in the plant is based on the gasification technology. Table 4-4 shows the notations used to develop the optimization model.  Table 4-4: Notations used in the models Set Definition  𝑇 Set of all periods 𝑇 = {1,2 … , 365} where each time period corresponds to one day 𝐵 Set of all biomass available 𝐵 = {ℎ𝑖𝑔ℎ 𝑞𝑢𝑎𝑙𝑖𝑡𝑦 𝑤𝑜𝑜𝑑 𝑐ℎ𝑖𝑝𝑠, 𝑤𝑜𝑜𝑑 𝑝𝑒𝑙𝑙𝑒𝑡𝑠, 𝑏𝑟𝑖𝑞𝑢𝑒𝑡𝑡𝑒𝑠} Parameter Definition 𝐶𝑏 Cost of biomass type 𝑏 ∈ 𝐵 on dry basis ($/odt) 𝐶𝑔 Cost of heat produced from natural gas ($/MWh) 𝐻𝐷𝑡 Heat demand from the campus on day 𝑡 ∈ 𝑇 (in MWh) 𝐸𝐸(𝑚𝑐𝑏𝑡) Efficiency of the existing gasification system as a function of the moisture content. 𝐸𝐸(𝑚𝑐𝑏𝑡) =(−6 ∗ 10−5 ∗ 𝑀𝐶𝑏𝑡3 + 0.0013 ∗ 𝑀𝐶𝑏𝑡2 − 0.1145 ∗ 𝑀𝐶𝑏𝑡 + 77.341)100 𝐸𝑁(𝑚𝑐𝑏𝑡) Efficiency of the new gasification system with a capacity of 12 MW.  𝐸𝑁(𝑚𝑐𝑏𝑡) =(−7 ∗ 10−5 ∗ 𝑀𝐶𝑏𝑡3 + 0.0018 ∗ 𝑀𝐶𝑏𝑡2 − 0.1948 ∗ 𝑀𝐶𝑏𝑡 + 82.936)100 𝐻𝑉𝑏𝑡 Heating value of biomass 𝑏 ∈ 𝐵 on day 𝑡 ∈ 𝑇 (MWh/bdt) 𝑀𝐶𝑏𝑡 Moisture content of biomass 𝑏 ∈ 𝐵 on day 𝑡 ∈ 𝑇 (% weight) 𝑁 Maximum number of truckloads received per day 𝑆 Maximum storage capacity (in m3) 𝑆𝑝 Maximum storage capacity of the silo (in m3) for storing wood pellets 𝑉𝐶𝑏 Vehicle capacity for biomass 𝑏 ∈ 𝐵 in terms of volume (in m3) 𝐻𝐶𝐸 Maximum heat capacity of the existing system in MWh 𝐻𝐶𝑁 Maximum heat capacity of the new system in MWh 𝐺𝐻𝐺𝑊𝑜𝑜𝑑 Emission ratio for wood in the existing system (tonnes of 𝐶𝑂2 per MWh of heat)   74  𝐺𝐻𝐺𝑁𝐺 Emission ratio for natural gas (tonnes of 𝐶𝑂2 per MWh of heat) 𝐷𝑏𝑡 Density of biomass 𝑏 ∈ 𝐵 at period 𝑡 ∈ 𝑇 (delivered tonne/m3) Decision Variable Definition 𝑥𝐸𝑏𝑡 Quantity of biomass type 𝑏 ∈ 𝐵 (m3) burned on day 𝑡 ∈ 𝑇 in the existing system from what is received on day 𝑡  𝑥𝑁𝑏𝑡 Quantity of biomass type 𝑏 ∈ 𝐵 (m3) burned on day 𝑡 ∈ 𝑇 in the new system from what is received on day 𝑡 𝑥𝐸𝑆𝑏𝑡 Quantity of biomass type 𝑏 ∈ 𝐵\{𝑝𝑒𝑙𝑙𝑒𝑡𝑠} (m3) burned on day 𝑡 ∈ 𝑇 in the existing system from what is stored on day 𝑡 − 1  𝑥𝑁𝑆𝑏𝑡 Quantity of biomass type 𝑏 ∈ 𝐵\{𝑝𝑒𝑙𝑙𝑒𝑡𝑠} (m3) burned on day 𝑡 ∈ 𝑇 in the new system from what is stored on day 𝑡 − 1  𝑛𝑏𝑡 Number of truckloads of biomass 𝑏 ∈ 𝐵 received on day 𝑡 ∈ 𝑇 𝑠𝑏𝑡 Quantity of biomass of type 𝑏 ∈ 𝐵 stored at the end of day 𝑡 ∈ 𝑇 (m3) 𝑔𝑡 Quantity of energy required to be produced from natural gas on day 𝑡 ∈ 𝑇 (MWh) The mixed-integer linear programming model, which determines the amount of feedstock and natural gas to be consumed each day and daily storage quantities of biomass at the plant, is described below.     Objective function The objective function, shown in the Eq. (4.13), minimizes the total cost including the purchase cost of biomass and natural gas. 𝑓 = ∑ ∑(𝑛𝑏𝑡 ∗ 𝑉𝐶𝑏 ∗ 𝐷𝑏𝑡 ∗ (1 − 𝑀𝐶𝑏𝑡) ∗ 𝐶𝑏𝑡)𝑏∈𝐵𝑡∈𝑇+ ∑(𝑔𝑡 ∗ 𝐶𝑔)𝑡∈𝑇 (4.13) Constraints Constraint (4.14) ensures that the daily demand of heat is met either by burning biomass or by using natural gas. Since wood residues and briquettes are stored in storage bays and pellets are stored separately in silos, there are separate inventory balance constraints for wood residues and briquettes, and pellets.    75  ∑ 𝑥𝐸𝑏𝑡 ∗ 𝐷𝑏𝑡 ∗ (1 − 𝑀𝐶𝑏𝑡) ∗ 𝐻𝑉𝑏𝑡 ∗ 𝐸𝐸(𝑀𝐶𝑏𝑡) + 𝑥𝑁𝑏𝑡 ∗ 𝐷𝑏𝑡 ∗ (1 − 𝑀𝐶𝑏𝑡) ∗ 𝐻𝑉𝑏𝑡𝑏∈𝐵∗ 𝐸𝑁(𝑚𝑐𝑏𝑡)+  ∑ 𝑥𝐸𝑆𝑏𝑡 ∗ 𝐷𝑏,𝑡−1 ∗ (1 − 𝑀𝐶𝑏,𝑡−1) ∗ 𝐻𝑉𝑏,𝑡−1 ∗ 𝐸𝐸(𝑀𝐶𝑏,𝑡−1)𝑏∈𝐵\{𝑝𝑒𝑙𝑙𝑒𝑡𝑠}+ 𝑥𝑁𝑆𝑏𝑡 ∗ 𝐷𝑏,𝑡−1 ∗ (1 − 𝑀𝐶𝑏,𝑡−1) ∗ 𝐻𝑉𝑏,𝑡−1 ∗ 𝐸𝑁(𝑀𝐶𝑏,𝑡−1) + 𝑔𝑡= 𝐻𝐷𝑡, ∀ 𝑡 ∈ 𝑇 (4.14) To ensure that biomass stored from day 𝑡 − 1 is first utilized completely before using biomass that is received on day 𝑡, the total biomass used from storage in both systems is equal to the total quantity of biomass in storage as shown constraint set (4. 15).  𝑥𝑁𝑆𝑏𝑡 + 𝑥𝐸𝑆𝑏𝑡 = 𝑠𝑏,𝑡−1, ∀ 𝑏 ∈ 𝐵\{𝑝𝑒𝑙𝑙𝑒𝑡𝑠}, 𝑡 ∈ 𝑇\{1} (4.15) Constraint set (4. 16) shows inventory balance constraints for wood residues and briquettes that are stored in storage bays.  𝑛𝑏𝑡 ∗ 𝑉𝐶𝑏 − 𝑥𝐸𝑏𝑡 − 𝑥𝑁𝑏𝑡 = 𝑠𝑏,𝑡 , ∀ 𝑏 ∈ 𝐵\{𝑝𝑒𝑙𝑙𝑒𝑡𝑠}, 𝑡 ∈ 𝑇 (4.16) Inventory balance constraints for pellets are shown in constraint set (4.17).  𝑠𝑏,𝑡−1 + 𝑛𝑏𝑡 ∗ 𝑉𝐶𝑏 − 𝑥𝐸𝑏𝑡 − 𝑥𝑁𝑏𝑡 = 𝑠𝑏,𝑡, 𝑏 = {𝑝𝑒𝑙𝑙𝑒𝑡𝑠}, ∀ 𝑡 ∈ 𝑇\{1} (4.17) Constraint set (4.18) imposes that the number of truckloads of all biomass types received by the plant on a given day does not exceed the maximum number of truckloads permitted daily. ∑ 𝑛𝑏𝑡𝑏∈𝐵≤ 𝑁, ∀ 𝑡 ∈ 𝑇 (4.18) Constraint sets (4.19) and (4.20) ensure that the stored biomass is not more than the available storage capacity at the plant.  ∑ 𝑠𝑏𝑡𝑏∈𝐵≤ 𝑆, ∀ 𝑏 ∈ 𝐵\{𝑝𝑒𝑙𝑙𝑒𝑡𝑠} 𝑡 ∈ 𝑇 (4.19) 𝑠𝑏𝑡 ≤ 𝑆𝑝, ∀ 𝑡 ∈ 𝑇, 𝑏 = {𝑝𝑒𝑙𝑙𝑒𝑡𝑠}  (4.20)   76  Constraint sets (4.21) and (4.22) ensure that heat generated at the plant does not exceed the heating capacities of the existing system (6 MW) and the new system (12 MW).  ∑ 𝑥𝑁𝑏𝑡 ∗ 𝐷𝑏𝑡 ∗ (1 − 𝑀𝐶𝑏𝑡) ∗ 𝐻𝑉𝑏𝑡 ∗ 𝐸𝑁(𝑀𝐶𝑏𝑡)𝑏∈𝐵+ ∑ 𝑥𝑁𝑆𝑏𝑡 ∗ 𝐷𝑏,𝑡−1 ∗ (1 − 𝑀𝐶𝑏,𝑡−1) ∗ 𝐻𝑉𝑏,𝑡−1 ∗ 𝐸𝑁(𝑀𝐶𝑏,𝑡−1)𝑏∈𝐵\{𝑝𝑒𝑙𝑙𝑒𝑡𝑠}≤ 𝐻𝐶𝑁, ∀ 𝑡 ∈ 𝑇 (4.21) ∑ 𝑥𝐸𝑏𝑡 ∗ 𝐷𝑏𝑡 ∗ (1 − 𝑀𝐶𝑏𝑡) ∗ 𝐻𝑉𝑏𝑡 ∗ 𝐸𝐸(𝑀𝐶𝑏𝑡)𝑏∈𝐵+  ∑ 𝑥𝐸𝑆𝑏𝑡 ∗ 𝐷𝑏,𝑡−1 ∗ (1 − 𝑀𝐶𝑏,𝑡−1) ∗ 𝐻𝑉𝑏,𝑡−1 ∗ 𝐸𝐸(𝑀𝐶𝑏,𝑡−1)𝑏∈𝐵\{𝑝𝑒𝑙𝑙𝑒𝑡𝑠}≤ 𝐻𝐶𝐸, ∀ 𝑡 ∈ 𝑇 (4.22) Constraint set (4.23) shows the sign restrictions and integrality restrictions. 𝑥𝑁𝐺𝑏𝑡 , 𝑥𝑁𝐶𝑏𝑡, 𝑥𝑁𝐺𝑆𝑏𝑡, 𝑥𝑁𝐶𝑆𝑏𝑡, 𝑥𝐸𝑏𝑡 , 𝑥𝐸𝑆𝑏𝑡, 𝑠𝑏𝑡 , 𝑔𝑡 ≥ 0, ∀ 𝑏 ∈ 𝐵, 𝑡 ∈ 𝑇 𝑛𝑏𝑡 ∈ 𝐼𝑛𝑡𝑒𝑔𝑒𝑟𝑠, ∀ 𝑏 ∈ 𝐵, 𝑡 ∈ 𝑇 (4.23) Consumption of biomass and natural gas result in carbon emission. The total emission from heat generation is shown in Equation (4.24). 𝑒 = ∑ ∑( 𝑥𝐸𝑏𝑡 ∗ 𝐷𝑏𝑡 ∗ (1 − 𝑀𝐶𝑏𝑡) ∗ 𝐻𝑉𝑏𝑡 ∗ 𝐸𝐸(𝑀𝐶𝑏𝑡) + 𝑥𝑁𝑏𝑡 ∗ 𝐷𝑏𝑡 ∗ (1 − 𝑀𝐶𝑏𝑡)𝑏∈𝐵𝑡∈𝑇∗ 𝐻𝑉𝑏𝑡 ∗ 𝐸𝑁(𝑀𝐶𝑏𝑡) ) ∗ 𝐺𝐻𝐺𝑊𝑜𝑜𝑑  + ∑ ∑ ( 𝑥𝐸𝑆𝑏𝑡 ∗ 𝐷𝑏,𝑡−1 ∗ (1 − 𝑀𝐶𝑏,𝑡−1) ∗ 𝐻𝑉𝑏,𝑡−1 ∗ 𝐸𝐸(𝑀𝐶𝑏,𝑡−1) + 𝑥𝑁𝑆𝑏𝑡𝑏∈𝐵{𝑝𝑒𝑙𝑙𝑒𝑡𝑠}𝑡∈𝑇∗ 𝐷𝑏,𝑡−1 ∗ (1 − 𝑀𝐶𝑏,𝑡−1) ∗ 𝐻𝑉𝑏,𝑡−1 ∗ 𝐸𝑁(𝑀𝐶𝑏,𝑡−1) ) ∗ 𝐺𝐻𝐺𝑊𝑜𝑜𝑑   + (𝑔𝑡 ∗ 𝐺𝐻𝐺𝑁𝐺) (4.24) The objective function of the optimization model shown in Expression (4.13) does not include the cost of carbon emission. The developed optimization model can be modified to incorporate the cost of carbon using carbon pricing policies by including the price of carbon in the objective function as shown in Section 4.2.    77  4.7 Results This section shows the results from solving the optimization models developed in Section 4.6 with different carbon pricing policies. The models are solved using CPLEX 12.7.1 solver within the AIMMS 4.46 software on a computer with Intel (R) Core (TM) i7-6700 CPU at 3.4 GHz processor to achieve an optimality gap of 0.05%. The optimization models are first solved for minimizing the total feedstock cost and emissions separately. The feedstock cost minimization model minimizes the cost function shown in expression (4.13) and emission minimization model minimizes the expression in Eq. (4.24). Both these models have the same set of constraints as shown in expressions (4.14) to (4.23). Table 4-5 shows the total cost, emission, and optimal feedstock mix for both of these models. The optimal cost of the cost minimization model is around $2.3 M and the resultant emissions are 3,945.5 tonnes of CO2-eq. The optimal cost and emissions of the emissions minimization model are approximately $3.1 M and 1,169.9 tonnes of CO2-eq., respectively. The feedstock cost minimization model prescribes using only wood residues as the feedstock in the district heating plant and around 16,300 MWh of heat is generated from burning natural gas. However, if the cost of natural gas increases, the optimum solution may prescribe using more briquettes and pellets to offset the increase in the total cost. The emission minimization model suggests using a mix of all biomass types and no heat is generated from natural gas. The emission minimization model prescribes the use of higher density feedstock (briquettes and pellets) at the plant compared to the cost minimization model.  Table 4-5: Total cost, emission, and optimal feedstock mix per year from feedstock cost minimization and emission minimization models Model Total cost per year (CDN $) Emission per year  (tonnes CO2-eq) # of truckloads per year Heat generated from natural gas per year (MWh) Wood residues Briquettes Pellets Feedstock cost minimization model 2,361,333.3 3,945.5 1,751 0 0 16,326.8 Emissions minimization model 3,107,281.3 1,169.9 996 172 360 0 For assessing the impact of carbon price on optimal solutions of models with carbon pricing policies, optimization model with each policy is solved using carbon prices varying between $0 per tonne of CO2-eq to $100 per tonne of CO2-eq with increments of $5 per tonne CO2-eq. Therefore, the model with each policy is solved 21 times using different carbon prices.    78  Apart from the carbon price, the initial allowance and the price of initial allowance are two other parameters in the model with the carbon cap-and-trade policy. To analyze the impact of the initial allowance on optimal solution of the carbon cap-and-trade model, two different values of initial allowance, 3000 tonnes and 2000 tonnes, are considered in the study. These values are selected as they lie in between the maximum and minimum emission values as determined by the cost minimization and emission minimization models, respectively. The impact of the initial allowance price is analyzed by considering the models with free initial allowance as well as an initial allowance price of $35 per tonne of CO2-eq. The price of $35 per tonne of CO2-eq. is used because it is the carbon tax rate in the province of British Columbia, Canada in 2018 (BC Ministry of Environment 2018). In total, four variants of the carbon cap-and-trade policy, with two different initial allowances and two prices of initial allowances, are considered. Similar to carbon cap-and-trade models, two values of compliance targets- 2000 tonnes and 3000 tonnes CO2-eq.- are considered for carbon offset models. Therefore, two variants of carbon offset policy are considered in this study.  Figure 4-2 shows optimal emissions from models with carbon tax, carbon cap-and-trade, and the two variants of carbon offset policies. The four variants of carbon cap-and-trade policies are not shown in Figure 4-2 because optimal emissions of the model with carbon cap-and-trade policy depends only on carbon price, and not on the initial allowance and the price of initial allowance (see, Proposition 4.6). Therefore, optimal emissions in models with the four variants of carbon cap-and-trade policies are equal.    79    Figure 4-2: Total emission in carbon tax (and carbon cap-and-trade) and carbon offset models for different carbon prices 4.7.1 Impact of carbon pricing policies on the total cost of the optimization models  Impact of carbon price on optimal cost of the carbon tax model  Total cost of the model with the carbon tax model for different carbon prices is shown in Figure 4-3.  0500100015002000250030003500400045000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Emission (tonnes of CO2-eq)Carbon price (CDN $/tonne CO2-eq)Carbon offset model with compliance target of 2000 tonnesCarbon offset model with a compliance target of 3000 tonnesCarbon tax and carbon cap-and-trade modelsPrice of initial allowance Emissions = 2927 tonnes CO2-eqEmissions = 2016 tonnes CO2-eq  80   Figure 4-3: Total cost in the carbon tax model for different carbon prices It can be observed from Figure 4-3 that the total cost of the model with carbon tax policy is non-decreasing with increasing carbon price. This observation is consistent with Proposition 4.1. Since total emissions is not nil, total cost is strictly increasing with increasing carbon price.  Impact of carbon price on optimal cost of the model with carbon cap-and-trade policy Figure 4-4 shows optimum costs of the four carbon cap-and-trade models. The price of initial allowance and the two carbon prices where the total emission of the model is almost equal to the initial allowance are shown in Figure 4-4.  2.32.352.42.452.52.552.62.652.70 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Total cost in the carbon tax model (CDN $)MillionsCarbon price (CDN $/tonne of CO2-eq)  81   Figure 4-4: Optimum costs of the four carbon cap-and-trade models for different carbon prices Figure 4-2, which shows total emission in the four carbon cap-and-trade models, suggests that for the carbon cap-and-trade model with initial allowance of 3000 tonnes, total emission is almost equal to 3000 tonnes at a carbon price of $55. According to Figure 4-4, the total cost of the carbon cap-and-trade models with an initial allowance of 3000 tonnes increases with increasing carbon trading price until the price of $55. For carbon price above $55, the total cost decreases with increasing carbon price. This observation is consistent with Proposition 4.2.  Similar to the model with initial allowance of 3000 tonnes, the carbon cap-and-trade models with an initial allowance of 2000 tonnes also shows non-monotonocity of total cost with respect to the carbon price. At a price of $75, total emissions of the model with an initial allowance of 2000 tonnes is almost equal to 2000 tonnes (see, Figure 4-2). Therefore, the total cost of the model increases with increasing carbon price until the price of $75. For carbon prices higher than $75, total cost of the model decreases with increasing carbon price.  Impact of the initial allowance on optimal cost of the model with carbon cap-and-trade policy 2.342.362.382.42.422.442.462.482.52.522.542.560 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Total cost in carbon cap-and-trade models (CDN $)MillionsCarbon price (CDN $/tonne CO2-eq)Carbon cap-and-trade model with free allowances and initial allowance of 2000 tonnesCarbon cap-and-trade model with free allowances and initial allowance of 3000 tonnesCarbon cap-and-trade model with initial allowance price of $35 and initial allowance of 2000 tonnesCarbon cap-and-trade model with initial allowance price of $35 and initial allowance of 3000 tonnesPrice of initial allowance   82  Figure 4-4 suggests that the optimal costs of the carbon cap-and-trade models are equal when carbon trading price and the price of initial allowance are equal to $35. When the carbon trading price is greater than $35 and the price of initial allowance is $35, the model with an initial allowance of 3000 tonnes has less cost compared to the model with an initial allowance of 2000 tonnes. On the contrary, when the carbon trading price is less than $35 and the price of initial allowances is $35, the cost of the model with an initial allowance of 3000 tonnes (green line) is greater than the cost of the model with an initial allowance of 2000 tonnes (purple line). These observations are consistent with Proposition 4.3.   Impact of carbon price on optimal cost of the model with carbon offset policy Optimum costs of the two carbon offset models are shown in Figure 4-5.   Figure 4-5: Optimum costs of carbon offset models for different carbon prices According to Figure 4-2, total emissions of the carbon offset model with a compliance target of 3000 tonnes is almost equal to 3000 tonnes at a carbon price of $55. Figure 4-5 suggests that the total cost of the carbon offset model increases with increasing carbon price until a price of $55. For carbon prices beyond $55, any increase in the carbon price does not impact the total cost of the model.  2.342.362.382.42.422.442.462.480 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Total cost in carbon offset models (CDN $)MillionsCarbon price (CDN $/tonne CO2-eq)Carbon offset model with compliance target of 2000 tonnesCarbon offset model with compliance target of 3000 tonnes  83  Similar to the model with a compliance target of 3000 tonnes, total emissions of carbon offset model with a compliance target of 2000 tonnes is almost equal to 2000 tonnes at a carbon price of $75. Therefore, according to Figure 4-5, total cost of the model with a compliance targer of 2000 tonnes increases with increasing carbon price until a price of $75. For carbon prices beyond $75, the total cost of the model decreases with increasing carbon price. These observations are consistent with Proposition 4.4.   Impact of the compliance target on optimal cost of the model with carbon offset policy It can be observed from Figure 4-5 that the total cost of the carbon offset model with a compliance target of 2000 tonnes per year is greater than that of the model with a target of 3000 tonnes per year. This observation is consistent with Proposition 4.5. 4.7.2 Pair-wise comparison of the optimization models with carbon pricing policies  Comparing optimal emissions of models with carbon tax and carbon cap-and-trade policies According to Figure 4-2, total emissions in the carbon tax model and all carbon cap-and-trade models are equal. This observation is consistent with Proposition 4.6. This also indicates that the optimal emission of carbon cap-and-trade models depend only on the carbon trading price, and not on the initial allowance and the price of initial allowance.  Comparing optimal cost of models with carbon tax and carbon cap-and-trade policies Optimal costs of the carbon tax and the four carbon cap-and-trade models are shown in Figure 4-6.    84   Figure 4-6: Optimal costs of the carbon tax and carbon cap-and-trade models When the cost of initial allowance is zero, i.e., when initial allowance is free of cost, Figure 4-6 suggests that the total cost of the carbon cap-and-trade model is less than that of the carbon tax model. However, the price of initial allowance is $35, the total cost of the carbon cap-and-trade model is greater than that of the carbon tax model for carbon prices below $35. On the other hand, when the carbon price is greater than $35, the total cost of the carbon cap-and-trade model is less than that of the carbon tax model. These results are consistent with Proposition 4.7. Comparing optimal emissions of models with carbon tax and carbon offset models According to Figure 4-2, the total emission of the carbon offset model with a compliance target of 3000 tonnes is equal to 3000 tonnes at the carbon price of $55. Therefore, for increasing carbon price, total emission of the carbon offset model decreases until a price of $55. For carbon prices beyond $55, total emissions of the carbon offset model remain constant. Therefore, confirming Proposition 4.8, total emissions of the carbon tax and carbon offset models are equal until the carbon price of $55. However, for 2.32.352.42.452.52.552.62.652.70 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Total cost in carbon cap-and-trade models (CDN $)MillionsCarbon price (CDN $/tonne CO2-eq)Carbon tax modelCarbon cap-and-trade model with free allowances and initial allowance of 2000 tonnesCarbon cap-and-trade model with free allowances and initial allowance of 3000 tonnesCarbon cap-and-trade model with initial allowance price of $35 and initial allowance of 2000 tonnesCarbon cap-and-trade model with initial allowance price of $35 and initial allowance of 3000 tonnesPrice of initial allowance  85  carbon prices beyond $55, total emissions of the carbon tax model is less than that of the carbon offset model with a compliance target of 3000 tonnes.  Similarly, the total emission of the carbon offset model with a compliance target of 2000 tonnes is approximately equal to 2000 tonnes at a carbon price of $75. Therefore, total emissions of the carbon tax model and carbon offset model with a compliance target of 2000 tonnes are equal for carbon prices less than $75. For carbon prices beyond $75, total emission of the carbon tax model is less than that of the carbon offset model with a compliance target of 2000 tonnes. These observations are consistent with Proposition 4.8. Comparing optimal cost of models with carbon tax and carbon offset models Figure 4-7 shows the optimal costs of the carbon tax and the two carbon offset models.   Figure 4-7: Optimal costs in the carbon tax and carbon offset models Consistent with Proposition 4.9, Figure 4-7 shows that the cost of carbon tax model is always greater than the cost of carbon offset model.  2.352.42.452.52.552.62.652.70 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Total cost (feedstcck cost + carbon cost)  (CDN $)(Millions)Carbon price (CDN $/tonne CO2-eq)Carbon tax modelCarbon offset model with compliance target of 2000 tonnesCarbon offset model with compliance target of 3000 tonnes  86  Comparing optimal cost of models with carbon cap-and-trade and carbon offset models Optimum costs of the two carbon offset models and the four carbon cap-and-trade models are shown in Figure 4-8.   Figure 4-8: Optimum costs of carbon cap-and-trade and carbon offset models for different carbon prices According to Figure 4-2, for the carbon offset model with a compliance target of 3000 tonnes, total emission is approximately equal to 3000 tonnes at a carbon price of $55. Similarly, at a carbon price of $55, total emissions of the carbon cap-and-trade model with an initial allowance of 3000 tonnes is approximately equal to 3000 tonnes. As shown in Figure 4-8, optimal costs of the carbon offset models and the carbon cap-and-trade model with free allowances with an initial allowance (and compliance target) of 3000 tonnes are equal until a carbon price of $55. For carbon prices above $55, the optimum cost of the model with carbon offset policy is greater than that of the model with carbon cap-and-trade policy.  2.342.362.382.42.422.442.462.480 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Total cost (CDN $)MillionsCarbon price (CDN $/tonne CO2-eq)Carbon cap-and-trade model with free allowances and initial allowance of 2000 tonnesCarbon cap-and-trade model with free allowances and initial allowance of 3000 tonnesCarbon offset model with compliance target of 2000 tonnesCarbon offset model with compliance target of 3000 tonnes  87  Similarly, the carbon offset model with a compliance target of 2000 tonnes and the carbon cap-and-trade model with free allowances and an initial allowance  of 2000 tonnes have equal optimum cost until a carbon price of $75. For carbon prices beyond $75, the total cost of the carbon cap-and-trade model is less than that of the carbon offset model. These observations are consistent Proposition 4.10.  4.7.3 Impact of carbon pricing policies on optimal feedstock mix at the district heating plant  Figure 4-9 shows the optimal mix of fuel obtained from the carbon tax and cap-and-trade models. Since optimal decisions of carbon tax and carbon cap-and-trade models depend only on carbon price (see, Proposition 4.6), models with all variants of carbon cap-and-trade policy and carbon tax policy result in the same optimal fuel mix.   Figure 4-9: Optimal decisions of the models with carbon tax and the four carbon cap-and-trade policies (optimal decisions of the models with carbon tax and carbon cap-and-trade policies are equal for a given carbon price) According to Figure 4-9, consumption of natural gas decreases with increasing carbon price. Since majority of emissions is from consumption of natural gas, decrease in natural gas consumption results in a decrease in total emission. It is prescribed that the district heating plant utilizes only wood residues for lower carbon prices. The proportion of denser forms of biomass such as briquettes and pellets increases and that of wood residues decreases with increasing carbon price. Figures 4-10 and 4-11 show the optimal fuel mix for carbon offset models with compliance targets of 2000 tonnes and 3000 tonnes for different carbon prices, respectively.  175117511751175117511748173317241706168216591604159115641538151314911460142914203 18 28 46 70 92 1461601862082332552813083174 8 1111 1212 1414 1717 18 22 26260200040006000800010000120001400016000180000200400600800100012001400160018000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Heat generated from natural gas (MWh)# of truckloads of biomassCarbon price (CDN $/ tonne CO2-eq)Wood residues Briquettes Pellets Natural gas  88   Figure 4-10: Annual biomass and natural gas consumption prescribed by the model with carbon offset policy with a compliance target of 2000 tonnes CO2-eq for different carbon prices  Figure 4-11: Annual biomass and natural gas consumption prescribed by the model with carbon offset policy with a compliance target of 3000 tonnes CO2-eq for different carbon prices As shown in Figure 4-10, the optimal fuel mix of the carbon offset model is the same as that of the carbon tax model when carbon price is lesser than $75 per tonne CO2-eq. when the compliance target is 2000 175117511751175117511748173317241706168216591604159115641538153615361536153615363 18 28 46 70 92 1461601862082102102102102104 8 1111 1212 1414 1717171717170200040006000800010000120001400016000180000200400600800100012001400160018000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Energy generated from natural gas (MWh)# of truckloads of biomassCarbon price (CDN $/tonne CO2-eq)Wood residues Briquettes Pellets Natural Gas175117511751175117511748173317241706168216701670167016701670167016701670167016703 18 28 46 70 828282828282828282824 8 1111 121212121212121212120200040006000800010000120001400016000180000200400600800100012001400160018000 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Energy generated from natural gas (MWh)# of  truckloads of biomassCarbon price (CDN $/tonne CO2-eq)Wood residues Briquettes Pellets Natural Gas  89  tonnes. As carbon price increases beyond $75, the optimal fuel mix remains constant as it was at carbon price of $75 for the model with carbon offset policy. This is because total emissions in the carbon offset model are close to the compliance target of 2000 tonnes when the carbon price is $75. Similar observation can be made in Figure 4-11, where the optimal fuel mix remains constant for carbon prices beyond $55.  4.8 Discussion Results of the optimization models for the biomass-fed district heating plant suggest that all carbon pricing models result in reduced emission. Due to the limitation on the number of trucks received by the plant daily, the optimal feedstock mix includes higher proportion of denser biomass types such as briquettes and pellets and lower proportions of biomass such as wood residues that have lower density with increasing carbon price. Therefore, with fewer truckloads, more energy could be produced using denser forms of biomass. Consumption of natural gas and total emissions decreases as carbon price increased in all carbon pricing models. However, instead of a sudden 100% transition, gradual transition from natural gas to biomass could be more economical for the emitters (Figure 4-9). This is because the feedstock cost of denser forms of biomass is more than the cost of natural gas when the cost of emissions is not considered. With increasing carbon price, the cost of emissions would increase, and the total cost of natural gas would increase. As a result, a portion of natural gas would be replaced with biomass to offset the increase in the total cost. However, replacing the entire natural gas with biomass for lower carbon prices would result in higher overall cost due to the higher feedstock cost.   Figures 4-9, 4-10, and 4-11 suggest that carbon prices above $30 per tonne CO2-eq. result in the replacement of natural gas with biomass. However, the literature highlights that the carbon price should be as high as $200 per tonne CO2-eq. to achieve emission reduction targets set by different countries (Vass 2016). The carbon price at which natural gas is replaced with biomass to achieve emissions reduction in the considered case study is less than that reported in the literature because this study does not include the cost of setting up and expanding the biomass-fed district heating plant. If the cost of setting up the plant was included in this study, the carbon price required to invest in the biomass-fed district heating plant would be higher than $30 per tonne CO2-eq. emissions. As a result, emissions reduction would happen at a higher carbon price.  The optimization models with carbon tax and carbon cap-and-trade policies result in equal emissions for equal carbon prices (Proposition 4.6 and Figure 4-2). While carbon prices are set by regulatory authorities in a carbon tax system, they are determined by the carbon trading market in carbon cap-and-trade systems (Goulder and Schein 2013). Carbon prices in real carbon cap-and-trade systems are identified to be lower than carbon tax rates in carbon tax systems in practice (Haites 2018). As reported in Haites (2018), the average carbon price in carbon cap-and-trade systems is more than 50% lower than that in the carbon tax   90  systems. As a result, the emissions in carbon cap-and-trade model could be greater than emissions in the carbon tax model if carbon prices implemented in practice are considered in optimization models. Therefore, it is important from a policy maker’s standpoint to regulate carbon trading prices either by imposing floor and ceiling values for the carbon prices or by restricting the total available emission allowances. Such hybrid carbon cap-and-trade model could ensure that emission reduction targets are met (Goulder and Schein 2013). Previous studies highlighted that the cap-and-trade model has less cost compared to the other models due to the possibility to make additional revenue by selling any remaining carbon allowances in the carbon market (e.g., Marufuzzaman et al. (2014a) and Marufuzzaman et al. (2014b)). However, all previous studies considered free initial emission allowance. This study suggests that the carbon cap-and-trade model is cost-effective only if the carbon trading price is more than the price of initial allowance (Proposition 4.7 and Figure 4-6). It is to be noted that transaction costs associated with trading carbon allowances are not considered in the calculation of total cost of the carbon cap-and-trade model. Although the literature and the results of this study suggest that the carbon cap-and-trade model with free initial allowance has the least cost compared to the other models (Figure 4-6), implementing the carbon cap-and-trade policy is more complex as governments should take on the function of a bank (Wittneben 2009). On the other hand, while the carbon tax model is relatively less complex to implement, it could result in high total costs (Figures 4-6 and 4-7). The carbon offset policy could combat the negative effects of uncertainties associated with the carbon cap-and-trade policy and can be implemented with less complexity (Commonwealth of Australia 2018). Moreover, the carbon offset policy has less cost than the carbon tax policy (Figure 4-7). However, emission reduction could be limited in the carbon offset model as total emissions are bounded by the compliance target (Figure 4-2). Therefore, from a policy maker’s standpoint, setting appropriate carbon emission baselines to emitters could be important to meet their emission reduction targets in the carbon offset systems.  4.9 Conclusions  Recent studies analyzed the impact of carbon pricing policies on the optimum cost and emissions of biomass supply chain optimization models. However, these studies focused on specific case studies, and the analyses conducted were pertinent to the considered case studies. Therefore, the results reported in these studies may not be generalized to other cases (Zakeri et al. 2015). This chapter analyzed the impact of carbon pricing policies on optimal cost and emissions of biomass logistics models using case-independent optimization models. The three main carbon policies, namely, carbon tax, carbon cap-and-trade, and carbon offset policies, were considered. Several propositions describing the impact of carbon pricing on optimal cost and   91  emissions of optimization models with different carbon pricing policies were made. These propositions were proved based on mathematical properties of the models. The propositions were verified numerically using the case of a biomass-fed district heating system at the University of British Columbia, Canada. An optimization model that determines the optimal mix of biomass feedstock and natural gas at the district heating plant was developed and extended to include carbon pricing policies. The impact of carbon pricing policies on optimal cost, emissions, and decision variables of the models was analyzed.  The results suggested that carbon pricing prescribes a gradual replacement of natural gas with biomass instead of a 100% transition to biomass. Furthermore, with increasing carbon price, the optimization models prescribed the usage of higher-density biomass such as pellets and briquettes compared to lower density fuels such as wood residues. Models with the carbon tax and the carbon-and-trade policies result in equal optimum decision variables and emissions for equal carbon prices. Optimum emissions in the carbon cap-and-trade model depend only on the carbon price, and not on the initial allowance and the price of initial allowance. The carbon cap-and-trade model has less cost than the carbon tax model only if the carbon price is more than the price of initial allowance. The carbon tax model has more cost than the carbon offset model for all carbon prices. However, emissions in the carbon offset model are bounded by the compliance target making the emissions and the cost of the carbon offset model sensitive to the compliance target.   The optimization models developed in this chapter combined cost and emissions into one objective function. As a result, they fail to capture the trade-off between total cost and emissions of the models. To understand the trade-off between total cost and emissions, the single-objective optimization model developed in the current chapter are extended to bi-objective models under different carbon pricing policies in the next chapter.     92  Chapter 5: Multi-objective biomass supply chain optimization models considering carbon pricing policies  5.1 Synopsis Cost and emissions are two separate objectives of biomass logistics optimization models that are conflicting in nature. However, most of the studies on biomass logistics optimization either considered only the cost objective function without considering emissions, or combined the cost and emission objectives into a single objective function in their models. Studies that combined the cost and emission objectives into a single objective added the cost of emissions to the total cost objective function following carbon pricing policies. The goal of these studies was to analyze the impacts of carbon pricing policies on the cost and emissions of biomass supply chain optimization models. However, with this type of modelling, the information about the trade-off between the cost and emissions of optimization models would be lost. The information about the trade-off between different objectives could be important for decision makers to incorporate their preferences into decision making. This necessitates the development of bi-objective optimization models considering carbon pricing policies. In this chapter, the single-objective optimization model developed for the biomass-fed district heating plant in Chapter 4 are extended to bi-objective models. The two objectives considered in the models include the minimization of  the total cost and minimization of  total emissions. First, a bi-objective model, which does not consider carbon pricing, is solved. In this model, the cost objective function includes only the feedstock cost. Next, bi-objective models with the three carbon pricing policies are solved. In these models that consider carbon pricing policies, the cost objective function includes the cost of feedstock as well as the cost of emissions defined by the carbon pricing policy. The results indicate that the number of candidate solutions available to decision makers reduces with increasing carbon prices. Moreover, the set of optimum solutions to the models with carbon pricing policies is a subset of the set of solutions to the model without carbon pricing. Solving bi-objective optimization models to obtain the set of trade-off solutions can be computationally time-consuming because it involves solving several single-objective models separately. To reduce the computational effort, a new algorithm is proposed in this chapter to obtain the set of optimum solutions for bi-objective models with carbon pricing policies from the set of solutions for the bi-objective model without carbon pricing. The algorithm is based on mathematical properties of optimum solutions of bi-objective models with and without carbon pricing policies. These properties are provided and proved mathematically. The developed algorithm is applied to the case of the biomass-fed district heating system.   93  5.2 Bi-objective optimization models Let 𝑓 be the feedstock cost shown in Eq. (4.13) and let 𝑒 be the total emissions shown in Eq. (4.24). The feasible region of the single-objective model developed in Chapter 4 is define by constraint sets shown in Equations (4.14)-(4.23). The bi-objective model which does not include carbon pricing is shown below.  Objective 1: Minimize Total feedstock cost, 𝑓 Objective 2: Minimize Total emissions, 𝑒  Subject to: Eq. (4.14)-(4.23) The bi-objective model with the carbon tax policy is shown below. In this formulation, 𝑝 denotes the carbon tax rate.  Objective 1: Minimize Total cost under the carbon tax policy, 𝑧𝑇 𝑍𝑇 = 𝑓 + 𝑝 ∗ 𝑒 Objective 2: Minimize Total emissions, 𝑒  Subject to: Eq. (4.14) - (4.23) The bi-objective model with the carbon cap-and-trade policy is shown below. In this formulation, 𝑝 is the carbon price, 𝑒𝐶&𝑇+  is the positive deviation from the initial allowance, and 𝑒𝐶&𝑇−  is the negative deviation from the initial allowance.  Objective 1: Minimize Total cost under the carbon cap-and-trade policy, 𝑧𝐶&𝑇 𝑍𝐶&𝑇 = 𝑓 + 𝑝 ∗ 𝑒𝐶&𝑇+ − 𝑝 ∗ 𝑒𝐶&𝑇−  Objective 2:  Minimize  Total emissions, 𝑒  Subject to: Eq. (4.14) - (4.23)   𝑒 − 𝑒𝐶&𝑇+ + 𝑒𝐶&𝑇− = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒   𝑒𝐶&𝑇+ , 𝑒𝐶&𝑇− ≥ 0 The bi-objective model with the carbon offset policy is shown below. In the model, 𝑝 is the carbon offset price,  𝑒𝐶𝑂+  is the positive deviation from the compliance target, and 𝑒𝐶𝑂−  is the negative deviation from the compliance target.    94  Objective 1: Minimize Total cost under the carbon offset policy, 𝑧𝐶𝑂 𝑍𝐶𝑂 = 𝑓 + 𝑝 ∗ 𝑒𝐶𝑂+  Objective 2:  Minimize Total emissions, 𝑒  Subject to: Eq. (4.14) - (4.23)   𝑒 − 𝑒𝐶𝑂+ + 𝑒𝐶𝑂− = 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡   𝑒𝐶𝑂+ , 𝑒𝐶𝑂− ≥ 0 For illustrating the impact of carbon pricing policies on the optimum solutions of bi-objective models, carbon price of $50 per tonne CO2-eq. emissions, and an initial allowance /compliance target of 2000 tonnes of CO2-eq. emissions are considered.  5.3 Results  Results from solving the bi-objective optimization models with and without carbon pricing policies for the case of the biomass-fed district heating plant are shown in this section.  Being one of the most common approaches, the 𝜖-constraint method is used to solve the bi-objective optimization models in this study. In the 𝜖-constraint method, the bi-objective optimization model is reformulated as a single-objective model by keeping one of the objectives as the objective function and the other objective as a constraint of the model limited (Haimes et al. 1971). The objective that is kept as a constraint is restricted by user-defined 𝜖 values (Haimes et al. 1971). In this study, the total cost is arbitrarily considered as the objective function and total emission is constrained by 𝜖 values. For the considered case study, total emission from the emission minimization model is approximately 1,200 tonnes of CO2-eq and that from the feedstock cost minimization model is approximately 4,000 tonnes of CO2-eq (see, Chapter 4). Therefore, 𝜖 values used for emissions constraint in the 𝜖-constraint method are varied between 1200 tonnes and 4000 tonnes, with increments of 200 tonnes. A Pareto-optimum solution is found for each 𝜖 value, therefore, a total of 15 Pareto-optimum solutions are obtained. Moving from one solution to other in this set of solutions, either emissions increase and cost decreases or emissions decrease and cost increases. Therefore, these solutions capture the trade-off between cost and emissions of the model. Table 5-1 shows the set of 15 Pareto-optimum solutions for the bi-objective models without carbon pricing policies. For each Pareto-optimum solution, the total feedstock cost, total emission, the number of truckloads of each biomass type required per year, and the amount of energy generated from natural gas are shown in Table 5-1.    95  Table 5-1: Pareto-optimum solutions obtained for the bi-objective model without carbon pricing Solution alternative 𝝐 value   Feedstock cost  (CDN $)   Emissions (tonnes CO2-eq) Number of truckloads/years Energy generated from natural gas (MWh) Wood residues Pellets Briquettes 1 1,200 2,535,477.8 1,199.7 1,421 320 26 164.6 2 1,400 2,516,369.9 1,399.5 1,451 293 23 1,339.7 3 1,600 2,498,471.9 1,599.5 1,478 269 19 2,516.2 4 1,800 2,481,237.3 1,799.7 1,509 241 17 3,694.1 5 2,000 2,465,595.4 1,998.3 1,539 211 17 4,862.3 6 2,200 2,451,035.4 2,199.2 1,566 186 15 6,044.0 7 2,400 2,437,585.5 2,399.3 1,593 160 14 7,221.4 8 2,600 2,425,510.0 2,597.0 1,619 135 13 8,384.3 9 2,800 2,414,298.0 2,793.9 1,646 108 13 9,542.2 10 3,000 2,403,261.8 2,995.4 1,672 83 12 10,727.5 11 3,200 2,393,437.9 3,195.2 1,698 57 12 11,903.1 12 3,400 2,384,314.8 3,396.1 1,720 35 11 13,084.8 13 3,600 2,376,630.8 3,594.4 1,730 24 6 14,251.0 14 3,800 2,369,361.6 3,800.0 1,741 13 1 15,460.3 15 4,000 2,365,250.0 3,951.9 1,754 0 0 16,354.0 Solutions 1 to 15 shown in Table 5-1 are the Pareto-optimum solutions obtained for values of 𝜖 ranging between 1200 and 4000. For increasing values of 𝜖, total emissions of the Pareto-optimum solutions increases and total feedstock cost decreases. Since the total emissions of the models are constrained by 𝜖 values, as shown in Table 1, smaller 𝜖 values result in the use of more energy dense biomass feedstock such as pellets and briquettes compared to larger 𝜖 values. Furthermore, with smaller 𝜖 values, the models prescribe the use of less natural gas compared to larger 𝜖 values. Solution alternative 15 obtained with the highest 𝜖 value corresponds to the least-cost alternative for the case study and the solution alternative 1 obtained with the lowest 𝜖 value corresponds to the least-emissions alternative.  To demonstrate the impact of carbon pricing policies on the trade-off between total cost and emissions, bi-objective models considering different carbon pricing policies are solved. The bi-objective models are solved using the 𝜖-constraint method with 𝜖 values ranging between 1200 and 4000. Figure 5-1 shows the set of Pareto-optimum solutions and the trade-off curves obtained for the bi-objective models with and without carbon pricing policies.    96    Figure 5-1: Pareto-optimum solutions for the model without carbon pricing and the model with the carbon tax model (tax = $50 per tonne CO2-eq), the carbon cap-and-trade policy (initial allowance = 2000 tonnes and price=$50 per tonne CO2-eq), and carbon offset policy (compliance target = 2000 tonnes and price=$50 per tonne CO2-eq) Figure 5-1 suggests that the number of Pareto-optimum solutions obtained for the models with carbon pricing policies is less than that for the model without carbon pricing. Models with carbon pricing have 11 optimum solutions while the model without carbon pricing has 15 optimum solutions for the considered 𝜖 values. This is because carbon pricing results in lower optimum emissions compared to the models without carbon pricing (see, Chapter 4). The least-cost alternative for the model without carbon pricing has more emissions than the least-cost alternative for the model with carbon pricing. Total emissions of the feedstock cost-minimization model is around 4000 tonnes of CO2-eq. (see, Table 5-1) while that of the cost 1234567891011121314 1512345 6 7 8 9 10 1612346 7 8 9 10 162.352.402.452.502.552.602.651000 1500 2000 2500 3000 3500 4000Total cost (CDN $)(Millions)Emissions (tonnes of CO2-eq)Model without carbon pricingCarbon tax model with carbon price = $50 per tonne of CO2-eqCarbon cap-and-trade model with an initial allowance of 2000 tonnes and price=$50 per tonne CO2-eqCarbon offset model with a compliance target of 2000 tonnes and price=$50 per tonne CO2-eq  97  minimization model with the carbon tax policy is approximately 3,100 tonnes of CO2-eq. Therefore, the total emissions of the Pareto-optimum solutions for the model without carbon pricing range between 1200 and 4000 tonnes of CO2-eq. while the total emissions of the Pareto-optimum solutions for the model with the carbon tax policy range between 1200 and 3100 tonnes of CO2-eq. Thus, the overall spread of the Pareto-optimum frontier for the model without carbon pricing is more than that for the models with carbon pricing. This implies that decision makers would have fewer alternatives to choose from when carbon pricing is implemented.  As shown in Figure 5-1, the bi-objective models with and without carbon pricing policies result in the same Pareto-optimum solutions for 𝜖 values up to 3000. Therefore, solution alternatives 1 to 10 are common for all models. However, solution alternatives 11 to 15 are obtained only for the model without carbon pricing. Since the maximum emissions of the models with carbon pricing policies is around 3100 tonnes of CO2-eq, 𝜖 values between 3200 and 4000 result in a single solution represented as alternative 16 in Figure 5-1.  Figure 5-1 shows that bi-objective optimization models with and without carbon pricing policies result in the same Pareto-optimum solutions for 𝜖 values between 1200 and 3000. For a given 𝜖 value between 1200 and 3000, the bi-objective models with and without carbon pricing result in the same optimum decision variables. Therefore, these alternatives have equal emissions. However, based on the considered carbon pricing policy, the total cost of these alternatives would be different for different policies. For example, when 𝜖 is 1200, all bi-objective models result in alternative solution 1. However, the total cost of alternative 1 is the highest under the carbon tax policy, and it is the least under the carbon cap-and-trade policy.  Since all emissions are priced in the carbon tax policy, alternatives of the carbon tax model have higher cost than those for the model without carbon pricing. The total cost of the alternatives of the carbon cap-and-trade model is less than that for the model without carbon pricing when 𝜖 is less than 2000 (the initial allowance). This is because additional revenue can be made from selling the remaining emission allowances to the carbon market when total emission is less than the initial allowance. On the other hand, when 𝜖 is greater than the initial allowance, total cost of the solutions of the carbon cap-and-trade model is more than the cost of the solutions for the model without carbon pricing.  Under the carbon offset policy, the total cost of alternatives with and without carbon pricing are equal when 𝜖 is less than 2000 tonnes (the compliance target). However, when 𝜖 is greater than the compliance target, the alternatives for the carbon offset model have more cost than the cost of the alternatives for the model without carbon pricing. Since the total cost of the solutions depends on the adopted carbon pricing policy, the slopes of the Pareto-optimal curves for different carbon pricing policies are different. Therefore, as   98  Figure 5-1 demonstrates, carbon pricing policies impact the trade-off between cost and emissions of the bi-objective models.  5.4 Mathematical properties of bi-objective optimization models with carbon pricing policies The results presented in Section 5.3 indicate that for 𝜖 values between 1200 and 3000, bi-objective optimization models with and without carbon pricing policies result in the same Pareto-optimum solutions for the considered case study. However, the number of Pareto-optimum solutions for the models with carbon pricing is less than that for the model without carbon pricing. This suggests that there are some linkages between the optimum solutions of bi-objective models with and without carbon pricing policies. In this section, different mathematical properties are proposed for describing the relationship between the optimum solutions of the bi-objective optimization models with and without carbon pricing policies. These properties are proved using case-independent bi-objective optimization models. The notations used in the case-independent optimization models are shown in Table 5-2.  Table 5-2: Notations used in the case-independent bi-objective optimization models Notation Definition 𝑥, 𝑥𝑖, 𝑥𝑗, 𝑥𝑘 Feasible solutions to the considered supply chain optimization models  𝑋 Set of all feasible solutions to the supply chain optimization model 𝑓(𝑥) Cost of a solution 𝑥 ∈ 𝑋 without including the cost of emissions 𝑒(𝑥) Total emission of a solution 𝑥 ∈ 𝑋 𝑝  Carbon price ($/tonne CO2-eq emissions) under all carbon pricing policy 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒 Initial emission allowance (tonnes of CO2-eq.) allocated in the carbon cap-and-trade policy 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 Emissions compliance target (tonnes of CO2-eq.) assigned in the carbon offset policy 𝑒𝐶&𝑇+  Positive deviation of total emissions from the initial allowance. This is the quantity of additional emission allowances that must be purchased from the carbon market under the carbon cap-and-trade policy.  𝑒𝐶&𝑇−  Negative deviation of total emissions from the initial allowance. This is the quantity of emission allowances that can be sold in the carbon market under the carbon cap-and-trade policy. 𝑒𝐶𝑂+  Positive deviation of the total emissions from the compliance target. This is the quantity of carbon offsets purchased in the carbon offset policy.  𝑒𝐶𝑂−  Negative deviation of the total emissions from the compliance target 𝑍𝑇 Objective function value of the carbon tax model 𝑍𝑇 = 𝑓(𝑥) + 𝑝 ∗ 𝑒(𝑥)   99  𝑍𝐶&𝑇 Objective function value of the carbon cap-and-trade model. 𝑍𝐶&𝑇 = 𝑓(𝑥) + 𝑝 ∗ (𝑒𝐶&𝑇+ − 𝑒𝐶&𝑇− ) 𝑍𝐶𝑂 Objective function value of the carbon offset model.  𝑍𝐶𝑂 = 𝑓(𝑥) + 𝑝 ∗ 𝑒𝐶𝑂+  Case-independent bi-objective optimization models with and without carbon pricing policies are shown below. Each of these models has two objective functions: 1) total cost, and 2) total emissions. The second objective function that deals with minimizing total emissions, 𝑒(𝑥), is common for all the models. However, the first objective function that minimizes the total cost is different depending on whether carbon emissions are priced.    The formulation for the case when carbon emissions are not priced, which is represented by 𝑀𝑂𝑂𝑁𝐶, is shown below.  𝑀𝑂𝑂𝑁𝐶 : Minimize Total feedstock cost, 𝑓(𝑥)  Minimize Total emissions, 𝑒(𝑥)  Subject to: 𝑥 ∈ 𝑋 The formulation for the model with the carbon tax policy, which is represented by 𝑀𝑂𝑂𝐶𝑇, is shown below. The model 𝑀𝑂𝑂𝐶𝑇 differs from the model 𝑀𝑂𝑂𝑁𝐶 only in the total cost objective function.   𝑀𝑂𝑂𝐶𝑇: Minimize Total cost under the carbon tax policy, 𝑧𝑇 𝑧𝑇 = 𝑓(𝑥) + 𝑝 ∗ 𝑒(𝑥)  Minimize Total emissions, 𝑒(𝑥)  Subject to: 𝑥 ∈ 𝑋 The formulation 𝑀𝑂𝑂𝐶&𝑇 represents the model with the carbon cap-and-trade policy. This model has additional constraints to determine the total emissions that is above or below the initial allowance. The objective function has terms to penalize emissions above the initial allowance and reward emissions below the emission allowance. 𝑀𝑂𝑂𝐶&𝑇: Minimize Total cost under the carbon cap-and-trade policy, 𝑧𝐶&𝑇 𝑧𝐶&𝑇 = 𝑓(𝑥) + 𝑝𝑒𝐶&𝑇+ − 𝑝𝑒𝐶&𝑇−   Minimize Total emissions, 𝑒(𝑥)   100   Subject to: 𝑒(𝑥) − 𝑒𝐶&𝑇+ + 𝑒𝐶&𝑇− = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙𝐴𝑙𝑙𝑜𝑤𝑎𝑛𝑐𝑒   𝑥 ∈ 𝑋   𝑒𝐶&𝑇+ , 𝑒𝐶&𝑇− ≥ 0 The formulation 𝑀𝑂𝑂𝐶𝑂 represents the model with the carbon offset policy. Similar to the 𝑀𝑂𝑂𝐶&𝑇 model, the carbon offset model determines the deviation of total emissions from the compliance target. The cost objective function of the model includes the penalty for total emissions exceeding the compliance target. Unlike the carbon cap-and-trade model, the carbon offset model does not reward emissions that are below the compliance target.  𝑀𝑂𝑂𝐶𝑂: Minimize Total cost under the carbon offset policy, 𝑍𝐶𝑂 𝑍𝐶𝑂 = 𝑓(𝑥) + 𝑝𝑒𝐶𝑂+   Minimize Total emissions, 𝑒(𝑥)  Subject to: 𝑒(𝑥) − 𝑒𝐶𝑂+ + 𝑒𝐶𝑂− = 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡   𝑥 ∈ 𝑋   𝑒𝐶𝑂+ , 𝑒𝐶𝑂− ≥ 0 Since total emissions above the compliance target is penalized in the carbon offset model, the carbon offset model is similar to the carbon tax model when the compliance target is zero. Therefore, the carbon offset model is a generalization of the carbon tax model.  The literature highlights that from an optimization perspective, supply chain optimization models under the carbon tax and the carbon cap-and-trade policies result in equal optimal decision variables for equal carbon prices (see, Proposition 4.6 of Chapter 4). Due to this property, the models with the carbon tax and the carbon cap-and-trade policies have the same set of optimal solutions.  Since the carbon offset model is a generalization of the carbon tax model, and the carbon tax and the carbon cap-and-trade policies have the same set of Pareto-optimal solutions, mathematical properties of Pareto-optimal solutions the carbon offset model are valid for the carbon tax and the carbon cap-and-trade models. Therefore, in the following parts of this section, several mathematical properties of the optimal solutions of the carbon offset model are proposed and proved. These properties would also be true for the carbon tax and the carbon cap-and-trade models.    101  As shown in Figure 5-1, solution alternatives 1 to 10 that are Pareto-optimum to the model with the carbon offset policy are also Pareto-optimum to the model without carbon pricing. This indicates that the set of Pareto-optimum solutions of the model with the carbon offset policy is a subset of the set of Pareto-optimum solutions of the model without carbon pricing. Proposition 5.1 states that this relationship between Pareto-optimum solutions to the models with and without carbon pricing is true independent of the considered case study.  Proposition 5.1  Every Pareto-optimum solution to the model with the carbon offset policy is also Pareto-optimum to the model without carbon pricing.  Proof of Proposition 5.1 Let 𝑥𝑖 be Pareto-optimum for the model with the carbon offset policy. This means that for any other solution 𝑥𝑗 that is feasible for the carbon offset model, if one of the objective functions improves at 𝑥𝑗, the other objective function deteriorates. Let us consider two cases as follows: Case 1: Let 𝑍𝐶𝑂(𝑥𝑗) < 𝑍𝐶𝑂(𝑥𝑖) and 𝑒(𝑥𝑗) > 𝑒(𝑥𝑖) Depending on the value of the compliance target, carbon emissions are priced if emissions are more than the compliance target. Consider the following three sub-cases.  Sub-case 1a: Let 𝑒(𝑥𝑗) > 𝑒(𝑥𝑖) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 In this case, since emissions at both 𝑥𝑖 and 𝑥𝑗 are more than the carbon cap, emissions are taxed in both these solutions. Therefore, 𝑍𝐶𝑂(𝑥𝑖) = 𝑓(𝑥𝑖) + 𝑝 ∗ (𝑒(𝑥𝑖) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡) and 𝑍𝐶𝑂(𝑥𝑗) = 𝑓(𝑥𝑗) +𝑝 ∗ (𝑒(𝑥𝑗) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡). Since 𝑍𝐶𝑂(𝑥𝑗) < 𝑍𝐶𝑂(𝑥𝑖), 𝑓(𝑥𝑗) + 𝑝 ∗ 𝑒(𝑥𝑗) < 𝑓(𝑥𝑖) + 𝑝 ∗ 𝑒(𝑥𝑖). Following the assumption that 𝑒(𝑥𝑗) > 𝑒(𝑥𝑖), we arrive at the conclusion that 𝑓(𝑥𝑗) < 𝑓(𝑥𝑖).  Sub-case 1b: Let 𝑒(𝑥𝑗) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 > 𝑒(𝑥𝑖) In this case, emissions are taxed only at the solution 𝑥𝑗. Therefore, 𝑍𝐶𝑂(𝑥𝑖) = 𝑓(𝑥𝑖) and 𝑍𝐶𝑂(𝑥𝑗) =𝑓(𝑥𝑗) + 𝑝 ∗ (𝑒(𝑥𝑗) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡). Since 𝑍𝐶𝑂(𝑥𝑗) < 𝑍𝐶𝑂(𝑥𝑖), we arrive at a relationship that 𝑓(𝑥𝑗) + 𝑝 ∗ (𝑒(𝑥𝑗) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡) < 𝑓(𝑥𝑖). Therefore, 𝑓(𝑥𝑗) − 𝑓(𝑥𝑖) < 𝑝 ∗(𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 − 𝑒(𝑥𝑗) < 0. Therefore, 𝑓(𝑥𝑗) < 𝑓(𝑥𝑖).  Sub-case 1c: 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 > 𝑒(𝑥𝑗) > 𝑒(𝑥𝑖) In this case 𝑍𝐶𝑂(𝑥𝑖) = 𝑓(𝑥𝑖) and 𝑍𝐶𝑂(𝑥𝑗) = 𝑓(𝑥𝑗). Therefore, 𝑓(𝑥𝑗) < 𝑓(𝑥𝑖). Case 2: Let 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖) and 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖)   102  Consider the following three sub-cases. Sub-case 2a: Let 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 < 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖) In this case, emissions are taxed at both 𝑥𝑖 and 𝑥𝑗. 𝑍𝐶𝑂(𝑥𝑖) = 𝑓(𝑥𝑖) + 𝑝 ∗ (𝑒(𝑥𝑖) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡) and 𝑍𝐶𝑂(𝑥𝑗) = 𝑓(𝑥𝑗) + 𝑝 ∗ (𝑒(𝑥𝑗) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡). Since 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖), 𝑓(𝑥𝑗) + 𝑝 ∗𝑒(𝑥𝑗) > 𝑓(𝑥𝑖) + 𝑝𝐶𝑂𝑒(𝑥𝑖). This implies that 𝑓(𝑥𝑗) − 𝑓(𝑥𝑖) > 𝑝 ∗ (𝑒(𝑥𝑖) − 𝑒(𝑥𝑗)) > 0. Therefore, 𝑓(𝑥𝑗) > 𝑓(𝑥𝑖). Sub-case 2b: Let 𝑒(𝑥𝑗) < 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 < 𝑒(𝑥𝑖) In this case, 𝑍𝐶𝑂(𝑥𝑖) = 𝑓(𝑥𝑖) + 𝑝 ∗ (𝑒(𝑥𝑖) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡) and 𝑍𝐶𝑂(𝑥𝑗) = 𝑓(𝑥𝑗). Since 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖), 𝑓(𝑥𝑗) >  𝑓(𝑥𝑖) + 𝑝 ∗ (𝑒(𝑥𝑖) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡). This results in the inequality 𝑓(𝑥𝑗) − 𝑓(𝑥𝑖) > 𝑝 ∗ (𝑒(𝑥𝑖) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡) > 0. Therefore, 𝑓(𝑥𝑗) > 𝑓(𝑥𝑖). Sub-case 2c: Let 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖) < 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 In this case 𝑍𝐶𝑂(𝑥𝑖) = 𝑓(𝑥𝑖) and 𝑍𝐶𝑂(𝑥𝑗) = 𝑓(𝑥𝑗). Therefore, 𝑓(𝑥𝑗) > 𝑓(𝑥𝑖). From cases 1 and 2, we can conclude that if 𝑥𝑖 is Pareto-optimum for the model with carbon offset policy, it is also Pareto-optimum for the model without carbon pricing.  Figure 5.1 suggests that solution alternatives 11 to 15 that are Pareto-optimum to the model without carbon pricing are not Pareto-optimum to the model with the carbon offset policy. This indicates that a subset of optimum solutions to the model without carbon pricing are not Pareto-optimum to the model with the carbon offset policy. The following propositions describe the conditions under which optimum solutions to the model without carbon pricing policies are optimum to the model with the carbon offset policy.  Proposition 5.2 Let 𝑥1 be Pareto-optimum for the model without carbon pricing such that 𝑒(𝑥1) is the minimum possible emission. Then, 𝑥1 is also Pareto-optimum for the model with a carbon pricing policy. Proof of Proposition 5.2 This proposition will be proved using the concept of proof by contradiction. Assume that 𝑥1 is not Pareto-optimum for the model with carbon offset policy. Consider the following two cases.  Case 1: 𝑒(𝑥1) < 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 If 𝑥1 is not Pareto-optimum for the model with carbon offset policy, there must exist some other solution 𝑥𝑘 such that 𝑒(𝑥𝑘) = 𝑒(𝑥1) and 𝑍𝐶𝑂(𝑥𝑘) < 𝑍𝐶𝑂(𝑥1). Since 𝑒(𝑥1) < 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡, 𝑍𝐶𝑂(𝑥𝑘) =  103  𝑓(𝑥𝑘) and 𝑍𝐶𝑂(𝑥1) = 𝑓(𝑥1). This implies that 𝑓(𝑥𝑘) < 𝑓(𝑥1) which contradicts the assumption that 𝑥1 is Pareto-optimum for the model without carbon pricing.  Therefore, if 𝑥1 is Pareto-optimum for the model without carbon pricing, 𝑒(𝑥1) is the minimum emission, and 𝑒(𝑥1) < 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡, then 𝑥1 is Pareto-optimum for the model with carbon offset policy. Case 2: 𝑒(𝑥1) ≥ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 If 𝑥1 is not Pareto-optimum for the model with carbon offset policy, there must exist some other solution 𝑥𝑘 such that 𝑒(𝑥𝑘) = 𝑒(𝑥1) and 𝑍𝐶𝑂(𝑥𝑘) < 𝑍𝐶𝑂(𝑥1). Since 𝑒(𝑥1) ≥ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡, 𝑍𝐶𝑂(𝑥1) =𝑓(𝑥1) + 𝑝 ∗ [𝑒(𝑥1) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡] and 𝑍𝐶𝑂(𝑥𝑘) = 𝑓(𝑥𝑘) + 𝑝 ∗ [𝑒(𝑥𝑘) − 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡]. This implies that 𝑓(𝑥𝑘) < 𝑓(𝑥1) which contradicts the assumption that 𝑥1 is Pareto-optimum for the model without carbon pricing.  Therefore, if 𝑥1 is Pareto-optimum for the model without carbon pricing, 𝑒(𝑥1) is the minimum emission, and 𝑒(𝑥1) ≥ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡, then 𝑥1 is Pareto-optimum for the model with carbon offset policy. From cases 1 and 2, it is shown that if a solution 𝑥1 is Pareto-optimum to the model without carbon pricing and 𝑒(𝑥1) is the minimum emission, then  𝑥1 is Pareto-optimum to the model with the carbon offset policy.  Proposition 5.3: Let 𝑥𝑖 Pareto-optimum solution to the model without carbon pricing policy. Then 𝑥𝑖 is Pareto-optimum to the model with the carbon offset policy if 𝑒(𝑥𝑖) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡. Proof of Proposition 5.3:  This proposition will be proved using the concept of proof by contradiction. Assume that 𝑥𝑖 is not Pareto-optimum for the model with carbon offset policy.  If 𝑥𝑖 is not Pareto-optimum to the model with the carbon offset policy, there must exist another solution 𝑥𝑘 such that 𝑒(𝑥𝑘) ≤ 𝑒(𝑥𝑖), 𝑍𝐶𝑂(𝑥𝑘) ≤ 𝑍𝐶𝑂(𝑥𝑖), and 𝑥𝑘 is Pareto-optimum to the model with the carbon offset model.  Since 𝑒(𝑥𝑖) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡,  and 𝑒(𝑥𝑘) ≤ 𝑒(𝑥𝑖), 𝑒(𝑥𝑘) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡. Thus, 𝑍𝐶𝑂(𝑥𝑘) =𝑓(𝑥𝑘) and 𝑍𝐶𝑂(𝑥𝑖) = 𝑓(𝑥𝑖). Since 𝑍𝐶𝑂(𝑥𝑘) ≤ 𝑍𝐶𝑂(𝑥𝑖), then 𝑓(𝑥𝑘) ≤ 𝑓(𝑥𝑖). This is a contradiction to the assumption that 𝑥𝑖 is Pareto-optimum to the model without carbon pricing.  Therefore, if 𝑒(𝑥𝑖) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 and 𝑥𝑖 is Pareto-optimum to the model without carbon pricing, then 𝑥𝑖 is Pareto-optimum to the model with the carbon offset policy.    104  According to the definition of Pareto-optimality, if two solutions 𝑥𝑖 and 𝑥𝑗 are Pareto-optimum to the model with the carbon offset policy model, then these solutions exhibit trade-off between their costs and emissions. This implies that if 𝑥𝑖 and 𝑥𝑗 are Pareto-optimum to the model with the carbon offset policy, then either 𝑍𝐶𝑂(𝑥𝑖) < 𝑍𝐶𝑂(𝑥𝑗) and 𝑒(𝑥𝑖) > 𝑒(𝑥𝑗) or 𝑍𝐶𝑂(𝑥𝑖) > 𝑍𝐶𝑂(𝑥𝑗) and 𝑒(𝑥𝑖) < 𝑒(𝑥𝑗). However, the converse of this definition may not be true. That is, if any two feasible solutions 𝑥𝑖 and 𝑥𝑗 exhibit trade-off between total cost and emissions, then they may or may not be Pareto-optimum to the model with the carbon offset policy.  Consider two solutions 𝑥𝑖 and 𝑥𝑗. Assume that it is known that 𝑥𝑗 is Pareto-optimum to the model with the carbon offset policy. According to the definition of Pareto-optimality discussed above, if solutions 𝑥𝑖 and 𝑥𝑗 exhibit trade-off between their total cost and emissions, it does not imply that 𝑥𝑖 is Pareto-optimum to the model with the carbon offset policy. However, Proposition 5.4 proves that if solution 𝑥𝑖 exhibits trade-off with all Pareto-optimum solutions of the carbon offset policy that are found so far, then 𝑥𝑖 will also be Pareto-optimum to the model with the carbon offset policy. On the contrary, if there is at least one Pareto-optimum solution 𝑥𝑖  to the model with the carbon offset policy with which the solution 𝑥𝑗 does not exhibit trade-off between cost and emissions, then Proposition 5.4 states that 𝑥𝑖 is not Pareto-optimum to the model with the carbon offset policy.   Proposition 5.4 Let 𝑥𝑖 be the Pareto-optimum solution to the model without carbon pricing policy. Then 𝑥𝑖 is Pareto-optimum for the model with the carbon offset policy if and only if 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖) for every Pareto-optimum solution 𝑥𝑗 to the model with carbon offset policy that satisfies 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖). Proof of Proposition 5.4:  This proposition has two parts to it. In the first part, it says that if 𝑥𝑖 is Pareto-optimum to the model with the carbon offset policy and 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖), then 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖). In the second part, it states that if 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖) and 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖), then 𝑥𝑖 is Pareto-optimum to the model with the carbon offset policy. Proofs for both of these parts are shown below.  Part 1: If both 𝑥𝑖 and 𝑥𝑗 are Pareto-optimum to the model with carbon offset policy and 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖), then according to the definition of Pareto-optimality, 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖). This proves the first part of the Proposition.  Part 2:    105  This part is proved using the concept of proof using contradiction. Assume that  𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖) for every Pareto-optimum solution 𝑥𝑗 to the model with carbon offset policy which satisfies 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖).  Now, assume that 𝑥𝑖 is not Pareto-optimum for the model with carbon offset policy. It means that there exists another solution 𝑥𝑘 that is Pareto-optimum to the model with carbon offset policy such that 𝑒(𝑥𝑘) ≤𝑒(𝑥𝑖) and 𝑍𝐶𝑂(𝑥𝑘) ≤ 𝑍𝐶𝑂(𝑥𝑖) (i.e., solution 𝑥𝑘 improves both objectives compared to the solution 𝑥𝑖). Consider the following cases: Case 1: 𝑒(𝑥𝑘) = 𝑒(𝑥𝑖) and 𝑍𝐶𝑂(𝑥𝑘) ≤ 𝑍𝐶𝑂(𝑥𝑖) Since 𝑒(𝑥𝑖) = 𝑒(𝑥𝑘) and 𝑍𝐶𝑂(𝑥𝑘) ≤ 𝑍𝐶𝑂(𝑥𝑖), 𝑓(𝑥𝑘) ≤ 𝑓(𝑥𝑖). However, this is a contradiction to the assumption that 𝑥𝑖 is Pareto-optimum for the model without carbon pricing.  Case 2: 𝑒(𝑥𝑘) < 𝑒(𝑥𝑖) and 𝑍𝐶𝑂(𝑥𝑘) ≤ 𝑍𝐶𝑂(𝑥𝑖) It is assumed that 𝑥𝑘 is Pareto-optimum to the model with carbon offset policy. According to the assumption of this part of the Proposition, if  𝑒(𝑥𝑘) < 𝑒(𝑥𝑖) and 𝑥𝑘 is Pareto-optimum to the model with the carbon offset policy, then 𝑍𝐶𝑂(𝑥𝑘) > 𝑍𝐶𝑂(𝑥𝑖). This is a contradiction to the conditions of this case.  From cases 1 and 2, it can be seen that 𝑥𝑖 will be Pareto-optimum to the model with carbon offset policy if 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖) for every solution Pareto-optimum solution 𝑥𝑗 to the model with carbon pricing which satisfies 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖).  5.5 Algorithm for solving bi-objective optimization models with carbon pricing policies Based on the propositions described in Section 5.4, an algorithm is developed in this section to determine the set of Pareto-optimum solutions to the model with the carbon offset policy. The input to the algorithm is the set of Pareto-optimum solutions to the model without carbon pricing. For each Pareto-optimum solution to the model without carbon pricing, the algorithm is used to determine whether it is optimum to the model with the carbon offset policy. The algorithm can be applied for carbon tax and carbon cap-and-trade policies as well. The pseudocode of the algorithm is shown in Algorithm 5-1, and the algorithm is described below.     106  Algorithm 5-1: Pseudocode of the algorithm to determine Pareto-optimum solutions of the models with the carbon offset policy from Pareto-optimum solutions of the model without carbon pricing Input:  Set of Pareto-optimum solutions {𝑥1, 𝑥2, … , 𝑥𝑛} for the model without carbon pricing such that 𝑒(𝑥1) < 𝑒(𝑥2) < ⋯ < 𝑒(𝑥𝑛) Output: Set of Pareto-optimum solutions to the model with the carbon offset policy 1. Solution 𝑥1 is Pareto-optimum to the carbon offset model  // Result of Proposition 5.2         2. for each 𝑖 ∈ {2,3 … , 𝑛} 3.       if 𝑒(𝑥𝑖) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 4.                𝑥𝑖 is Pareto-optimum to the carbon offset model  // Result of Proposition 5.3        5.       endif 6. endfor 7. for each 𝑖 ∈ {2,3, … , 𝑛} and 𝑗 ∈ {1,2, … , 𝑖 − 1} // Since 𝑗 < 𝑖, 𝑒(𝑥𝑗) < 𝑒(𝑥𝑖)       8.       if 𝑒(𝑥𝑖) > 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 and 𝑥𝑗 is Pareto-optimum to the carbon offset model  9.                 if 𝑍𝐶𝑂(𝑥𝑗) > 𝑍𝐶𝑂(𝑥𝑖)  10.                       𝑥𝑖 is Pareto-optimum to the carbon offset model;  // Result of Proposition 5.4        11.                       continue in the loop;               12.               else  13.                         𝑥𝑖 is not Pareto-optimum to the carbon offset model; // Result of Proposition 5.4        14.                      exit the loop;     15.               endif  16.        endif  17. endfor  Let {𝑥1, 𝑥2, … , 𝑥𝑛} be the set of Pareto-optimum solutions to the model without carbon pricing. Let these solutions be sorted in ascending order of their emissions. Therefore, 𝑒(𝑥1) ≤ 𝑒(𝑥2) ≤ ⋯ ≤ 𝑒(𝑥𝑛). Alternative 𝑥1 is the least emissions alternative and alternative 𝑥𝑛 is the least cost alternative. According to Proposition 5.2, since 𝑥1 is the least emissions alternative, 𝑥1 is Pareto-optimum to the model with the carbon offset policy. This is shown in line 1 of the algorithm.   107  If, for any Pareto-optimum solution 𝑥𝑖 to the model without carbon offset policy and 𝑒(𝑥𝑖) ≤𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡, then according to Proposition 5.3, solution 𝑥𝑖 is Pareto-optimum to the model with the carbon offset policy. Therefore, for of all solutions 𝑥𝑖 ∈ {𝑥2, 𝑥3, … , 𝑥𝑛}, the algorithm is used to check if 𝑒(𝑥𝑖) ≤ 𝐶𝑜𝑚𝑝𝑙𝑖𝑎𝑛𝑐𝑒𝑇𝑎𝑟𝑔𝑒𝑡 in line 4 of the algorithm.  Until this stage, a set of Pareto-optimum solutions to the model with the carbon offset policy is found. This set includes 𝑥1, which has the least emissions, and any other solutions whose emissions are less than the compliance target. For each of the remaining solutions 𝑥𝑖 in the set of Pareto-optimum solutions to the model without carbon pricing, the algorithm is used to determine if 𝑥𝑖 is Pareto-optimum to the model with the carbon offset policy. For this part, the algorithm is based on the conditions described in Proposition 4. The algorithm is used to evaluate if the solution 𝑥𝑖 exhibits trade-off in total cost and emissions with all solutions that are identified to be Pareto-optimum to the model with the carbon offset policy. If 𝑥𝑖 exhibits trade-off in cost and emissions with all Pareto-optimum solutions to the model with the carbon offset policy, then it is Pareto-optimum to the model with the carbon offset policy. If 𝑥𝑖 fails to exhibit trade-off with even one Pareto-optimum solution to the model with the carbon offset policy, then 𝑥𝑖 is not Pareto-optimum to the model with the carbon offset policy.  Due to the relationship between the models with the carbon offset, carbon tax, and carbon cap-and-trade policies, the algorithm can be used to determine Pareto-optimum solutions for the models with all three policies. 5.6 Illustration of the algorithm This section illustrates how to use the algorithm described in Section 5.5 to determine the Pareto-optimum solutions for the models with carbon pricing policies. The algorithm can be employed to solve bi-objective optimization models under all carbon pricing policies. The algorithm is demonstrated using the case study of the biomass-fed district heating plant described in Section 4.5.  The algorithm uses the set of Pareto-optimum solutions for the model without carbon pricing as its input. This set comprises of 15 solutions as shown in Table 5-1. For each solution alternative, the algorithm is employed to determine whether it is optimum to the models with different carbon pricing policies. The algorithm involves calculating the cost of each of the 15 alternatives under different carbon pricing policies. For the purpose of illustration, consider a carbon price to be $50 per tonne of CO2-eq. emissions in the carbon tax model, a carbon price to be $50 per tonne of CO2-eq. and an initial allowance of 2000 tonnes in the carbon cap-and-trade model, and a carbon price to be $50 per tonne of CO2-eq. and a compliance target of 2000 tonnes in the carbon offset model. The cost of each of the 15 solutions under   108  each carbon pricing policy is shown in Table 5-3. The solutions shown in Table 5-3 are sorted according to their total emissions in an ascending order. For two solutions 𝑖 and 𝑗 such that 𝑖 < 𝑗, total emissions of solution 𝑖 is less than total emissions of solution 𝑗.  Table 5-3: Costs of all Pareto-optimum solutions to the model without carbon pricing under different carbon pricing policies Solution alternative Cost of the model without carbon pricing (CDN $) Emissions (tonnes CO2-eq.) Cost under the carbon tax policy (CDN $) Cost under the carbon cap-and-trade policy (CDN $) Cost under the carbon offset policy (CDN $) 1 2,535,477.8 1,199.7 2,595,462.2 2,495,462.2 2,535,477.8 2 2,516,369.9 1,399.5 2,586,343.3 2,486,343.3 2,516,369.9 3 2,498,471.9 1,599.5 2,578,445.0 2,478,445.0 2,498,471.9 4 2,481,237.3 1,799.7 2,571,222.5 2,471,222.5 2,481,237.3 5 2,465,595.4 1,998.3 2,565,510.7 2,465,510.7 2,465,595.4 6 2,451,035.4 2,199.2 2,560,995.2 2,460,995.2 2,460,995.2 7 2,437,585.5 2,399.3 2,557,552.5 2,457,552.5 2,457,552.5 8 2,425,510.0 2,597.0 2,555,362.0 2,455,362.0 2,455,362.0 9 2,414,298.0 2,793.9 2,553,992.5 2,453,992.5 2,453,992.5 10 2,403,261.8 2,995.4 2,553,030.8 2,453,030.8 2,453,030.8 11 2,393,437.9 3,195.2 2,553,200.2 2,453,200.2 2,453,200.2 12 2,384,314.8 3,396.1 2,554,121.6 2,454,121.6 2,454,121.6 13 2,376,630.8 3,594.4 2,556,349.8 2,456,349.8 2,456,349.8 14 2,369,361.6 3,800.0 2,559,359.6 2,459,359.6 2,459,359.6 15 2,365,250.0 3,951.9 2,562,844.5 2,462,844.5 2,462,844.5 Consider the bi-objective model with the carbon tax policy. Finding the set of Pareto-optimum solutions to this model using the algorithm is described below.  According to Proposition 5.2, solution 1 is Pareto-optimum to the model with the carbon tax policy. The carbon offset model reduces to the carbon tax model when the compliance target is equal to zero. Therefore, all solution alternatives have their emissions greater than the compliance target. Therefore, line 4 of the algorithm is redundant for the carbon tax model. Thus, the algorithm is used to consider solution alternatives 2 to 15 to determine whether they are Pareto-optimum to the model with the carbon tax policy.  According to the algorithm, the trade-off between solutions 2 and 1 are evaluated in the first step. It is for this reason that the rows in Table 5-4 start with alternative 2. According to the emissions and total cost of solutions under the carbon tax policy shown in Table 5-3, solutions 1 and 2 have trade-off between their total cost and emissions. Therefore, according to the algorithm, solution 2 is Pareto-optimum to the model   109  with the carbon tax policy. In the next iteration, the algorithm is used to check if trade-off between total cost and emissions exist for solution pairs 3 and 1, and 3 and 2.   In general, for each of the solution 𝑖 ∈ {2, 3, … , 15}, the trade-off between the solution 𝑖 and the set of Pareto-optimum solutions to the carbon offset model in the set {1, 2, … , 𝑖 − 1} is evaluated. The results of this evaluation are shown in Table 5-4. In Table 5-4, a “Y” mark in the cell (𝑖, 𝑗) indicates that there is a trade-off between the cost and emissions of the corresponding solutions 𝑖 and 𝑗, and an “N” mark indicates that there is no trade-off between the cost and emissions of the corresponding solutions 𝑖 and 𝑗.  Table 5-4: Matrix evaluating the pairwise trade-off between total cost and emissions of the Pareto-optimum solutions of the model without carbon pricing under the carbon tax policy (tax = $50 per tonne CO2-eq.) and the carbon cap-and-trade policy (carbon price = $50 per tonne CO2-eq. and initial allowance = 2000 tonnes of CO2-eq.)  Alternative 𝑗 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Alternative i 2  Y               3 Y Y              4 Y Y Y             5 Y Y Y Y            6 Y Y Y Y Y           7 Y Y Y Y Y Y          8 Y Y Y Y Y Y Y         9 Y Y Y Y Y Y Y Y        10 Y Y Y Y Y Y Y Y Y       11 Y Y Y Y Y Y Y Y Y N      12 Y Y Y Y Y Y Y Y N N N     13 Y Y Y Y Y Y Y N N N N N    14 Y Y Y Y Y Y N N N N N N N   15 Y Y Y Y Y N N N N N N N N N   Y indicates that solutions 𝑥𝑖 and 𝑥𝑗 exhibit trade-off between total cost and emissions N indicates that solutions 𝑥𝑖 and 𝑥𝑗 do not exhibit trade-off between total cost and emissions Table 5-4 shows that all solution pairs (𝑖, 𝑗) for 𝑖 ∈ {1,2, … , 10} and 𝑗 ∈ {1,2, … , 𝑖 − 1} exhibit trade-off between total cost and emissions. Therefore, solutions 1 to 10 are Pareto-optimum to the model with the carbon tax policy. However, for 𝑖 ∈ {11, 12, 13, 14, 15}, there is at least one 𝑗 ∈ {1, 2, … , 𝑖 − 1} for which the solutions pair (𝑖, 𝑗) does not exhibit trade-off between cost and emissions. For instance, consider solution 11. It exhibits trade-off with solutions 𝑗 ∈ {1, 2, … , 9}. However, total cost and emissions of solution 11 are both greater than those of solution 10. Therefore, solutions 10 and 11 do not exhibit trade-off between their total cost and emissions. As a result, according to Proposition 5.3, solution 11 is not   110  Pareto-optimum to the model with the carbon tax policy. Similarly, solutions 12, 13, 14 and 15 are not Pareto-optimum to the model with the carbon tax policy.  Running the Algorithm 1 for the carbon cap-and-trade with carbon price of $50 per tonne CO2-eq. and an initial allowance of 2000 tonnes of CO2-eq. results in the same trade-off matrix as shown in Table 5-4. Therefore, the algorithm identifies that solutions 11, 12, 13, 14, and 15 are not Pareto-optimum to the model with the carbon cap-and-trade policy. This confirms the observation that the models with the carbon tax and the carbon cap-and-trade policies have the same set of Pareto-optimum solutions. However, due to the differences in the cost of the models with carbon tax and the carbon cap-and-trade policies, the shape of the trade-off curve is different for the two policies. This can be observed in Figure 5-1 and Table 5-3 as well.  Consider the carbon offset policy with carbon price of $50 per tonne of CO2-eq. emissions and a compliance target of 2000 tonnes of CO2-eq. emissions. In the considered case study, solution alternatives 1, 2, 3, 4, and 5 have emissions less than the compliance target. Therefore, these are optimum to the model with the carbon offset policy according to Proposition 5.3.  The algorithm is used to evaluate whether solution alternatives 6, 7, … , 15 are Pareto-optimum to the model with the carbon offset policy. The trade-off evaluation matrix for solution alternatives 6, 7, … , 15  is shown in Table 5-5.  Table 5-5: Matrix evaluating the pairwise trade-off between total cost and emissions of the Pareto-optimum solutions of the model without carbon pricing under the carbon offset policy (carbon price = $50 per tonne CO2-eq. and compliance target = 2000 tonnes of CO2-eq.)  Alternative 𝑗 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Alternative i 6 Y Y Y Y Y -          7 Y Y Y Y Y Y -         8 Y Y Y Y Y Y Y -        9 Y Y Y Y Y Y Y Y -       10 Y Y Y Y Y Y Y Y Y -      11 Y Y Y Y Y Y Y Y Y N -     12 Y Y Y Y Y Y Y Y N N N -    13 Y Y Y Y Y Y Y N N N N N -   14 Y Y Y Y Y Y N N N N N N N -  15 Y Y Y Y Y N N N N N N N N N -  Y indicates that solutions 𝑥𝑖 and 𝑥𝑗 exhibit trade-off between total cost and emissions N indicates that solutions 𝑥𝑖 and 𝑥𝑗 do not exhibit trade-off between total cost and emissions According to Table 5-5, solution alternatives 6, 7, 8, 9, and 10 exhibit trade-off in total cost and emissions with all previously found Pareto-optimum solutions to the model with the carbon offset policy. Therefore,   111  they are Pareto-optimum to the model with the carbon offset policy. However, solution alternatives 11, 12, 13, 14, and 15 do not exhibit trade-off between total cost and emissions with at least one Pareto-optimum solution to the model with the carbon offset policy. Therefore, solution alternatives 11, 12, 13, 14, and 15 are not Pareto-optimum to the model with the carbon offset policy. The results of the algorithm correspond to the observations made in Figure 5-1. 5.7 Discussion The results shown in Section 5.3 suggest that for the selected 𝜖 values, the model without carbon pricing policies has 15 optimum solutions, while the models with carbon pricing policies have 11 optimum solutions when the carbon price is $50 per tonne CO2-eq. emissions (see, Figure 5-1). This suggests that the number of Pareto-optimum solutions obtained for the models with carbon pricing policies is less than that for the model without carbon pricing. Solution alternatives 1 to 10 are common for all models, whereas solution alternatives 11 to 15 are obtained only for the model without carbon pricing. This is because the overall range of emissions for the models decrease when emissions are priced. For example, the maximum optimal emissions possible for the model without carbon pricing is around 4000 tonnes, while that for the models with carbon pricing policies is around 3000 tonnes. As a result, decision makers would have fewer optimum alternatives to choose from when carbon emissions are priced. Furthermore, the number of Pareto-optimum solutions decreases with increasing carbon price.  Trade-off between the total cost and emissions for carbon pricing models The Pareto-optimum solutions shown Figure 5-1 illustrate the when the total cost of solution alternatives decreases, then the total emissions increase. This suggests a trade-off between the total cost and emissions of optimum solutions for models with and without carbon pricing policies.  Since optimum solutions 1-10 and 16 are common for all carbon pricing models, the emissions of these solutions are equal under all carbon pricing policies. Therefore, emissions of Pareto-optimum solutions are equal under all carbon pricing policies. However, depending on the policy, the cost of the Pareto-optimum solutions may vary. As a result, different carbon pricing policies have different impacts on the trade-off between cost and emissions of the bi-objective models.  Comparing solution alternatives 16 and 1, the total decrease in emissions is 1902 tonnes for all carbon pricing policies. For mitigating the total emissions by 1902 tonnes, the total increase in the cost of the carbon tax model is $42,490.50 (see, Figure 5-1). For mitigating the same amount of emissions, the increase in the cost of the carbon cap-and-trade model is $42,604.90, and the increase in the cost of the carbon offset   112  model is $82,624. This suggests that the increase in total cost for mitigating the emissions by a given quantity is the highest for the carbon offset model.  Pairwise comparison of carbon pricing models  According to Figure 5-1, all Pareto-optimum solutions 1-10 and 16 have more cost under the carbon tax policy compared to the carbon cap-and-trade and the carbon offset policies. Since all emissions are priced in the carbon tax policy, and emissions are priced in the carbon cap-and-trade and the carbon offset policies only when the total emissions exceed the emission allowance and the compliance target, respectively, the carbon tax model results in the highest cost compared to the other two policies. For the models with the carbon cap-and-trade and the carbon offset policies, total emissions of solution alternatives 1, 2, 3, and 4  (see, Figure 5-1) are less than the initial allowance (and compliance target in the case of carbon offset policy) of 2000 tonnes. Since the total emissions are less than the initial allowance, additional revenue can be gained in the carbon cap-and-trade model by selling the remaining emission allowances to the carbon market. The additional revenue reduces the total cost of the carbon cap-and-trade model. However, under the carbon offset policy, emissions less than the compliance target do not yield additional revenue. Therefore, the total cost of the carbon cap-and-trade model is less than that of the carbon offset model when the total emissions are less than the initial allowance/compliance target.  The total emissions of solution alternatives 6-10 and 16 in Figure 5-1 exceed the initial allowance/compliance target of 2000 tonnes. Since emissions that exceed the initial allowance/compliance target are priced in both carbon cap-and-trade and carbon offset policies, respectively, these solutions have equal cost under both policies. Impact of using the algorithm for solving bi-objective models with carbon pricing The Pareto-optimum solutions shown in Figure 5-1 for the models with and without carbon pricing were obtained using 15 𝜖 values. Therefore, 15 single-objective optimization models were solved for each bi-objective model with and without carbon pricing policies. Overall, 60 single-objective optimizations were solved to generate the Pareto-optimum solutions shown in Figure 5-1. While the single-objective optimization models for the considered case study were solved within one minute, generating Pareto-optimum frontiers for other bi-objective models may have required significant computational effort and time if the single-objective models were large and complex.  Employing the algorithm reduces the computational effort involved in determining the solutions of the bi-objective models with carbon pricing policies. For the considered case study, the bi-objective model without carbon pricing is solved by solving 15 single-objective models. The resultant set of 15 Pareto-optimum   113  solutions are used to determine the solutions of the models with carbon pricing policies. Therefore, bi-objective models with carbon pricing policies are solved by solving only 15 single-objective models as opposed to solving 60 single-objective models when the algorithm was not used. In addition, sensitivity analysis for varying carbon prices and initial allowances can be conducted without re-solving the bi-objective models.    5.8 Conclusions The literature highlights that different carbon pricing policies have different impacts on the optimal cost and emissions of biomass supply chain optimization models. Previous studies that investigated the impact of these policies on optimum cost and emissions of biomass supply chains optimization models combined the cost and emissions into a single objective function. Due to this type of modeling, the information about the trade-off between cost and emissions is not captured (Wu et al. 2010). Understanding the trade-off between these objectives is important for decision makers to be able to make decisions based on their preferences. Therefore, it is important to develop multi-objective optimization models considering carbon pricing policies and to obtain the set of trade-off solutions.   In this chapter, bi-objective optimization models considering carbon pricing policies were developed for the case of a biomass-fed district heating plant described in Chapter 4. The bi-objective optimization models with and without carbon pricing policies were solved using the 𝜖-constraint method. Results indicated that there are fewer candidate solutions available to decision makers under carbon pricing policies than no carbon policy optimization models. In addition, the trade-off between the total cost and the emissions depends on the considered carbon pricing policy. To reduce the total emissions by a given quantity, the carbon offset model results in the highest increase in total cost compared to other policies.  Solving bi-objective optimization models involves solving several single-objective optimization models separately. This could involve significant computational effort especially if the single-objective models are large and complex to solve. Moreover, to incorporate different carbon pricing policies, bi-objective optimization models have to be solved for each carbon pricing policy separately. To reduce this computational effort, an algorithm was developed in this study to determine the optimum solutions for the models with carbon pricing policies. The algorithm was developed based on the mathematical properties of optimum solutions of case-independent bi-objective models with and without carbon pricing policies. The mathematical properties were proposed and proved. The algorithm, which takes the set of optimum solutions of the bi-objective model without carbon pricing as its input, was used to determine the set of optimum solutions for the models with different carbon pricing policies. Therefore, instead of solving several bi-objective models for different carbon pricing policies, one bi-objective model without carbon   114  pricing can be solved and the algorithm can be run. The applicability of the algorithm was demonstrated using the case study of the biomass-fed district heating plant. Since the algorithm is independent of the considered case study, it can be applied to other cases and industries as well.    115  Chapter 6: Conclusions  6.1 Summary and conclusions Biomass is a low-carbon source of energy that has emerged as an attractive alternative for replacing fossil fuels. However, high logistics cost is one of the barriers that impede the widespread utilization of biomass. These logistics operations also contribute to total emissions from using biomass for energy generation. Biomass logistics is comprised of harvest/collection, storage, pre-processing and transportation activities. These decisions are taken at both tactical and operational levels. Carbon pricing policies, which aim at mitigating carbon emissions, impact optimal cost and emissions of biomass logistics. The overall goal of this dissertation was to optimize forest-based biomass logistics at the operational level and to analyze the impact of carbon pricing policies on optimal cost and emissions of biomass logistics optimization models, independent of the underlying case study. To achieve this goal, several optimization models were developed for biomass logistics at the operational level and the models were applied to relevant case studies.  Chapter 2 presented a review of important features of biomass logistics operations and how different studies incorporated these features in their optimization models. The studies were categorized into those that dealt with tactical or operational level planning. Furthermore, studies that analyzed the impact of carbon pricing policies on biomass supply chain optimization models were reviewed.  The review highlighted that most of the literature on biomass logistics focused on economic optimization at the tactical level. Number of studies dealing with the operational level planning was limited and no previous study considered biomass storage, pre-processing, flow, and truck routing decisions simultaneously at the operational level. Studies that analyzed the impact of carbon pricing policies on optimal cost and emissions of biomass supply chain models considered specific case studies. Since these studies focused on specific case studies, the results reported in these studies might not be applicable to other cases. Moreover, these studies combined cost and emissions into a single objective function by adding the cost of emissions to the cost of the supply chain. With this type of modelling, the information about the trade-off between cost and emissions objectives would be lost (Wu et al. 2010). Decision makers prefer having a set of trade-off solutions over a single cost-minimization alternative for making choices based on their preferences (Konak et al. 2006; Deb and Deb 2014). In fact, it would be difficult to obtain decision makers’ goals and preferences without the information about the trade-off between different objectives. Therefore, multi-objective optimization models should be developed for biomass logistics considering carbon pricing policies.    116  Chapter 3 focused on developing optimization models for forest-based biomass logistics at the operational level considering biomass flow, storage, pre-processing, and truck routing decisions. Transportation of different types of biomass was carried out using different types of trucks. The problem was solved by decomposing it into transshipment and routing problems. The transshipment model was solved to determine weekly storage, pre-processing and flow decisions. The results of the transshipment model were used in the routing model to determine the routes for each truck on a daily basis. The models were applied to a large biomass logistics company based in Lower Mainland region, British Columbia, Canada. The results indicated a potential to improve up to 12% of total logistics cost, and up to 11.7% of total fuel consumption for operating trucks and loaders. The models suggested direct delivery of biomass from suppliers to customers where possible, and larger trucks were used more for the direct delivery of biomass. The results of the routing model indicated a reduction in relative cost of empty truck travel compared to the relative cost of loaded truck travel, where as it was the opposite in the company’s original routes. A decision support tool based on the transshipment model was provided to the company.  The impact of carbon pricing policies on optimal cost and emissions of case-independent optimization models was analyzed in Chapter 4. The three main carbon policies, namely, carbon tax, carbon cap-and-trade, and carbon offset policies, were considered in this chapter. Several properties describing the impact of carbon pricing policies on optimal cost and emissions of case-independent optimization models were proposed. These properties were proved based on mathematical properties of the models. An optimization model that determines the optimal mix of biomass feedstock and natural gas to meet the heat demand of a biomass-fed district heating plant at the University of British Columbia, Canada was developed. The model was extended to include carbon pricing policies. The impact of different carbon pricing policies on optimal cost, emissions, and decision variables of the models was analyzed. The results obtained from the models developed for the case study were consistent with the propositions made in this chapter. The analysis conducted in Chapter 4 suggested that optimization models with carbon tax and carbon cap-and-trade policies result in equal emissions for equal carbon prices. The results suggested that the carbon cap-and-trade model has less cost than the carbon tax model only if the carbon trading price is more than the price of initial allowances. The models with the carbon offset policy have less cost than the models with the carbon tax policy. However, the emissions reduction in the carbon offset models are restricted by the value of the compliance target while no such restriction exists in the carbon tax models. Therefore, the carbon tax model could have less emissions than the carbon offset model. Results of the optimization models developed for the case study suggested that all carbon pricing models resulted in emissions mitigation. As the carbon price increased, the optimal feedstock mix at the district heating plant included   117  higher proportion of denser biomass types such as briquettes and pellets and lower proportions of biomass such as wood residues that have lower density. Consumption of natural gas and total emissions decreased as carbon price increased in all carbon pricing models. However, instead of a sudden 100% transition, gradual transition from natural gas to biomass could be more economical for the emitters.  In Chapter 5, bi-objective optimization models considering carbon pricing policies were developed for the case of the biomass-fed district heating plant that is described in Chapter 4. The bi-objective optimization models with and without carbon pricing policies were solved using the 𝜖-constraint method. Results indicated that there are fewer candidate solutions available to decision makers under carbon pricing policies compared to the models without carbon pricing. In addition, the trade-off between the total cost and the emissions depends on the considered carbon pricing policy. For mitigating emissions by a given quantity, the carbon offset model results in highest increase in total cost compared to other policies. A new algorithm was proposed to determine the optimum solutions of the bi-objective optimization models with carbon pricing policies using the optimum solutions of the bi-objective model without carbon pricing. Therefore, instead of solving several bi-objective optimization models separately for different carbon pricing policies, one bi-objective model without carbon pricing can be solved and the algorithm can be used to solve the remaining models. The applicability of the algorithm was demonstrated using the case of the biomass-fed district heating plant.   6.2 Strengths From the modelling perspective, the main strength of the models developed in Chapter 3 is that biomass flow, storage, pre-processing, and truck routing decisions were considered simultaneously in the models. Transportation of different types of biomass using different types of trucks was considered. This was the first attempt in the literature to incorporate complexities related to multiple types of biomass, multiple types of trucks, compatibilities between trucks and biomass types, and compatibilities between trucks and biomass supply and demand locations in biomass logistics optimization modeling.  Another strength of the work presented in Chapter 3 is that the models were applied to a real case of a large biomass logistics company in BC. Real data were collected from the company and the results of the optimization models were compared with the actual solutions implemented by the company. The results of the models were validated by the manager of the company. A decision support tool was developed and given for the company’s use. The decision support tool was based on the optimization models presented in Chapter 3.  The analyses conducted in Chapter 4 were based on case-independent optimization models. Therefore, the propositions made in Chapter 4 hold true for other optimization models that consider carbon pricing policies   118  irrespective of the underlying case study. Previous studies that attempted to analyze the impact of carbon pricing policies conducted their analyses using specific case studies. Therefore, whether the results obtained in previous studies were generalizable was not clear. In this regard, the analysis conducted in Chapter 4 is applicable to other case studies and other industries as well. The work presented in Chapter 4 was the first attempt in the literature to study the impacts of the price of the initial allowance in optimization models with the carbon cap-and-trade policy. Previous studies considered that the initial allowance was allocated free of cost, therefore, the price of initial allowance was zero (e.g., (Palak et al. 2014; Marufuzzaman et al. 2014a). It was reported that the models with the carbon cap-and-trade policy have less cost than the models with the carbon tax policy. However, the analysis in Chapter 4 of this dissertation suggested that the models with carbon cap-and-trade policy have less cost than the models with the carbon tax policy only if the carbon price is more than the price of initial allowance.  The work presented in Chapter 5 was the first attempt to study bi-objective supply chain models considering carbon pricing policies in the general supply chain optimization literature. Several propositions that were made on the properties of Pareto-optimum solutions of the bi-objective models with carbon pricing policies were based on case-independent optimization models. These propositions were proved mathematically using the concepts from multi-objective optimization theory. Therefore, these results are applicable to other case studies and industries as well. In addition, the propositions were numerically verified using the case study of a biomass-fed district heating plant.  Another strength of the work presented in Chapter 5 was the new algorithm that was developed to obtain the optimum solutions of bi-objective optimization models with carbon pricing policies. Solving bi-objective optimization models to generate the set of Pareto-optimum solutions requires solving several single-objective optimization models separately (Miettinen 1998). This may require significant computational effort, especially if the single-objective optimization models are large and complex to solve. The algorithm developed in this study uses the set of optimum solutions for the model without carbon pricing and determines the set of optimum solutions for the models with different carbon pricing policies. Therefore, one bi-objective optimization model without carbon pricing can be solved and the algorithm can be used to solve the remaining bi-objective optimization models for different carbon pricing policies. The algorithm was developed based on the mathematical properties of optimum solutions of case-independent bi-objective optimization models. Therefore, the algorithm can be applied to other case studies and other industries as well.     119  6.3 Limitations One of the limitations of the models developed in Chapter 3 for optimizing forest-based biomass logistics at the operational level is related to the assumption that biomass supply and demand quantities were known with certainty. While the logistics company, whose data were used to evaluate the optimization models, had a general idea about the supply and demand quantities of biomass from their prior experience, the actual supply and demand quantities are prone to variations over the week. Consideration of variations in biomass supply and demand quantities would require additional information about these parameters and would require modifications to the models.  Another limitation of the models developed in Chapter 3 is that they do not consider dynamic events like truck breakdowns and traffic jams. During such events, the routes developed for drivers must be updated in real time. This requires the development of efficient solution strategies to cope with dynamics in the transportation network. Incorporating dynamics into the models would require installing GPS systems on the trucks, which must be integrated with the decision support tool.  The biomass logistics problem described in Chapter 3 was solved using a decomposition approach. While this was the simplest and the most logical way to solve the problem, the decomposition approach may not result in the global optimal solution. While the developed models can be applied to other cases which involve the transportation of products in full truckloads, the models may require modifications to incorporate constraints specific to the cases.  The models developed in Chapter 4 assumed constant carbon prices while they are prone to fluctuations in practice. Therefore, fluctuations in carbon prices should be considered to compare the effectiveness of carbon tax and carbon cap-and-trade policies from cost and emissions reduction perspectives. This would require incorporating uncertainties into the developed models.  Another aspect of carbon pricing policies that has not been considered in this study are transaction costs. Transaction costs pose a major barrier to the economic effectiveness of carbon cap-and-trade systems (Yuan et al. 2018). Transaction costs could have both fixed and variable cost components (Stavins 1995). Incorporating transaction costs into the models would require identifying trends in transaction costs from the literature and adding them to the objective function of the carbon cap-and-trade models.  Although the optimization models developed in Chapters 4 and 5 considered variations in biomass quality characteristics, they were assumed to be deterministic. However, biomass quality characteristics are prone to uncertainties. Therefore, the optimum biomass feedstock mix for the district heating plant prescribed by   120  the models in Chapters 4 and 5 may be different from the optimum solutions when uncertainties in biomass quality characteristics are incorporated.  6.4 Future work Future developments of the models for optimizing biomass logistics at the operational level could incorporate dynamics in the system such as traffic jams and truck breakdowns. Incorporating such dynamics would make the biomass logistics optimization problem a variant of the dynamic vehicle routing problem (Pillac et al. 2013). This would require access to data about real-time locations of trucks, and under any dynamic disruptions, the optimization models would re-organize the truck routes to carry out the transportation operations. In addition to system dynamics, incorporating uncertainties in supply and demand quantities of biomass could enhance the applicability of the models developed in Chapter 3.  Another area for future work is to incorporate fluctuations in carbon price in the models. The risk associated with fluctuations in carbon price and its impact on the solution of optimization models could be important to analyze. Banking and borrowing carbon emission allowances is identified as a strategy to mitigate price volatility in carbon cap-and-trade systems (Goulder and Schein 2013). Future studies could consider this aspect of incorporating banking and borrowing of carbon emission allowances between different carbon trading periods in the models. In addition, future models could include transactions cost in carbon cap-and-trade policy to compare the carbon cap-and-trade policy with the other policies.  The optimization models developed in Chapters 4 and 5 for the biomass-fed district heating plant can be extended to incorporate uncertainties in biomass quality characteristics. The impact of uncertainties in biomass quality characteristics on the replacement of fossil fuel with biomass under different carbon pricing policies could be assessed in the future models. Modelling techniques such as stochastic programming and robust optimization could be used for this purpose. However, this would require data about the probability distributions and ranges for biomass quality parameters. Furthermore, the bi-objective optimization models developed in Chapter 5 can be extended to include uncertainties to analyze the impacts of uncertainties on the set of Pareto-optimum solutions.      121  References Aguayo, Maichel M., Subhash C. Sarin, John S. 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