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UBC Theses and Dissertations

Value chain optimization of a forest biomass power plant considering uncertainties Shabani, Nazanin 2014

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VALUE CHAIN OPTIMIZATION OF A FOREST BIOMASS POWER PLANT CONSIDERING UNCERTAINTIES by Nazanin Shabani B.Sc., Civil Engineering, Sharif University of Technology, 2003   M.Sc., Civil Engineering, Iran University of Science and Technology, 2005  M.A.Sc., Civil Engineering, University of British Columbia, 2009   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) April 2014  © Nazanin Shabani, 2014ii  Abstract  Mathematical modeling has been employed to improve the cost competitiveness of forest bioenergy supply chains. Most of the studies done in this area are at the strategic level, focus on one part of the supply chain and ignore uncertainties. The objective of this thesis is to optimize the value generated in a forest biomass power plant at the tactical level considering uncertainties. To achieve this, first the supply chain configuration of a power plant is presented and a nonlinear model is developed and solved to maximize its overall value. The model considers procurement, storage, production and ash management in an integrated framework and is applied to a real case study in Canada. The optimum solution forecasts $1.74M lower procurement cost compared to the actual cost of the power plant. Sensitivity analysis and Monte Carlo simulation are performed to identify important uncertain parameters and evaluate their impacts on the solution.  The model is reformulated into a linear programming model to facilitate incorporating uncertainty in the decision making process. To include uncertainty in the biomass availability, biomass quality and both of them simultaneously, a two-stage stochastic programming model, a robust optimization model and a hybrid stochastic programming-robust optimization model are developed, respectively. The results show that including uncertainty in the optimization model provides a solution which is feasible for all realization of uncertain parameters within the defined scenario sets or uncertainty ranges, with a lower profit compared to the deterministic model. Including uncertainty in biomass availability using the stochastic model decreases the profit by $0.2M. In the robust optimization model, there is a trade-off between the profit and the selected range of biomass quality. Profit decreases by up to $3.67M when there are ±13% variation in moisture content and ±5% change in higher heating value. The hybrid model takes advantage of iii  both modeling approaches and balances the profit and model tractability. With the cost of only $30,000, an implementable solution is provided by the hybrid model with unique first stage decision variables. These models could help managers of a biomass power plant to achieve higher profit by better managing their supply chains.    iv  Preface  This dissertation is original and presents the work of Nazanin Shabani during her Ph.D. program. The research was conducted by the author under the supervision of her academic adviser, Dr. Taraneh Sowlati. Dr. Sowlati advised Shabani during the process of defining the research topic, gathering data and information, developing and validating the models and preparing manuscripts. This thesis presents a background on the research topic, research objectives, a review of the literature, several optimization models to achieve the research objectives with their application to a real case study in Canada, analysis of the obtained results, and the main findings and conclusions. The author visited the power plant several times, had close collaboration with the managers of the power plant, obtained information and detailed data on the power plant supply chain, presented the model results to the power plant managers and had the model validated by them. Five scientific papers were generated from this research, and in all of them Shabani was the first author. The list of papers generated from this research is provided below.  A version of Chapter 2 is published. Shabani, N., Akhtari, S., Sowlati, T. 2013. Value chain optimization of forest biomass for bioenergy production: A review. Renewable and Sustainable Energy Reviews, 110(3): 280-290.   A version of Chapter 3 is published. Shabani N, Sowlati T. 2013. A mixed integer non-linear programming model for tactical value chain optimization of a wood biomass power plant. Applied Energy, 104:353-361.  A version of Chapter 4 is submitted for publication. Shabani N, Sowlati T.Evaluating the impact of uncertainty and variability on the value chain optimization of a forest biomass power plant using Monte Carlo Simulation. v   A version of Chapter 5 is submitted for publication. Shabani N, Sowlati T., Ouhimmou M., Rönnqvist M. Tactical supply chain planning for a forest biomass power plant under supply uncertainty.  A version of Chapter 6 is submitted for publication. Shabani N, Sowlati T. A hybrid stochastic programming-robust optimization model for maximizing the value chain of a forest biomass power plant under uncertainty.   vi  Table of Contents Abstract ........................................................................................................................................... ii Preface............................................................................................................................................ iv Table of Contents ........................................................................................................................... vi List of Tables ................................................................................................................................. ix List of Figures ................................................................................................................................ xi Acknowledgements ...................................................................................................................... xiii Dedication ..................................................................................................................................... xv Chapter 1 Introduction .................................................................................................................. 1 1.1 Background ...................................................................................................................... 1 1.2 Research objectives .......................................................................................................... 6 1.3 Organization of the dissertation ....................................................................................... 7 Chapter 2 Literature review ........................................................................................................... 9 2.1 Synopsis ........................................................................................................................... 9 2.2 Deterministic optimization models .................................................................................. 9 2.2.1 Power plants ............................................................................................................ 10 2.2.2 District heating plants ............................................................................................. 13 2.2.3 Co-generation plants ............................................................................................... 16 2.2.4 Biofuel plants .......................................................................................................... 20 2.3 Optimization models with uncertainties ......................................................................... 23 2.3.1 Modeling approaches .............................................................................................. 27 2.3.2 Sensitivity analysis and Monte Carlo simulation.................................................... 31 2.3.3 Stochastic programming ......................................................................................... 34 2.3.4 Robust optimization model ..................................................................................... 38 vii  2.4 Discussion and conclusions ............................................................................................ 41 Chapter 3 Deterministic model ................................................................................................... 43 3.1 Synopsis ......................................................................................................................... 43 3.2 The power plant supply chain ........................................................................................ 43 3.3 The optimization model ................................................................................................. 49 3.4 Case study ...................................................................................................................... 55 3.5 Results ............................................................................................................................ 60 3.5.1 Scenario analysis ..................................................................................................... 61 3.5.2 Sensitivity analysis.................................................................................................. 63 3.6 Discussion and conclusions ............................................................................................ 65 Chapter 4 Monte Carlo simulation .............................................................................................. 68 4.1 Synopsis ......................................................................................................................... 68 4.2 Uncertainty and Monte Carlo simulation ....................................................................... 68 4.2.1 Uncertainty in biomass quality ............................................................................... 70 4.2.2 Uncertainty in biomass availability and cost and electricity price ......................... 76 4.3 Results ............................................................................................................................ 77 4.4 Discussion and conclusions ............................................................................................ 82 Chapter 5 Stochastic programming ............................................................................................. 85 5.1 Synopsis ......................................................................................................................... 85 5.2 The mixed integer programming model of the power plant supply chain ..................... 85 5.3 The stochastic mixed integer programming model of the power plant supply chain .... 88 5.4 Managing the risk ........................................................................................................... 92 5.4.1 Variability index ..................................................................................................... 93 5.4.2 Downside risk ......................................................................................................... 94 5.5 Results ............................................................................................................................ 95 viii  5.5.1 Result of deterministic models................................................................................ 95 5.5.2 Results of the stochastic model ............................................................................... 95 5.5.3 Results for the variability index ............................................................................ 101 5.5.4 Results for the downside risk ................................................................................ 103 5.6 Discussion and conclusions .......................................................................................... 103 Chapter 6 Hybrid stochastic programming-robust optimization model .................................... 106 6.1 Synopsis ....................................................................................................................... 106 6.2 Robust optimization formulation ................................................................................. 106 6.3 Hybrid stochastic programming-robust optimization model ....................................... 111 6.4 Results .......................................................................................................................... 113 6.5 Discussion and conclusions .......................................................................................... 118 Chapter 7 Conclusions, strength points, limitations and future research .................................. 121 7.1 Conclusions .................................................................................................................. 121 7.2 Strengths points ............................................................................................................ 122 7.3 Limitations ................................................................................................................... 124 7.4 Future research ............................................................................................................. 125 References ................................................................................................................................... 127        ix    List of Tables  Table 2-1: Summary of studies on deterministic optimization of forest biomass power plants ... 13 Table 2-2: Summary of studies on deterministic optimization of district heating plants ............. 15 Table 2-3: Summary of studies on deterministic optimization of co-generation plants ............... 19 Table 2-4: Summary of studies on deterministic optimization of biofuel plants .......................... 23 Table 2-5: Summary of studies on sensitivity analysis, scenario analysis and Monte Carlo simulation applied to bioenergy supply chain with uncertainty ................................................... 33 Table 2-6: Summary of studies on stochastic programming of forest and bioenergy supply chains....................................................................................................................................................... 38 Table 2-7: Summary of studies on robust optimization of forest and bioenergy supply chain .... 41 Table 3-1: List of indices and decision variables used in the optimization model ....................... 49 Table 3-2: List of parameters used in the optimization model ..................................................... 50 Table 3-3: Characteristics of the case study ................................................................................. 56 Table 3-4: Variables and parameters of the case study ................................................................. 58 Table 3-5: Results of cost, revenue and profit for the optimization model (in $M) ..................... 60 Table 3-6: Total profit and biomass procurement cost for four different scenarios ..................... 63 Table 4-1: Product type of suppliers and their contract type ........................................................ 70 Table 4-2: Average and standard deviation of bark, sawdust and shavings MC for Suppliers 1 to 5..................................................................................................................................................... 71 Table 4-3: Average and standard deviation of biomass MC for Supplier 6 ................................. 73 Table 4-4: Average and standard deviation of HHV for different biomass types ........................ 75 Table 4-5: Minimum, maximum, average and standard deviation of profit for considering uncertainty in different parameters ............................................................................................... 78 Table 4-6: Results of Monte Carlo simulation-optimization model for scenarios of electricity price and biomass availability and cost ........................................................................................ 79 Table 4-7: Probability of having profit within different ranges when considering uncertainty in different model parameters ........................................................................................................... 81 x  Table 4-8: Ranges of biomass purchase from suppliers without contract, biomass consumption and storage levels when considering uncertainty in different parameters (1000 green tonnes) ... 82 Table 5-1: Decision variables of the linear programming model ................................................. 85 Table 5-2: Stochastic model decision variables ............................................................................ 90 Table 5-3: Expected value of profit for scenario analysis, stochastic and average scenario models ($M) .............................................................................................................................................. 96 Table 5-4: Biomass procurement cost for each scenario of stochastic and deterministic models ($M) .............................................................................................................................................. 99 Table 5-5: Average profit if the first stage decision variables of each scenario is implemented and other scenarios happen ($M) ................................................................................................ 100 Table 5-6: Average and standard deviation of the monthly biomass consumption for deterministic model with scenario analysis and stochastic models (1000 green tonnes) ........... 101 Table 6-1: Profit ($M) for different ranges of MCs,p,t  and HHVs,p,t used in the robust optimization model........................................................................................................................................... 114 Table 6-2: Profit for different ranges of MCs,p,t and HHVs,p,t used in the robust optimization and hybrid models.............................................................................................................................. 118         xi  List of Figures  Figure 3-1: Schematic of supply chain configuration of a forest biomass power plant ............... 43 Figure 3-2: The amount of firm and surplus electricity production in each month ...................... 61 Figure 3-3: Optimum amount of biomass stored, purchased and consumed in each month based on the 2011 data ............................................................................................................................ 61 Figure 3-4: Variations in profit with  20% change in different parameters ................................ 64 Figure 3-5: Variations in profit for different initial storage levels ............................................... 64 Figure 4-1: Histogram and probability distribution of MC of bark (a), sawdust (b), shavings (c)72 Figure 4-2: Histogram and probability distribution for MC of RLD in January (a), February (b), March (c), April (d), June (e), July (f), August (g), September (h), October (i), November (j), and December (k) ................................................................................................................................ 75 Figure 4-3: Histogram and probability distribution of HHV of sawdust (a) and RLD (b) ........... 76 Figure 4-4: Histogram of profit when MC varies ......................................................................... 78 Figure 4-5: Histogram of profit when HHV varies ....................................................................... 78 Figure 4-6: Histogram of profit when electricity price and biomass availability and cost vary for  a) low, b) average and c) high scenarios ....................................................................................... 80 Figure 5-1: Histogram of profit distribution for the deterministic model with first stage decisions based on average scenario and the stochastic model .................................................................... 97 Figure 5-2: Profit mean and standard deviation for different weights (ρ) .................................. 102 Figure 5-3: Histogram of profit distribution for different weights (ρ) associated with variability index ............................................................................................................................................ 102 Figure 5-4: Histogram of profit distribution for before and after managing the downside risk (Ω=$14M) ................................................................................................................................... 103 Figure 6-1: Solution of the robust optimization model for different ranges of moisture content114 Figure 6-2: Solution of the robust optimization model for different ranges of higher heating value..................................................................................................................................................... 115 Figure 6-3: Solution of the robust optimization model for different ranges of energy value ..... 115 Figure 6-4: The optimum storage level in different months from the robust optimization model with HHVs,p,t∈[8100,8900] and MCs,p,t∈[26,34] and the deterministic model ........................... 116 xii  Figure 6-5: The optimum biomass consumption level in different months from the robust optimization model with HHVs,p,t∈[8100,8900] and MCs,p,t∈[26,34] and the deterministic model..................................................................................................................................................... 117 Figure 6-6: Optimum storage level of 27 scenarios for the first five months  for a) robust optimization model, and b) hybrid stochastic programming-robust optimization model when HHVs,p,t∈[8100,8900] and MCs,p,t∈[26,34] ................................................................................ 118      xiii  Acknowledgements   First of all, I offer my enduring gratitude to my dear supervisor, Dr. Taraneh Sowlati, for her help, guidance, and consideration during my Ph.D. studies. She put endless effort into supervising and motivating me, and never seemed to be tired or reluctant to help. Her enthusiasm and dedication for research always inspire me.  I am indeed grateful to my supervisory committee, Dr. Farrokh Sassani and Dr. Philip D. Evans for their time and invaluable feedback on my research during my PhD studies. Special thanks to Dr. John D. Nelson, Dr. Harish Krishnan and Dr. Chander K. Shahi for critically reviewing the thesis and providing constructive comments on it. I am also grateful to Dr. Mustapha Ouhimmou and Dr. Mikael Rönnqvist for their comments and collaboration on reformulating the non-linear model to a linear model presented in Chapter 5.  I would also like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for providing the funding for this research. I also acknowledge the partial funding provided by the power plant and sincerely thank the Fiber Supply Manager and Finance Manager for their time and support in providing the required data and information for the modeling and validating the results.  I express my appreciation to my lab-mates and friends in the Industrial Engineering Research Group for their comments and discussion during my Ph.D. studies. I am also thankful of my friends at UBC for their kindness and support during my stay in Canada and creating so much fun in my life. Special thanks to my dear friend, Nina, who assisted me in proofreading the thesis. xiv  I would like to express my deepest gratitude to my beloved family in Iran. My sincere appreciation is due to my lovely parents for their unconditional love and support throughout my life.  Last but not least, I express my profound affection and appreciation to my love and my best friend, Hamid, for his continuous love, kindness, encouragement and support during my years of education. This accomplishment would not have been done without him.       xv  Dedication  To my mother, Faranak, who taught me how to be happy by immersing myself in every joyful moment of my life. To my father, Mohammad, my polestar, who cultivated in me a hunger for learning.    1  Chapter 1 Introduction 1.1 Background  Using forest biomass as an energy source is not a new idea; before the twentieth century, wood was one of the main sources of energy, but it has been substituted partly by coal, oil and natural gas during the past century (Bowyer et al., 2007). Recently, there has been an interest in the utilization of more sustainable and secure energy sources such as forest biomass due to the high levels of emissions associated with fossil fuels, increasing energy demands and volatility of international energy market (Ragauskas et al., 2006). Biomass is defined as biological material derived from living, or recently living organisms. Depending on the source of biomass, there are three main biomass categories: forest (woody) biomass, agricultural biomass, and bio-wastes (animal waste and municipal solid waste) (Rentizelas et al., 2009b). This research focuses on utilizing forest biomass to generate electricity in power plants using direct combustion.  Although the conversion and transportation of forest biomass for energy generation affects the air quality negatively (Hall & Scrase, 1998), there is a potential to decrease emissions significantly when it is used as a substitute for fossil fuels (Hall (2002), Ahtikoski et al. (2008), International Energy Agency (2012)). Generally, if managed (produced, transported and used) in a sustainable manner, converting biomass to energy or fuel is considered to be a carbon neutral process, meaning that the amount of CO2 released during biomass combustion is equal to the amount of CO2 taken from the atmosphere by the plant during the growing stage (Saidur et al., 2011). Increasing the contribution of biomass in energy generation is an important step in developing sustainable communities, managing greenhouse gas emissions effectively according 2  to Ragauskas et al. (2006), and decreasing the gap between the actual emission levels and the international protocol targets such as those in the Kyoto and Copenhagen Accords (Bradley D., 2010). In areas covered with large forest lands and large forest industries, such as Canada, a large amount of forest biomass is available for energy generation (Bradley D., 2010). In Canada, after hydro, biomass has the highest share in the production of renewable energy (2.9% in 2008 according to Ralevic (2008)). Forest biomass is a flexible energy source, capable of generating electricity, heat, biofuels or a combination of them. Compared to other renewable energy sources such as wind or solar, the advantage of using forest biomass for energy generation is that it can be stored and used on demand (Hall & Scrase (1998), Demirbaş (2001)). It also has the potential to: increase the recovery of forest residues that would otherwise be disposed of in landfills or incinerated, create jobs, and provide local and sustainable energy for communities which in turn will decrease their dependency on the international fuel market (BIOCAP Canada, 2008).  Forest biomass for the purpose of energy generation can be supplied from forest residues including branches and tops left in the harvest areas, by-products of other forest product mills, such as sawdust, bark and shavings (Demirbaş, 2001), and fast growing crops such as poplar and willow grown specifically for energy purposes (Rockwood et al., 2004). Chipping, handling, transporting, storing and pre-processing operations, such as drying for improving the quality of biomass, are usually needed before using forest biomass for energy generation (Flisberg et al., 2012). Forest biomass can be used for energy generation either directly, such as in direct heat and power generation, or indirectly such as in biofuels (pellet, bioethanol) production. The energy production process depends on the conversion technology used in the 3  energy plant (Rentizelas et al., 2009b). The conversion technologies include pelletization, combustion, co-combustion, gasification, pyrolysis, digestion and fermentation (Demirbaş (2001), Frombo et al. (2009a)). Different types of products are then sent to customers through the grid, networks or channels of distributors, wholesalers and retailers. Profit in each stage of the forest bioenergy supply chain is a function of procurement, transportation, operating, capital and other costs, and depends also on the availability and quality of biomass (Gunn, 2009).  Despite the advantages of using forest biomass for energy generation, there are several barriers to its efficient utilization, including biomass availability, cost and quality, conversion efficiency, transportation cost, and the efficiency of the supply logistics system. Forest biomass is a bulky material with relatively low density (400-900 kg/m3 (Demirbaş, 2001)) and high moisture content (Hall, 2002). The quality of raw material plays an important role in the performance of the production process (Rentizelas et al., 2009b). It is difficult to collect, transport, handle and store low-density materials. Moreover, unlike fossil fuels, forest biomass is usually spread over large areas rather than being concentrated. Transportation could contribute as much as 50% to the total biomass cost in some cases (Allen et al., 1998) and may involve the use of a large amount of equipment and different transportation modes (Hall & Scrase, 1998). Inaccessibility of forests in some months during the year, when energy demand is quite high, raises concerns about the secure supply of biomass to energy plants. Therefore, storage, which affects the quality of material (Fuller, 1985), is also important in this supply chain. Comminution and storage of residues can take place either in the forest, at the plant or at an intermediate point. Another challenge in using forest biomass for energy generation is the existence of uncertainty in this supply chain due to several factors, such as market instability, 4  natural disasters, policy and climate change as well as the nature of the industry (for example heterogeneous raw material and unpredictable quality (Hall, 2002)). Uncertainty makes this supply chain volatile and risk vulnerable, which in turn makes proper planning difficult.  All these challenges contribute to a higher cost of energy generation from forest biomass compared to that of other sources of energy. Utilizing more advanced technologies, for example to improve raw material quality or system efficiency, is one way to deal with some of these challenges. A complementary approach to reduce the cost of energy produced from forest biomass and increase its competitiveness is to improve its supply chain and optimize its design and production planning (Bowyer et al., 2007). A supply chain model can be developed to help decision makers in their decisions and manage the supply chain more efficiently. Operations research and mathematical programming have been used in forest biomass supply chain planning and management. Modeling is effective particularly if it integrates different parts of the supply chain, such as procurement, production, transportation, and distribution, and at different decision levels, such as strategic, tactical and operational levels. Using optimization techniques in designing and managing forest bioenergy supply chains could result in better performance which could help to make this energy source economically viable according to Bowyer et al. (2007). Optimization models have been developed and used in the literature to determine the optimal material flow, transportation, storage and chipping location of energy systems, mainly heating plants (Eriksson & Björheden (1989), Gunnarsson et al. (2004), Kanzian et al. (2009), Freppaz et al. (2004) and Van Dyken et al. (2010)). There are also some studies that evaluated the conversion technology and the possibility of co-generation in the design of district heating 5  systems using mathematical programming (Nagel (2000), Frombo et al. (2009a), Difs et al. (2010), Wetterlund & Söderström (2010), Börjesson & Ahlgren (2010) and Keirstead et al. (2012)). Biomass supply chains for generating biofuels have also been studied (Chinese & Meneghetti (2009), Ekşioğlu et al. (2009), Kim et al. (2011a), and  a  iba e -Aguilar et al. (2011)). Although using forest biomass for electricity generation is not as common as using it for generating heat, there are some studies that considered the supply chain design of biomass power plants in an optimization framework, i.e. Reche et al. (2008), Alam et al. (2009 and 2012a) and  Vera et al. (2010). These studies focused on the strategic design of a forest biomass power plant and did not consider the tactical planning with multiple time steps in their models. Alam et al. (2012b) suggested an optimization model for biomass procurement to meet the monthly electricity demand of a forest biomass power plant over a one-year time horizon. Decision variables of the model included monthly harvesting levels from several forest cells. This study focused on the procurement of biomass and did not include the whole supply chain in an integrated framework. Moreover, none of the above mentioned studies included the impact of biomass quality on the electricity production and its cost. Specifically, the quality of different types of biomass in different months of the year, the quality of the mix of biomass in the storage and the impact of storing biomass on the amount and cost of generated electricity have not been studied previously.  There are some studies in the literature that consider uncertainty in biofuel supply chain optimization models. Some of them evaluated the impact of uncertainty on the model solution through sensitivity analysis and Monte Carlo simulation (Kim et al. (2011a), Rauch & Gronalt (2011)). More advanced optimization techniques that incorporate uncertainty in the modeling, e.g. stochastic programming (Kim et al. (2011b), Chen & Fan (2012), Awudu & Zhang (2013), 6  Kazemzadeh & Hu (2013)) and robust optimization (Tay et al. (2013), Bredström et al. (2013)), have also been used in this supply chain. Uncertainty in the supply chain of forest biomass power plants and its effect on electricity production cost are ignored in previous studies. Alam et al. (2012b) have only addressed uncertainty in the supply chain of a forest biomass power plant through sensitivity analysis and concluded that the impact of uncertainty in moisture content on the production cost was significant. However, there is no study in this area that include uncertainty into the decision making process. Instead, optimization models in the literature provided results only based on the expected value of the uncertain parameters. Ignoring uncertainty in deterministic optimization models may result in non-optimal and/or infeasible solutions for real world case studies. Hence, optimization models need to be extended to incorporate uncertainty and variations in the input parameters of the supply chain. 1.2  Research objectives  The main objective of this research is to develop tools (models) to help managers of biomass power plants to achieve greater efficiencies (profit) by better managing their supply chains, especially by considering uncertainty in the supply chain. The specific objectives of this study are as follows: 1. To optimize the supply chain of a forest biomass power plant considering biomass supply, storage and electricity production in an integrated framework. To achieve this objective a new mathematical programming model is developed to provide decisions on the amount of biomass to be purchased, stored and consumed in each month over a one-year time horizon to maximize the profit.  2. To apply the developed model to a real case study in Canada. 7  3. To evaluate the impact of changes in uncertain parameters on the solution of the mathematical programming model. This objective is achieved by examining historical data and performing sensitivity analysis, scenario analysis and Monte Carlo simulation. 4. To incorporate uncertainty in different parameters into the supply chain decision making process. This objective is reached by developing stochastic programming and robust optimization models, and a hybrid model. Using different modeling approaches allowed the inclusion of uncertainty in different parameters, i.e. biomass quality and availability. The hybrid model should incorporate uncertainties in different parameters in the model simultaneously.  1.3 Organization of the dissertation  In addition to the introduction chapter, this dissertation includes a chapter on the literature review, a chapter on the deterministic optimization model, a chapter on the Monte Carlo simulation model, a chapter on stochastic programming model, a chapter on hybrid stochastic programming-robust optimization model and a chapter on conclusions, limitations of the study, and suggestions for future research.  The previous studies on forest bioenergy supply chain in the literature are discussed in Chapter 2. They are categorized based on the inclusion of uncertainty.  In Chapter 3, an optimization model is proposed for the supply chain planning of a forest biomass power plant over one year. Later in this Chapter, the model is applied to a real case study in Canada and different scenarios are evaluated and sensitivity analysis is also performed to assess the impact of different scenarios as well as variations in input parameters on the generated profit.  8  Monte Carlo simulation along with the optimization model is performed and presented in Chapter 4 to provide the ranges of generated profit and its distribution when input parameters vary. Historical data on input parameters (biomass quality, cost, availability and electricity price) as well as data analysis are presented in this Chapter. From the results of the Monte Carlo simulation model, risks of having low profit and low or high storage levels associated with uncertainty in model parameters are identified.  In Chapter 5, uncertainty in the available monthly supply is incorporated in the decision making by developing a two-stage stochastic model and two different risk measures, variability index and downside risk, are also considered.  In Chapter 6, uncertainty in biomass quality is also added to the decision making process through developing a robust optimization model. Then, a hybrid stochastic programming-robust optimization model is proposed to include uncertainty in different parameters simultaneously. The final conclusions, strengths and limitations of the study and some suggestions for future research are presented in the last chapter, Chapter 7.   9  Chapter 2 Literature review  2.1 Synopsis  In several previous studies, optimization techniques have been employed to manage the forest bioenergy supply chain for heat, electricity and biofuels production from strategic, tactical and operational points of view. Most of these studies were deterministic and ignored uncertainty, while there are examples that included uncertainty in the supply chain models especially during the past few years. This chapter covers major relevant studies on optimization of forest bioenergy supply chains. It also discusses the issue of uncertainty in this supply chain, uncertainty sources and the methods used for dealing with it. The studies are categorized into two groups: 1) studies that used deterministic mathematical programming for modeling forest bioenergy supply chains, and 2) studies that considered uncertainty in the analysis of forest biomass supply chains. Studies that used deterministic models are categorized based on the type of bioenergy plants into those related to power plants, district heating plants, co-generation plants and biofuel plants. Studies that considered uncertainty are categorized based on the modeling approaches. Some examples from other forest product industries as well as other biofuel industries that included uncertainty in their supply chain optimization are also presented. The strengths and shortcomings of the relevant literature are highlighted at the end of this chapter.   2.2 Deterministic optimization models Different optimization techniques, such as linear programming (LP) and mixed integer linear programming (MILP), have been used for supply chain design and management. LP is a mathematical method which includes a set of variables to be determined, a linear objective function to be optimized, and a set of linear equality or inequality constraints to be met. The 10  main advantages of using this optimization method are its ability to solve large scale problems, its assured convergence to global optimum solutions, having no need to have an initial solution and its use of a well-developed duality theory for sensitivity analysis and the ease of problem formulation (Labadie, 1997). If some of the variables in LP are integers, the model is called mixed integer linear programming (MILP). In this section, the studies that optimized the supply chain of electricity plants, district heating systems, co-generation plants and biofuel plants using forest biomass are reviewed. 2.2.1 Power plants Forest biomass can be used in power plants directly or in combination with fossil fuels for generating electricity. It can be burnt at a constant rate in a boiler furnace to heat water and produce steam. The steam is then carried through the furnace using pipes to raise its temperature and pressure further. Finally, the steam passes through the multiple blades of a turbine, spinning the shaft, which runs an electricity generator which produces an alternating current to use locally or to supply the national grid (Demirbaş, 2001).  The optimal supply area and location of a forest biomass power plant in a distributed power generation system was determined by Reche et al. (2008). The objective function was to maximize the profitability index as a function of the net present value of benefits from the sale of electrical energy minus the initial investment, collection, transportation, maintenance and operating costs. The authors used an artificial intelligence method, called particle swarm optimization. They concluded that it is important in distributed generation systems to consider the technical constraints of the network and the voltage regulation. Finally, they evaluated the model performance using simulation.  11  Alam et al. (2009) constructed a three–objective model for optimizing the amount of each individual type of biomass from each of the harvesting zones, and then applied their model to a 50 MWh biomass power plant using both harvesting residues and poplar trees collected from three management zones in Northwestern Ontario, Canada. To optimize the supply chain of energy plants, it is sometimes necessary to formulate a problem with more than one objective since single objective models cannot always represent the problem accurately. The objectives are often in conflict (minimizing and maximizing objectives) and it might not be possible to achieve an optimal solution that optimizes all the objectives simultaneously. In this situation, the trade-off between objectives can be shown and the most efficient solution is selected. In Alam et al. (2009), pre-emptive goal programming was applied to give priority to the objectives as follows: 1) minimizing the procurement cost of feedstock, 2) minimizing the transportation distance of biomass supply to the plant, and 3) minimizing the feedstock moisture content. Alam et al. (2012b) developed a GIS based integrated optimization model to optimize the supply chain of the forest biomass power plant with 230 MW capacity. The power plant was fed by two biomass types: harvesting residues (leftover tops, branches and other parts of the trees harvested mainly for lumber and pulp and paper industries) and unutilized biomass (non-merchantable trees, and trees damaged by wildfire, wind and insects). GIS data were used to estimate transportation costs from each forest cell to the power plant. The decision variables were the harvest levels of two types of woody biomass in each month. The objective function was to minimize the total piling, processing, felling, extraction and transportation costs.  Finding the optimal size, location, supply area and net present value of an electricity plant in Spain was studied in Vera et al. (2010). The raw material of the power plant was olive tree pruning residues and the technology for electricity generation was gasifier with gas turbine. 12  The authors used GIS data for the location and number of olive trees per km2, roads, topographical features, electric line locations, etc. Different plant sizes and locations were considered and the optimal one with the highest net present value was determined using three metaheuristic methods. These methods were Genetic Algorithms (GA), Binary Honey Bee Foraging (BHBF) and Binary Particle Swarm Optimization (BPSO). It was concluded that BHBF algorithm converged to the optimal solution better than BPSO and GA. The results indicated that the optimal plant size was 2 MW and the predicted optimal location of the plant was in the area with highest available biomass.  Pérez-Fortes et al. (2014) developed an optimization model to determine location-allocation and the selection/capacity of different pre-treatment technologies for feeding biomass to an already existing coal combustion power plant. They included biomass transportation and storage in their model. Different pre-treatments technologies were considered including torrefaction, torrefaction combined with pelletization, pelletization, fast pyrolysis and fast pyrolysis combined with char grinding. Changes in biomass quality (moisture content, dry matter, energy density and bulk density) through the use of pretreatment processes were also studied.  Table 2-1 summarizes the studies on optimization models used for modeling forest biomass power plant supply chain.     13  Table 2-1: Summary of studies on deterministic optimization of forest biomass power plants  Author-Year- Region Objective Function Decision Variables Method Reche et al. (2008) Spain Maximizing profitability index (net present value of revenue from selling electricity minus initial investment, biomass collection and transportation costs, and maintenance and operation costs)  Location and supply area of the biomass power plant Particle swarm optimization  Vera et al. (2010) Spain Maximizing net present value (revenue from the sale of electrical energy minus initial investment and collection, transportation, maintenance and operation costs)  Plant size and location  Supply area  Several metaheuristic methods  Alam et al. (2009) Canada Minimizing total biomass procurement cost  Minimizing total distance for procurement of biomass Maximizing the quality of biomass (minimizing moisture content) Quantity of biomass procured from each supply location to each plant Biomass procurement zone selection (Binary variable) Multi Objective Programming Alam et al. (2012b) Canada Minimize the total piling, processing, felling, extraction and transportation costs. The harvest levels of two types of woody biomass in each month Non-linear dynamic programming Pérez-Fortes et al. (2014) Spain Minimizing net present value of investment and operational costs Maximizing the environmental impact of adding biomass to a coal power plant Location/allocation and selection/capacity of pre-treatment technology  Fraction of coal replaced by biomass Material flow Multi-objective Mixed Integer Programming (MIP)  2.2.2 District heating plants  Forest biomass for energy generation is mainly used in district heating systems. These systems consist of a central plant producing heat which is sent to a group of users (customers) through a network of pipelines in the form of hot water or steam (Gilmour & Warren, 2007). Several authors developed optimization models for supply chain design and management of heating 14  plants. Eriksson & Björheden (1989) developed a model with decision variables related to storage and the chipping locations for a heating plant. Gunnarsson et al. (2004) developed a mixed integer programming model for tactical-strategic supply chain management of forest fuel used in a heating plant in Sweden by focusing on supply procurement decisions rather than the production process. Multiple time steps were considered in this model. It was used to solve six generated problems rather than being applied to a real case study. The results of using different solution methods (LP and IP, and IP heuristic) to solve the problems were compared in this work. Strategic decisions such as plant size and location were studied in Chinese & Meneghetti (2005) and Schmidt et al. (2010). The most profitable configuration (plant size) of a multi-source biomass district heating plant in Italy was considered in Chinese & Meneghetti (2005). The model developed by Kanzian et al. (2009) included 16 combined heat and power plants and 8 terminal storages in Austria. Optimum locations of bio-energy plants were studied in Schmidt et al. (2010) with a case study in Austria. In another study, done by Van Dyken et al. (2010), a linear mixed-integer model was developed for biomass supply chains with transportation, storage and processing operations over 12 weekly time steps considering supply, constant demand, three different biomass products and two demand loads for chips and heat. This study focused on operational supply chain planning and the developed model was not applied to a real case study. A truck scheduling optimization model was developed in Han & Murphy (2012) for transportation of four types of forest biomass to energy plants in Oregon, US. This study only considered the transportation part of the supply chain.  Table 2-2 summarizes all of these studies with their objective functions and decision variables.  15  Table 2-2: Summary of studies on deterministic optimization of district heating plants  Author -Year- Region Objective Function Decision Variables Method Eriksson & Björheden (1989) Sweden Minimizing forest biomass supply cost (chipping, storing and transportation costs) Flow of biomass direct or via storage Chipping location Linear Programming (LP)  Nagel (2000) Germany Maximizing annual profit (revenue from sale of energy  minus investment cost, fixed and variable costs, fuel cost and waste disposal cost) Level of heat produced by each boiler at each time period The capacity of the system  Whether or not to integrate a boiler into the heating system (Binary variable) Mixed Integer Programming (MIP) Gunnarsson et al. (2004) Sweden Minimizing biomass supply cost (transportation, chipping and storage costs) Flow of biomass within the supply network  Quantity of biomass chipped and stored at roadside and terminal If biomass is forwarded to or is chipped at each roadside location,  each sawmill is contracted or not, each terminal is used or not (Binary variables) Mixed Integer Programming (MIP) Chinese & Meneghetti (2005) Italy Maximizing profit (revenues from sale of energy and charging customers with connection fees minus boiler investment, construction and operating costs)  Heat produced by each boiler at each time period The capacity of the system  If a boiler would integrate to the heating system or not (Binary variable) Mixed Integer Programming (MIP) Frombo et al. (2009a) Italy Maximizing net annual profit (revenue from sale of heat and power minus felling and processing, skidding, highway transportation, plant installation and management costs) Annual quantity of biomass harvested from each supply area The plant capacity for different conversion technologies  Linear Programming (LP)   16  Author -Year- Region Objective Function Decision Variables Method Frombo et al. (2009b) Italy Maximizing net annual profit (revenue from sale of heat and power minus felling and processing, skidding, highway transportation, plant installation and management costs) The quantity of biomass harvested at each harvesting location and to be used at each plant location. The capacity of each plant Selection of the conversion technology (Binary variables) Mixed Integer Programming (MIP) Kanzian et al. (2009) Austria Minimizing biomass supply cost to the heating plants (chipping, storing and transporting costs) Volume of wood chips transported from each terminal to each plant   Location of terminals and plants (Binary variable) Mixed Linear Programming (MIP) Van Dyken et al. (2010) Norway  Minimizing the present value of the costs (investment and operating costs and salvage value) Biomass, product and energy flow within the supply network  Emissions from storing and drying biomass  Biomass input and output moisture content to and from dryer  Biomass storage duration (Binary variable) Linear and Mixed Integer Programming (LP and MIP) Keirstead et al. (2012) UK Minimizing system cost (biomass purchase, storage, transportation and conversion costs) Optimal capacity of boilers  Whether chipped forest biomass should be imported from neighbor area or non-chipped residues should be imported and then chipped within the area (Binary variable) Mixed Integer Programming (MIP) Han & Murphy (2012) US Minimize the weighted sum of transportation costs  Minimize the total working time Truck schedules  Simulated Annealing  2.2.3 Co-generation plants Combined heat and power (CHP) systems are energy plants that use cogeneration technology to produce both heat and power in a district heating system (Gilmour & Warren, 2007). In some studies, mathematical programming was used to compare the cost of generating either energy or biofuels from biomass and evaluate the possibility of co-generation.  17  Some studies indicated that utilizing biomass for energy generation is more cost effective than for biofuel production. Azar et al. (2003) concluded that utilizing biomass for generating heat was the most economical scenario. Wahlund et al. (2004) showed that using wood biomass for pelletization would have a lower cost and higher CO2 reduction than using it for biofuel production. Feng et al. (2010) investigated the possibility of having bioenergy facilities (they called them biorefineries) in typical existing sawmills, pulp and paper mills, wood panel facilities, biochemical, energy, and pellet facilities. Then, the authors developed a mathematical model to design this integrated supply chain optimally.  A methodology for optimizing the utilization of distributed biomass resources for energy production was proposed by Alfonso et al. (2009). The main focus of the paper was to optimize the logistics, but economic and environmental analyses of different bioenergy alternatives were also performed. The authors indicated that the methodology would provide the optimal locations of the biomass plant, energy application (electricity, heat and/or standardized biofuels such as pellets), and the employed technology.  This methodology was applied to three districts in Spain. Based on the results, the authors concluded that the shortest payback period and highest CO2 savings were attained from cogeneration plants, followed by pellet plants. The least ranked option was power-only power plants.  Difs et al. (2010) analyzed different biomass gasification scenarios, and determined the optimum configuration with the current fossil fuel price and green energy policies. Wetterlund & Söderström (2010) considered two scenarios of co-generating Synthetic Natural Gas (SNG) and district heat, and co-generation of heat and power. The authors determined the policy support levels (tradable biofuel certificates) that would make the SNG scenario cost competitive with CHP, while maximizing the annual profit over a 20-year time period. The 18  cost-effectiveness of different applications of biomass gasification was analyzed by Börjesson & Ahlgren (2010). The focus of this study was to determine whether CHP generation in biomass integrated gasification combined cycle (BIGCC) plants, and biofuels production in biomass gasification biorefineries in a case study in Sweden were cost efficient. Schmidt et al. (2010) used a mixed integer linear programming model for optimizing the location of bioenergy plants using forest biomass in Austria. The bioenergy plants included integrated gasification combined cycle (IGCC) system and biomass CHP plants with carbon capture storage (CCS), pellet mill, and transportation fuel (methanol and ethanol) plants.  The problem of indicating whether to produce electricity in addition to heat at biomass combustion plants was studied by Freppaz et al. (2004). A decision support system (DSS) was presented by Rentizelas et al. (2009a) to optimize a multi-biomass energy conversion system to generate electricity, heating and cooling in an area in Greece. The authors concluded that considering multi-biomass supply chain reduced the cost by decreasing warehousing requirements, especially for seasonal types of biomass. The developed model was non-linear and a hybrid optimization method was used to solve that. Rauch & Gronalt (2011) developed a model for designing a forest fuel CHP plant supply chain in Austria.  The summary of studies on modeling co-generation plants is provided in Table 2-3.      19  Table 2-3: Summary of studies on deterministic optimization of co-generation plants Author- Year- Region Objective Function Decision Variables Product Method Freppaz et al. (2004) Italy Maximizing annual profit (revenues from sale of energy minus harvesting,  transportation, installation and maintenance, and energy distribution costs)  Annual quantity of biomass harvested at each collection area and transported from each collection area to each of six district energy systems Capacity and the percentage of thermal energy generated at each plants If the plant produces electricity or not  (Binary variable) Heat/ Electricity Mixed Integer Programming (MIP) Alfonso et al. (2009) Spain Minimize transport duration, optimize the location, etc.  Biomass resources, logistics structure, bioenergy plants size and location, technology type, etc.  Co-generation Did not mention Rentizelas et al. (2009a) Greece Maximizing the financial yield of the investment Location and size of the bioenergy facility The biomass types and quantities  The maximum collection distance for each type Co-generation  Hybrid optimization Börjesson & Ahlgren (2010) Sweden Not discussed in the paper The optimal production capacity at different subsidy levels.   Selection of alternative technologies for district heat generation (Binary variable) Biofuel/ Heat Mixed Integer Programming (MIP) Difs et al. (2010) Sweden Maximizing annual profit (revenues from sale of energy products minus investment, fuel and maintenance costs)  Capacity of new investment Selection of investment alternatives for future (Binary variable) Co-generation  Mixed Integer Programming (MIP) 20  Author- Year- Region Objective Function Decision Variables Product Method Schmidt et al. (2010) Austria Minimizing total cost of energy generation (costs of biomass supply and transportation, energy generation, carbon capture and storage, plant building and distribution network investment and distribution) The annual amount of energy commodities produced at plants: heat, power, pellets, and transportation fuels.   Plant size and location, pipeline networks selection and district heating networks selection (Binary variables) Fuel/ Energy Mixed Integer Programming (MIP) Wetterlund & Söderström (2010) Sweden Maximizing annual profit (revenues from sale of electricity  and synthetic natural gas minus investment, fuel and maintenance costs)  The  optimal government support level (subsidy)  Selection of new investment alternatives (Binary variable)  Co-generation   Mixed Integer Programming (MIP) Rauch & Gronalt (2011) Austria Minimizing total procurement cost (transport, chipping investment, operations and maintenance costs)  The annual volume of fuel transported between districts, terminals, regional departure railway, and the CHP plant Open or close a terminal (binary variable) CHP  Mixed Integer Programming (MIP)  2.2.4 Biofuel plants Bioethanol is a type of fuel that is extracted from biomass through fermentation (Limayem & Ricke, 2012). The bioethanol production has increased in recent years in many countries, such as the U.S. Although most of the bioethanol is produced from agricultural biomass, the controversial issue of using plants as fuel instead of food made it necessary to look for more acceptable sources, namely forest biomass (Limayem & Ricke, 2012). Generating biofuels from forest biomass is still in the developing phase and has not been commercialized yet. The main challenges in commercialization of this technology include high energy or chemical consumption for woody biomass pretreatment, even when compared to agricultural biomass, 21  low system efficiency, process scalability and intensive capital investment (Zhu & Pan, 2010). In most of the studies presented here, forest biomass combined with agricultural biomass was used for biofuel production Some previous studies considered biomass supply chain management for generating biofuels. Chinese & Meneghetti (2009) considered a real-life problem of supplying a biofuel plant with forest fuel. A mixed-integer linear programming model was proposed to determine the optimal configuration of the supply chain. It was mentioned that the model could be useful in resolving trade-offs between decentralized early treatment of biofuels, resulting in lower transportation costs, and centralized final treatment, allowing to reap the benefits of economies of scale. It was therefore advised to apply integrated supply chain planning concepts to design biofuel logistics systems and to support policy making in the energy field. An MIP model was also developed by Ekşioğlu e  al. (2009) for designing the biorefinery supply chain producing cellulosic ethanol from agricultural and woody biomass. The model outputs were the number, size and location of biorefinery plants with the objective of minimizing the total cost of annual harvesting, storing, transporting and processing biomass; storing and transporting ethanol; and locating and operating bio-refineries. The model included constraints on biomass availability, flow, conversion, production and inventory capacities, and demand. The data from the State of Mississippi was used to validate the model. The author concluded that transportation costs, biomass availability, technology type, and planting and harvesting costs are important factors in supply chain design decisions.  Kim et al. (2011a) developed a mixed integer linear programming optimization model for the supply chain design of bio-gasoline and biodiesel production from six forestry resources (logging residuals, thinnings, prunings, inter-cropped grasses, and chips/shavings). The first set 22  of conversion plants could be from a set of candidate sites with four capacity options to convert biomass to three product types: bio-oil, char and fuel gas. These intermediate products could be used either as local fuel sources or as feedstock to produce final products (gasoline and biodiesel) at the second conversion plants, which could be from a set of candidate sites with four capacity options. There were several possible markets for the final products with certain maximum demands. The objective of the model was to maximize the overall profit by determining the number, location, and size of the conversion plants, biomass supply locations, the logistics and the amount of materials to be transported between the various nodes of the designed network, while satisfying the demand constraints. The considered case study was based on an industrial database related to a case in the Southeastern United States. The authors evaluated the trade-off between centralized and distributed network designs.  The trade-off between economic and environmental objectives in the optimal planning of a biorefinery in Mexico was evaluated in  a  iba e -Aguilar et al. (2011). The authors used a multi-objective optimization model for selecting the feedstock type, processing technology, and a set of products in a biorefinery supply chain. The raw material contained different types of agricultural biomass, wood chips, sawdust, commercial wood for producing ethanol, hydrogen, and biodiesel (generated only from agricultural biomass). The objectives were: 1) to maximize the profit considering the costs of feedstock, products, and processing, and 2) to minimize the life cycle environmental impacts. The authors applied their model to a case study in Mexico. The decision makers could select from the output the solutions that fit the specific requirements and compensate for both objectives simultaneously.  Table 2-4 summarizes the studies on optimization models used for modeling the supply chain of forest biofuel plants.  23  Table 2-4: Summary of studies on deterministic optimization of biofuel plants Author-Year- Region Objective Function Decision Variables Method Chinese & Meneghetti (2009) Italy Minimizing total cost of supply chain (harvesting, transportation, processing and facility installation costs) Flow of biomass within the supply network Whether to use a preprocessing equipment or not (Binary variable) Mixed Integer Programming (MIP) Ekşioğlu e  al. (2009) USA Minimizing total annual cost (investment, harvesting, storing and transportation costs) Number, size, and location of bio- refineries required Quantity of biomass harvested, shipped, processed and stored Whether a biorefinery and a collection facility with specific size are located in each site (Binary variables)  Mixed Integer Programming (MIP) Kim et al. (2011) US Maximizing the overall profit  Number, location, and size of the conversion plants Biomass supply locations Logistics and the amount of materials to be transported between the various nodes of the designed network Mixed Integer Programming (MIP)   a  iba e -Aguilar et al. (2011) Mexico Maximizing profit (revenue from sale of products minus investment, process, operating and transportation costs)  Minimizing environmental impacts The quantity of products produced from different biomass feedstock using different processing routes  The quantity of each biomass feedstock used for producing different products through different processing routes Multi-objective Programming  2.3 Optimization models with uncertainties Uncertainty refers to the lack of information or the lack of certainty in the validity of the information about the existing or future state of a system (Kangas & Kangas, 2004). It can result from measurement errors and ignorance, which is to some extent inevitable and might be reduced by further studies or investing in improved technology to acquire high quality data (Petrovic (2001), Ells et al. (1997)). It may result from variability in random future events due 24  to their inherent nature (such as feedstock characteristics) (Bowyer et al. 2012), which can be controlled to some extend by employing better forecasting methods and/or using expert judgment. It can also result from lack of reliable historical data or lack of certainty in historical data, for example lack of data on the demand of a new product. Other sources of uncertainty include imprecision in judgment, vagueness, and ambiguity related to the known objects, which belong to poorly defined sets so they cannot be classified well (Kangas & Kangas (2004), Petrovic (2001), Ells et al. (1997)). From the system boundary point of view, the source of uncertainty may exist outside the production process, called environmental uncertainty, such as uncertainty in demand and supply. It may also be within the production process, called system uncertainty, such as uncertainty in lead time due to machine failure (Chrwan-JYH, 1989).  In terms of time horizon, uncertainty may be the result of short term variations, such as day-to-day processing variations, cancelled/rushed orders and equipment failure, or long-term variations, such as raw material/final product unit price fluctuations, seasonal demand variations and technology changes. Therefore, uncertainty exists in supply chains at strategic, tactical and operational levels and should be considered in supply chain decisions.  There are several reasons why uncertainties exist in the biomass supply chain. Some of the sources of uncertainty in forest biomass supply chains are similar to those in other industries, such as economic fluctuation and instability, raw material supplies, manufacturing process time, machine breakdowns, reliability of transportation channels, and exchange rates. However, there are other sources of uncertainty that are related specifically to the characteristics of forest biomass supply chains which are summarized here:   Interdependency between different forest sectors: There are interdependencies between different sectors and markets within the forest industry supply chains. This means that 25  raw material of one sector could be the product of another sector. Consequently, variations in one part of the supply chain usually propagate into the other parts.   Variations in feedstock supply: The need for having a continuous supply of raw material for a bioenergy facility necessitates the use of a mixture of materials or even to have new sources of material.  Even when one type of biomass is used for energy production, the quality of biomass varies over time. Therefore, this industry must have a dynamic supply chain.   Wood is a heterogeneous natural material: Its physical and chemical characteristics affect the quality and quantity of the products (Bowyer et al. 2012). In the bioenergy industry, the moisture content and heating value of raw material play an important role in the amount of produced energy and its costs (Saidur et al., 2011). Heating value and moisture content vary from one tree stands or species to another (Demirbaş (2001), Carlsson et al. (2009), Demirbaş (2003)) and also differ in different types of biomass (e.g. bark, sawdust, shavings) (Lehtikangas, 2001). Wood properties may be affected by external factors, such as growth condition, climate, harvesting methods, storage and transportation methods. Biomass quality, such as moisture content, can also change during storage, production, and transportation.   Divergent production structure: Unlike most of other manufacturing industries, which have an assembly structure, forest products industries generally have a divergent production structure. This means that multiple products, by-products, and co-products are made from a single product simultaneously. Consequently, it is difficult to completely control the manufacturing processes. Moreover, it is challenging to forecast the quality and quantity of outputs due to this production structure and the use of 26  heterogeneous natural raw material in the production.  This fact can impact the amount of raw material available for bioenergy plants, which are supplied by other forest product mills.  Ambiguous values and objectives: Most forest areas include large areas with diverse geographical and ecological characteristics. In forestry, it is usually needed to i corpora e differe   values a d s akeholders’ prefere ces a d i  eres s which sometimes cannot be understood, interpreted or quantified completely (Ells et al., 1997). Therefore, it is likely to have vague factors, values and objectives which can also exist in the forest bioenergy supply chain. This aspect of uncertainty cannot be dealt with like other sources of uncertainty. To some extent, it is possible to spend time and money in some form of consulting with the stakeholders to get a better understanding of their preferences, opinions, and values. However, sometimes the stakeholders may not be able to express their preferences before a specific decision is made.   New markets and new production technologies: Investment grants, carbon and energy taxes, green certificate schemes, conversion technologies, and availability and quality of biomass resources may not be known with certainty (Mccormic, 2011). For example, in designing and planning a biomass power plant, it may be hard to estimate the long term availability, quality and cost of biomass. Alternatively, market demand for biofuels may not be lucid from the beginning. In general, uncertainty can be dealt with at the source, or it can be dealt with during the process of decision making. When uncertainty is ignored, decision making is based on the expected values of stochastic parameters, which may be different from their actual values and may lead to non-optimal or infeasible results and solutions. Considering uncertainty in decision making 27  usually helps companies safeguard against threats while simultaneously taking advantage of the opportunities that higher levels of uncertainty would provide. It also makes decisions robust and mitigates the effect of the variations and perturbation on the optimal solution. The modeling approaches for dealing with uncertainty in optimization models are presented next.   2.3.1 Modeling approaches  The method for dealing with uncertainty depends on the type of uncertain parameter, the source of uncertain parameter, available data on the uncertain parameter, the computational effort needed for each method, and the degree of sophistication that can be handled and accepted by the users and decision makers. In the literature, several approaches were used to incorporate uncertainty in the supply chain design including sensitivity analysis, scenario-based approaches, Monte Carlo simulation, stochastic programming, and robust optimization. Mula et al. (2006) provided an overview of the uncertainty in production planning. Based on their classification, the supply chain planning problems with uncertainty are usually solved by conceptual, analytical or artificial intelligent based approaches.  I  de ermi is ic models, a si gle bes  guess or a si gle fu ure “sce ario” represe  s a  uncertain parameter based on its expected values. Through sensitivity analysis, model sensitivities are tested to determine variations in model outcomes when input parameters fluctuate around best guesses. In the scenario analysis approach, multiple scenarios for uncertain parameters are generated and then the optimization model is solved for each individual scenario. This provides an extensive what-if analysis which also helps in evaluating the outcomes based on different realizations of the stochastic parameters. Scenario generation itself is a challenging task which can be done using historical data, forecasting methods, managerial and expert judgment, etc. (Benders (1962), Shapiro (2004)).  28  Similar to scenarios analysis, in Monte Carlo simulation a number of scenarios are generated, but one further step is taken in this method by considering every possible value that each stochastic parameter could take using its probability distribution. Moreover, each scenario is weighted by the probability of its occurrence. In other words, in this method the deterministic model is solved repeatedly with its stochastic input parameter based on a probability distribution function instead of a single value. The Monte Carlo simulation method 1) determines a possible distribution of the model outcomes, 2) evaluates model robustness and behavior in the presence of uncertainty in input parameters, 3) determines regions of input parameters that result in particular levels of the optimal solutions, and 4) identifies possible risks and opportunities that result from uncertainties in the system. The process of developing and implementing Monte Carlo simulation involve: 1) determining the ranges and distributions of each stochastic input parameter, 2) generating samples from the specified ranges and distributions, 3) running the model for these samples, and 4) evaluating and analyzing the outputs (Pannell (1997), Vose (2008), Saltelli et al. (2008), Kim et al. (2011b)).  The problem with all of these approaches is that they do not provide a single overall optimal solution for all scenarios. Stochastic programming is an approach which overcomes this problem. In stochastic programming, it is assumed that accurate probabilistic descriptions of the random variables such as probability distributions, densities or other probability measures are available. In this method, the expected objective value of different potential scenarios is optimized. In a two-stage stochastic model, decision variables are divided into two groups called the first stage variables (control, here-and-now), which are made before the realization of the uncertain parameters, and the second stage variables (state, wait-and-see), which are taken after the realization of the uncertain variables. The output of such a model is the optimal 29  single first-stage policy and a set of recourse decision rules that determine which second-stage action should be taken in response to each random variable. One of the advantages of developing stochastic programming models is in their capability to manage the risk associated with the supply chain performance (Birge & Louveaux (1997), Shapiro (2004)).  While stochastic programming seems to be an adequate and attractive method for addressing uncertainty when it is possible to define potential scenarios, it is computationally intractable when the value of an uncertain parameter covers a continuous range, unless it is approximated by a set of scenarios derived from discretizing the uncertainty sets. The problem, hence, is that even for a small number of discretized scenarios, the total number of scenarios will grow exponentially when one deals with a sequence of scenarios, e.g. a scenario tree. This again results in being computationally intractable and not being able to have a ready-to-use model (Ben-Tal et al. 2000). Approximation and decomposition methods are being used to address this issue (Benders, 1962). Another alternative, however, is robust optimization which is attractive since it can be solved effectively and efficiently using the current powerful solvers if a tractable uncertainty set is selected (Ben-Tal & Nemirovski, 2000). In stochastic programming, the objective is to find solutions that are feasible for all realization of uncertain parameters while optimizing the expected value of the objective function over all scenarios. In robust optimization, the objective is to find solutions that are feasible for all realization of uncertain parameters while optimizing the worst case performance of the system. Moreover, contrary to stochastic programming, to incorporate uncertainty in robust optimization only a range of uncertain parameters is required (Gabrel et al. 2013).  30  The formulation of robust optimization depends on the definition of a robust solution. A robust solution as defined by Mulvey et al. (1995) is a solution that is not far from the optimum solution. The authors developed a model based on scenario analysis with the objective of providing less sensitive solutions to the realization of the model data and solved it using goal programming. In another definition, a robust solution is a solution that must be feasible for any realization of the uncertain parameter, or it is a solution that its objective function value must be guaranteed. This approach was originally proposed by Soyster (1973) and later on was co sidered as “ul raco serva ive s ra egies”. Bertsimas & Sim (2003) and (2004) suggested an approach  ha  uses  he idea of “budge  of u cer ai  y”  o co  rol  he level of co serva ive ess. In this method, only some of the uncertain parameters deviate from their nominal values simultaneously. Using this definition, a constraint is immunized against uncertainty by de ermi i g  he si e of  he buffer or a “pro ec io  fu c io ” of i . This pro ec io  fu c io  is itself an optimization model and its dual is embedded in the original model. Given the linearity of the original problem, the robust counterpart is also a linear problem with a modified feasible region. Ben-Tal & Nemirovski (2000) suggested less conservative approaches which were nonlinear. In all of these methods, the solution is optimized based on the worst case, which is the most unfavourable realization of the uncertainty. The worst case can be selected differently too, either from a finite number of scenarios, such as historical data, or continuous, convex uncertainty sets, such as polyhedral or ellipsoids. For a recent review of robust optimization the reader is referred to Gabrel et al. (2013).   There are few studies which considered uncertainty in the forest biomass supply chain. These studies as well as some other studies which included uncertainty in other forest industries and biofuel supply chain are reviewed here.   31  2.3.2 Sensitivity analysis and Monte Carlo simulation Sensitivity analysis and scenario analysis are usually done after an optimization model is developed. One study which used sensitivity analysis in the forest biomass supply chain was conducted by Kim et al. (2011a). They ran their model for different demands (100%, 90%, 75% and 60%) to evaluate the effect of changes in demand on the optimal network design. The results showed the total profit for the distributed system was higher than that for the centralized design at 100% demand. When the demand decreased, the profit difference between the two systems reduced as well. In this study, although demand uncertainty was evaluated through a very simple sensitivity analysis method, it showed the importance of considering uncertainty as it affected the optimal result. Alam et al. (2012b) performed sensitivity analysis by testing sixteen scenarios for different harvesting levels, processing and felling costs, conversion efficiency, moisture content, energy density and equivalency of energy. They concluded that moisture content and conversion efficiency had more impact on the cost compared to the other parameters. This paper incorporated variations in biomass quality, however, a fixed range was used to capture variations and only sensitivity analysis was performed. In Rauch & Gronalt (2011), the effect of changes in forest fuel supply (domestic resources or imports), transport modes (truck only, truck and ship, or truck, ship and rail), energy price (increase by 0, 20% and 40%), and truck load capacity (50%, 40% and 30%) on the overall cost was evaluated. Eight scenarios were constructed and compared.   In a number of studies, Monte Carlo simulation was used in optimization of forest and agricultural bioenergy supply chains. Rozakis & Sourie (2005) developed an optimization model for eco omic a alysis of a biofuel supply chai  i  Fra ce. They de ermi ed  he efficie   tax exemption policies in presence of uncertainty in petroleum prices and feedstock prices 32  using Monte Carlo simulation. A mixed integer programming model was developed by Schmidt et al. (2009) to determine optimum locations of bioenergy gasification plants in Austria. The model considered the spatial distribution of biomass supply and biomass transportation costs. The authors used Monte Carlo simulation to incorporate uncertainties in 9 parameters: annualized district heating costs, biomass supply, biomass costs, plant setup costs, transportation costs, price local heat, carbon price, connection rate and power price. They used literature and expert opinion to assign ranges to the uncertain parameters. An optimization model of a biofuel supply chain was presented in the study done by Marvin et al. (2012). It determined the location and capacity of biorefineries, and the amount of harvested biomass to ship to biorefineries while maximizing the Net Present Value (NPV) of the entire supply chain. The authors performed sensitivity analysis and Monte Carlo simulation to evaluate the impact of uncertainty in costs, harvestable biomass and conversion factor on the robustness of the supply chain. The output of the Monte Carlo model was a range of internal rate of return (IRR) which was used to make conclusions about the probability of having IRR less than a certain level (10%) and consequently having a negative NPV. It was also concluded from Monte Carlo simulation results that it was not economical to construct any biorefineries in 21.5% of the trials. Sharmaa et al. (2013) studied the weather uncertainty in biomass supply chain through developing a scenario optimization model. The model had a one year planning horizon with monthly time steps and the objective function of minimizing the cost of biomass supply to biorefineries. Uncertainty in other parameters, e.g. yield, land rent and storage dry matter loss was analyzed by sensitivity analysis.  33  Table 2-5 summarizes the previous studies which incorporated uncertainty in the forest biomass supply chain and biofuel supply chain management and design through sensitivity analysis, scenario analysis and Monte Carlo simulation model. Table 2-5: Summary of studies on sensitivity analysis, scenario analysis and Monte Carlo simulation applied to bioenergy supply chain with uncertainty  Author/ Year Uncertain parameter Method Case Study Kim et al. (2011a) Demand Sensitivity analysis Biofuel plant in Southeastern US Alam et al. (2012b) Harvesting levels, processing and felling costs, conversion efficiency, moisture content, energy density and equivalency of energy Sensitivity analysis Forest biomass power plant in Ontario, Canada Rauch & Gronalt (2011) Forest fuel supply, transport modes, energy price, and truck load capacity   Scenario analysis CHP in Austria Rozakis & Sourie (2005) Petroleum price and feedstock prices Monte Carlo simulation Biofuel plant in France Schmidt et al. (2009) Annualized district heating costs, biomass supply, biomass costs, plant setup costs, transportation costs, price local heat, carbon price, connection rate and power price Monte Carlo simulation Bioenergy gasification plants in Austria Marvin et al. (2012) Costs, harvestable biomass and conversion factor Sensitivity analysis and Monte Carlo simulation Biorefinery plant in Midwestern US Sharmaa et al. (2013) Weather uncertainty Yield, land rent and storage dry matter loss Scenario analysis Sensitivity analysis Biorefinery in the US  34  2.3.3 Stochastic programming Some previous studies incorporated uncertainty in supply chain optimization of other industries such as chemical and lumber industries (e. g. Gupta & Maranas (2003), You et al. (2009), Kazemi Zanjani et al. (2010b)). The results of these studies mainly demonstrated that incorporating uncertainties in the decision making of real case scenarios using stochastic models provide more robust solutions compared to deterministic models. There are few studies that included uncertainty in bioenergy supply chain models. A number of them are reviewed and presented by Awudu & Zhang (2012).  Some of the studies used stochastic programming in modeling bioenergy and biofuel supply chains. Kim et al. (2011b) performed a global sensitivity analysis and two-stage stochastic programming on a biofuel supply chain model and evaluated the effect of uncertainty in different parameters on the final result using Monte Carlo simulation. The authors concluded that the most important uncertain parameters affecting the profit were the price of the final product, the conversion yield ratios of the two conversion processes, maximum demand and biomass availability. They then generated 33 scenarios from changing these five most important uncertain parameters by ± 20% and developed an optimal model to maximize the expected profit from all these scenarios plus the expected value scenario. The first stage decision variables were the size and location of the processing plants and the second stage recourse decision variables were flows of biomass and product within plants and markets. They implemented the robustness analysis and Monte Carlo global sensitivity analysis to compare the performance of the multiple scenario design with the single scenario design. It was concluded that the impact of variations in stochastic parameters on the optimal solution was mitigated in the model that optimized multiple scenarios. In this paper, uncertain parameters 35  were only changed within some range rather than using probability distributions. Moreover, it only modeled the supply chain in one time-step and ignored the possibility of having correlation between random parameters.   Strategic decisions regarding choosing investment options for heat savings and decreasing energy imports or increasing energy exports in pulp mills under market uncertainty was studied in Svensson et al. (2011).  The objective function of the developed model was to maximize the expected net present value of the investments. The decision variables were related to investment in heat saving and energy conversion technologies as well as distribution of the obtained heat from different energy conversion technologies. The uncertainty was considered in future energy prices and policy instruments through a scenario tree of five different combinations of several emission reduction policies, and electricity, lignin and bark prices. The original optimization model was a mixed-binary linear programming model and uncertainty was included using a multistage stochastic programming model. Each stage contained five years and the total planning horizon was assumed to be 30 years. Svensson & Berntsson (2011) introduced a methodology for making decisions about investment in new energy technologies in industrial plants. The main focus of the paper was to include uncertainty in the energy market. The authors used stochastic programming and scenario analysis to take into account the uncertainty in CO2 charge, and crude oil and electricity prices. They then applied the methodology to a pulp mill and presented examples of possible future investments to show the usefulness of the proposed approach.   Chen & Fan (2012) developed a mixed integer stochastic programming model to incorporate uncertainty in strategic planning of bioenergy supply chain systems. They considered bioethanol production, feedstock procurement, and fuel delivery in an integrated model to 36  minimize costs. The raw material included both agricultural and forest biomass. A two-stage stochastic programming model was developed and applied to a case study in California. Uncertainty was considered in available feedstock supply and fuel demand. A Lagrangian relaxation based decomposition solution algorithm, called the progressive hedging method, was used to reduce the computational effort needed to solve the stochastic model. This method decomposed a stochastic problem across scenarios by partitioning the original problem into manageable sub-problems. The first stage decision variables were refinery and terminal locations and sizes and the second stage decision variables were feedstock procurement and transportation, ethanol production and transportation. In the baseline scenario, four discrete demand scenarios with equal probabilities were considered and the results showed that the stochastic programming provided lower expected with lower variation total cost than deterministic solutions. In the second scenario, uncertainty in biomass supply was considered using ten scenarios of feedstock with equal probabilities and a fixed demand. The authors stated that the optimal solution of the bioethanol supply chain was not sensitive to the uncertainty in supply. This study did not integrate different sources of uncertainty within a single framework. It also ignored uncertainty in production yields and prices. In addition, the model had single annual time-step and did not consider multiple shorter time variations.  A two-stage stochastic MILP modeling approach was proposed by Kostina et al. (2012) to include uncertainty in the demand of an integrated ethanol-sugar supply chain. Different risk measures were studied in their model including value at risk, opportunity value and risk area ratio. Another MILP model with demand uncertainty was developed by Gebreslassie et al. (2012) which minimized the design and planning cost of biofuels network and risk simultaneously. Awudu & Zhang (2013) developed a stochastic production planning model for 37  a biofuel supply chain which included biomass suppliers, biofuel refinery plants and distribution centers. They incorporated uncertainty in the demand and price in a single period planning framework. Decision variables were the amount of raw materials purchased and consumed and the amount of products produced. Kazemzadeh & Hu (2013) determined the optimal design of biofuel supply chain with uncertainties in fuel market price, feedstock yield and logistics costs. They developed two stage stochastic programming models with two different objective functions (profit and conditional value at risk) and compared the results of the two models. In Giarolaa et al. (2013), an MIP model was developed to provide optimum design and planning decisions for a multi-period multi-echelon ethanol supply chain. The model had multi-objectives, considering both the economic and environmental performances as well as risk mitigation preferences. Moreover, uncertainty in feedstock cost and carbon cost was captured through developing a two-stage stochastic model. For controlling risks, two risk indices were considered: expected downside risk and value at risk.  Table 2-6 summarizes the previous studies, which incorporated uncertainty in the forest and bioenergy supply chains using the stochastic programming method.       38  Table 2-6: Summary of studies on stochastic programming of forest and bioenergy supply chains   Author/ Year Uncertain parameter Method Case Study Kim et al. (2011b) Price of the final product, the conversion yield ratios of the two conversion processes, maximum demand and biomass availability Two-stage stochastic programming Biofuel industry in Southeastern United States Svensson et al. (2011) Emission reduction policies, electricity, lignin and bark prices Multi-stage stochastic programming  Pulp industry in Sweden Svensson & Berntsson (2011)  CO2 charge, fuel and electricity prices Multi-stage stochastic programming Pulp industry in Sweden Chen & Fan (2012) Feedstock supply and demand Two-stage stochastic programming with decomposition Biofuel plants in California Kostina et al. (2012) Demand Stochastic programming Ethanol-sugar plants in Argentina Gebreslassie et al. (2012) Demand  Stochastic programming Biofuel plants in the US Awudu & Zhang (2013) Demand and price  Stochastic programming Biofuel plants Kazemzadeh & Hu (2013) Fuel market price, feedstock yield and logistics costs Two-stage stochastic programming models Biofuel plants Giarolaa et al. (2013) Feedstock cost and carbon cost   Two-stage stochastic model Ethanol plants  2.3.4 Robust optimization model Robust optimization has been used and applied in several fields of study, such as facility location and inventory management (Gülpı ar e  al. (2013), Solyali et al. (2012)), resource allocation and project management (Wiesemann et al. 2012), and in specific supply chain 39  optimization problems such as the refinery industry (Leiras et al. 2010). This method has also been used and applied in forest industry problems. Palma & Nelson (2009) used the robust optimization approach to incorporate uncertainty in volume and demand of two products over the entire planning horizon in a harvest scheduling decision making model. In their model, they assumed that the uncertain parameters were uniform and independently distributed within a symmetrical range of values. Their results showed that the robust optimization model with different protection levels provided lower objective function and infeasibility rates than the deterministic model. They then used robust optimization in a bi-objective planning model with random objective weights for forest planning problem (Palma & Nelson, 2010). The two objective functions were the amount of employment and the proportion of old forest through the planning horizon. Using the robust optimization method, no large changes in the weighted sum of the objectives were expected even when the weights changed over time. In both studies, they used Bertsimas formulation (Bertsimas & Sim (2003), (2004)). Kazemi Zanjani et al. (2010a) developed a robust optimization model based on Mulvey et al. (1995) formulation for sawmill production planning. Robust optimization was combined with a two-stage stochastic programming model and the uncertainty in raw material quality was included in the model. They used two variability measures: 1) solution robustness, which measured the variability of the recourse cost in the stochastic model for any of the scenarios, and 2) model robustness, which measured the feasibility of all scenarios. The result demonstrated the trade-off between variability in backorder/inve  ory cos  (pla ’s robus  ess) a d raw ma erial co sump io  a d expected backorder/inventory cost. Alvarez & Vera (2011) applied the robust optimization methodology to a sawmill planning problem with uncertainty in the yield coefficients associated with the cutting patterns. The authors employed Bertsimas formulation (Bertsimas 40  & Sim (2003), (2004)) and concluded that the solution provided by robust optimization is feasible for a large proportion of randomly generated scenarios with moderate reduction in the objective function value.  One of the by-products of the Kraft process in the pulp and paper industry is black liquor, which is used for energy production. Tay et al. (2013) considered uncertainties in raw material supply and product demand in integrated biorefineries using the robust optimization method based on Bertsimas formulation (Bertsimas et al., 2011). The model was mixed integer nonlinear programming with one time step. Different scenarios with associated probabilities were considered for supply and product demand. The results identified the optimum capacity of each process technology and its corresponding amount of biomass, intermediate and final products. Bredström et al. (2013) developed a formulation for robust optimization to be applied in the production planning problem with rolling time horizon. The method was based on the decomposition approach and recourse decision making framework so that some decisions were made after realization of uncertain parameters. A heuristic algorithm was proposed to solve this model iteratively. Their case study was a biofuel heating plant with uncertain energy demand over time. Carlsson et al. (2014) used the same approach in the pulp and paper industry considering uncertainty in customer demand. They concluded that using the robust optimization approach, building safety stock was not needed since the variation in the demand would be considered in the decision making process.  Table 2-7 summarizes the previous studies which incorporated uncertainty in the forest and bioenergy supply chains.   41  Table 2-7: Summary of studies on robust optimization of forest and bioenergy supply chain Author/ Year Uncertain parameter Case Study Palma & Nelson (2009)  Volume and demand of products Harvest scheduling in Canada Palma & Nelson (2010) Weight of objectives in a bi-objective optimization problem Forest planning in Canada Kazemi Zanjani et al. (2010a) Raw material quality Sawmill industry in Canada Alvarez & Vera (2011) Yield coefficients Sawmill planning in Canada Tay et al. (2013) Raw material supply and product demand Kraft pulp and paper industry Bredström et al. (2013) Energy demand Biofuel heating plant Carlsson et al. (2014) Demand Pulp and paper industry in Sweden  2.4 Discussion and conclusions Mathematical programming and optimization techniques have been used in the design and management of forest biomass supply chains. These models provided the optimum solution for decisions related to the network design including technology choices, plant size and location, storage location, mix of products and raw materials, logistics options, supply areas, and material flows. The objective functions included profit/cost, CO2 emissions, travel time, etc. Both deterministic and stochastic mathematical programming models were developed for forest biomass supply chains.  Deterministic models are necessary and helpful but not sufficient for capturing all aspects of forest biomass supply chains. Despite the interest and much effort in extending deterministic models, only a number of previous studies considered uncertainty in optimizing forest bioenergy supply chains. Understanding the problem characteristics, gathering sufficient data and choosing the appropriate methodology are important in developing stochastic models. The 42  appropriate methods to be implemented depend on the characteristics of the problem, data availability, form and type of uncertainties.  In the literature, sensitivity analysis and stochastic programming approach were used to deal with uncertainties in supply, demand, prices, conversion yields, carbon tax and emission reduction policies in forest biomass supply chains. None of the previous studies dealt with uncertainty in biomass quality, such as moisture content and heating value, and its effect on the produced energy and its costs. Moreover, most of the published papers did not consider supply chain planning in different time steps, and instead modeled the supply chain system in a single time-step optimization framework. The only study that provides a dynamic model is (Dal-Mas et al., 2011), which is related to a corn-ethanol supply chain. Therefore, the temporal uncertainty or seasonality of uncertain parameters in forest biomass power plant has not been studied in any of the previous studies.    43  Chapter 3 Deterministic model  3.1 Synopsis  In this chapter, an optimization model is developed to maximize the overall value of a forest biomass power plant supply chain and is applied to a real case study. The model has a one-year planning horizon with monthly time steps. It includes all parts of the supply chain from procurement, to storage, production and ash management. The effects of the quality (moisture content, energy value, ash content) of different types of biomass purchased and mixed in the storage area on the amount of generated ash, cost of ash handling, total production cost and total amount of generated electricity are considered in the model. The results of the model are analyzed and compared to the power plant situation in 2011. Moreover, scenario and sensitivity analyses are performed to evaluate the impact of variations on the model solution.  3.2 The power plant supply chain The supply chain of a power plant, including procurement of different raw materials from different suppliers, storage of biomass, electricity production and ash management, is considered in this research. Figure 3-1 shows the supply chain components of the considered power plant.      Figure 3-1: Schematic of supply chain configuration of a forest biomass power plant  Wood Residues Bark  Sawdust Shavings Roadside logging debris Storage Production  Ash  Electricity  Supplier 1  Supplier n ...... 44  Details of each component are explained below. Raw materials: The forest biomass can be supplied from forest residues, by-products of forest product mills, or fast growing crops grown specifically for energy purposes (Demirbaş, 2001). Forest residues include branches and tops left in the harvest areas after the logs have been transported to wood manufacturing facilities, as well as small diameter and infected trees not suitable for lumber production. By-products of forest product mills include wood chips, sawdust, bark and shavings (Demirbaş, 2001). Poplar and willow are examples of fast growing tree-crops grown specifically for energy purposes (Rockwood et al., 2004).  In addition to the long term availability of biomass, its quality is an important factor in economic feasibility of bioenergy projects and the amount of energy generated from it. The quality of biomass depends on a variety of factors such as Higher Heating Value (HHV), moisture content, physical, chemical and thermal properties. HHV is the amount of heat released from complete combustion of dry biomass under standard conditions. Different types of biomass (e.g. bark, sawdust, shavings, etc.) and different species have different HHV’s. The moisture content affects the biomass heat content since energy has to be used to evaporate water at the beginning of the combustion process (Saidur et al., 2011). Density, porosity, size and shape of biomass are other important physical properties of biomass that impact its utilization as fuel (Saidur et al., 2011). Different energy conversion technologies require different particle size ranges. For combustion, the particle size of biomass should be between 0.6 cm (Demirbaş, 2005) and 10 cm (Personal communication with power plant managers). Size and shape can be modified and improved through pre-processing operations such as chipping which can take place in the forest, a sort yard or at the power plant. Bark, sawdust and shavings usually need minor screening and chipping, while larger sized raw materials, such as 45  roadside logging debris, need to be chipped before they can be used in direct combustion. Biomass chemical properties include the ratio of chemical elements, such as carbon, hydrogen, oxygen, nitrogen, in biomass and its structural components, which is the amount of cellulose, lignin and hemicelluloses. These properties are different in different species and impact the HHV of the biomass as mentioned earlier. However, the impact of chemical composition on combustion is less significant compared to moisture content (Demirbaş, 2007). Important thermal properties are specific heat, thermal conductivity, and emissivity and vary with the moisture content and temperature (Saidur et al. (2011), Lehtikangas (2001), Demirbaş (2003), Demirbaş (2005)). Transportation: Different transportation modes, e.g. trucks, railcars, vessels and barge, can be used for transporting biomass to the power plant. However, forest biomass power plants are usually supplied from local suppliers and therefore road transport is more likely to be used among other possible methods. As forest biomass density is relatively low (400 and 900 kg/m3 (Demirbaş, 2001)), its transportation to the power plant requires a large number of trucks which increases the delivered cost of biomass and the complexity of its logistics system. Biomass can be transported directly to the power plants, or stored at a satellite storage facility and used later. The transportation cost of biomass depends on the power plant size, raw material availability, average transportation distance, biomass density, carrying capacity, and the travelling speed. Transportation and handling costs usually represent a significant proportion of the total biomass delivered cost (as high as 50% in some cases (Allen et al., 1998)). Storage: Storage is an important issue in a forest biomass supply chain. The storage site can be located either in the forest, at the power station or at an intermediate point. Usually, when 46  forest biomass is kept in a pile, it generates internal heat over time which is the result of respiration of living cells in wood (Fuller, 1985). The internal heat generated during storage makes the biomass more homogenous and warmer which makes it burns more easily. Thus, to improve the quality of biomass, it is better to keep it in storage for a period of time (1-2 months) (Personal communication with power plant managers) and not to let the amount of biomass in storage drop below a certain level. The storage level has to be kept within certain limits, which depend on the size of power plant. However, if the storage amount is higher than a certain level, its handling cost increases incrementally since there is a need for an extra operator and material handling equipment. The risk of fire and biomass deterioration are also higher when it is kept in large piles (Fuller, 1985).  Combustion: Direct combustion is a way to convert the energy embedded in biomass and derived from the sun and stored in biomass through photosynthesis into other forms of energy, mainly heat and then electricity (Demirbaş, 2001). Combustion is defined as “a series of chemical reactions resulting in carbon and hydrogen deoxidization” (Demirbaş, 2005). Biomass elements, moisture content, and air are critical components of wood combustion. The products of these reactions include CO, hydrocarbons, oxides of nitrogen, sulfur and inorganic species such as the alkali chlorides, sulfates, carbonates and silicates (Saidur et al., 2011). When wood is combusted, mass losses occur which is related to the combustion temperature. This can be depicted in a plot, called the burning profile, which shows the rate of weight loss against temperature. The first peak in the burning profile is related to the release of moisture content and a small amount of other absorbed gases. Then, as the temperature increases to 175°-225°, other volatiles chemicals start to be released and ignite. Then, the rate of mass losses increases drastically up to temperatures between 325° and 425°, when the mass losses 47  start to decrease. This decline will continue and eventually the weight will be almost constant. These temperatures are different for different types of biomass (Saidur et al., 2011). Production: The electricity production process depends on the technology used and the layout of the power plant (Rentizelas et al., 2009b). There are different biomass combustion technologies for energy generation such as fixed bed combustion, fluidized bed combustion and pulverized bed combustion (Saidur et al., 2011). The scale of forest biomass energy conversion plants can vary from very small scale (for domestic heating) up to a scale in the range of 100 MWe1. The main restriction on the power plant size is the availability of the local feedstock, which makes it difficult to have biomass power plants larger than 25 MW. If dedicated feedstock supplies are available, larger power plants can be built producing 50–75 MW. The net electrical efficiency depends on the scale of the power plant and ranges from 20% to 40%. Small sized biomass power plants have low efficiencies, while the large sized biomass power plants can be as efficient as fossil fuel systems, however, access to high volume of biomass throughout the year and high cost of transporting low density material are the issues with these types of plants (Demirbaş, 2001). In this study, the overall configuration of the forest biomass power plant is based on a conventional power cycle used in typical thermal utility plants. The power plant uses forest biomass which is transferred by conveyors to a boiler where it is burned to generate heat for steam production. The steam is then transferred into a turbine, where the thermal energy is converted to electrical power. The exhaust steam is converted to water by a condenser and used in the system again. The water is re-used for almost seven cycles and then it is discharged to the sewer network. Other equipment pieces might include a high voltage step up transformer, a                                                           1 Megawatt Electrical 48  solid fuel handling system, an ash removal/handling system and other steam cycle auxiliary pieces of equipment, multiple cyclones and an electrostatic precipitator.  Ash management: Ash management is an important task in a direct combustion power plant. Generally, wood ash properties are related to several factors such as: species of tree or shrub, part of the tree or shrub (bark, wood, and leaves), type of waste (wood, pulp, or paper residue), combination with other fuel sources, type and quality of soil, weather conditions and combustion process (Saidur et al., 2011). Forest biomass ash generally contains calcium (Ca) and potassium (K) (Saidur et al., 2011). Two kinds of ash are generated in the production process: one is derived from soil and rock contamination (called bottom ash) and the other one is derived from the minerals in the foliage or wood (called fly ash). Ash disposal is a challenge for most of the power plants and has economic, environmental and social costs for the power plant. High ash content biomass is less desirable as a fuel (Demirbaş, 2005). For example, sawdust has lower ash content than bark and logging residues (Lehtikangas, 2001). In order to design and manage an efficient supply chain for biomass power plants, all of the above mentioned components of the supply chain should be considered. It is important that the power plant receives the required amount of biomass at the right time with a competitive price to meet the electricity demand and maximize the profit. This can be determined by an optimization model. An optimum configuration of all processes in the supply chain can help decision makers run a better operation. The optimality is usually related to cost/profit, however other factors, such as fuel consumption, greenhouse gas emissions and customer satisfaction, are also important in the supply chain management. Usually there is a trade-off between these different objectives, which adds its own challenges to the model.  49  3.3 The optimization model  The model presented here considers multiple procurement sources, several storage options, different types of forest fuel, and several time periods. The overall objective of the optimization model developed in this study is to provide estimates of the amount of biomass to be purchased, stored and consumed in each month during a one-year planning horizon. Here, the main component of the model, decision variables, constraints and the objective function are defined. Table 3-1 shows the notation and definition of different sets and decision variables used in the developed model.   Table 3-1: List of indices and decision variables used in the optimization model Indices  Product type p∈{Bark, Sawdust, Shavings, Roadside Logging Debris (RLD)} Supplier s ∈ { Supplier1, ..., Supplier8} Time steps t ∈ {Jan, Feb, ..., Dec} Decision Variables        Average energy value of the mix of biomass in storage in month t (MW/green tonnes)    Amount of mixed biomass consumed in month t to produce electricity (green tonnes)    Amount of electricity generated in month t (MWh)       Amount of biomass purchased from supplier s in month t (green tonnes)    Amount of mixed biomass stored in month t (green tonnes)    Binary variable, 1 if storage is higher than the storage upper limit (SUL) in month t    Binary variable, 1 if storage is less than the storage lower limit (SLL) in month t Table 3-2 shows the notation and definition of different sets and decision variables and parameters used in the developed model.    50  Table 3-2: List of parameters used in the optimization model Model Parameters       Average ash content of mix of biomass (%)          Cost of ash handling ($/green tonnes)      Unit cost of biomass type p produced at supplier s ($/green tonnes)     Incremental cost of chemical used for power production ($/MWh)    Electricity demand from client in month t (MWh)             ys em’s overall efficie cy (%)         Electricity price in month t ($/MWh)        Energy value of biomass type p purchased from supplier s in month t (MWh/green tonnes)         Higher heating value of biomass type p purchased from supplier s in month t (MWh/dry tonnes)         Maximum biomass available from supplier s in month t (green tonnes)      Maximum absolute storage capacity (green tonnes)        Moisture content of biomass type p purchased from supplier   in month t (%)     Penalty cost if storage is above the storage upper limit ($)     The percentage of reduction in biomass quality if storage level is below SLL           The ratio of biomass type p produced at supplier s in month t (%)      Incremental cost of sewer used for power production ($/MWh)     Storage lower limit. If the storage level is below this level, the biomass quality decreases since there is not enough time for it to be mixed and produce internal heat (green tonnes)     Storage upper limit (green tonnes)         Target storage for the last month (green tonnes)       Biomass transportation cost for supplier s in month   ($/green tonnes)    Incremental cost of water used for power production ($/MWh) The objective function is to maximize the profit (Equation 3.1).                                                                                                                                                           (3.1) 51  Revenue from selling electricity to customer(s) is calculated by multiplying the electricity unit price by the amount of electricity produced by the power plant (Equation 3.2).           ∑                                                                                                              (3.2) Biomass procurement costs include the cost of purchasing biomass and its transportation costs (Equation 3.3). The biomass purchase cost is calculated by multiplying the biomass price by the ratio of each biomass type produced in each supplier and the amount of biomass purchased. The transportation costs depend on the distance between suppliers and the power plant, and the amount of purchased biomass. It is assumed that no chipping is done in this power plant. If this is not the case, the model can easily be modified to include that.                           ∑   ∑                                                       (3.3) Ash handling cost is equal to the average ash content of all biomass multiplied by the amount of biomass consumed for power production multiplied by the cost of handling ash (Equation 3.4). It should be noted that the average ash content is considered in this equation which can be substituted by more detailed parameter such as ash content of each type of biomass if enough data were available. Usually different types of wood are burnt together; therefore, the ash content of the mix of biomass is available.                      ∑                                                                               (3.4) The penalty storage cost is calculated by multiplying a fixed penalty cost by a binary variable if the decision variable related to storage is more than a certain level (SUL) (Equation  3.5).                      ∑                                                                                          (3.5) 52  The production cost contains water, sewer and chemical costs multiplied by the amount of power produced by the power plant (Equation 3.6).                             ∑                                                                      (3.6) Constraints of the model are listed below:  In month t, biomass purchased from supplier s has to be less than or equal to the maximum biomass produced by that supplier.                                                                                                          [For all s, t]        (3.7) Storage level in each month has to be less than or equal to the absolute maximum storage levels.                                                                                                              [For all t]         (3.8) Equation 3.9 implies the storage in the final month of the year has to be equal to a target storage level. This constraint guarantees a desired initial storage condition for the next year that can be set by managers.                                                                                                                                   (3.9)  Mass balance equation is considered in Equation  3.10, which indicates that storage in month   is equal to storage in month (t-1) plus the total biomass purchased from all suppliers minus biomass consumption in month t.         ∑                                                                                        [For all t]         (3.10) 53  Equation 3.11 shows that the power generated by the power plant in each month has to be equal  o  he cus omer’s mo  hly dema d.                                                                                                                [For all t]         (3.11)  Equation 3.12 relates the amount of electricity produced in each month to the energy value of biomass utilized in that month, efficiency of the system and the reduction in biomass quality if the storage level is less than the SLL (1-QRF×Zt).                                                                      [For all t]         (3.12)                                                                                The average energy value of biomass in storage in month t is calculated based on the weighted average of energy values of biomass purchased from suppliers in that month and the average energy value of stored biomass in month (t-1) as shown in Equation 3.13.         ∑      ∑                                     ∑                [For all t]        (3.13)       AveEVt is a decision variable since it is calculated based on variables Fs,t and St-1. EVs,p,t is calculated based on higher heating value HHVs,p,t and the corresponding moisture content MCs,p,t of biomass as implied in Equation  3.14 (Bowyer et al., 2007).                                                                                    [For all s, p, t]         (3.14)  And finally, all the continuous variables are non-negative as shown in Equation 3.15.                                                                                                    [for all t]         (3.15) 54  When the model is formulated, additional constraints have to be added for definition of binary variables Yt and Zt. Yt is defined as:    {                                                                                                                                                   (3.16) The following constraints (Equations 3.17 and 3.18) are equivalent to the above definition and replaced it in the model formulation, where M is a sufficiently large number.                                                                                       [For all t]          (3.17)                                                                                                  [For all t]          (3.18) The same constraints have to be added to the model for Zt. Zt is defined as:     {                                                                                                                                                   (3.19) It can be seen that the model is a nonlinear mixed integer programming (MINLP) model since Equations 3.12 and 3.13 contain non-linear terms and the model contains both continuous and binary variables.  MINLP is a combination of two theoretically difficult to solve categories of problems particularly for large scale problems: mixed integer programs (MIP) and nonlinear programs (NLP). Therefore, it is not straightforward to solve MINLP problems and get precise results (Bussieck & Pruessner, 2003). However, several methods have been developed and improved in the past few years which make it possible to solve MINLP problems more precisely under special circumstances (convexity, etc.), i.e. terminate with a guaranteed optimal solution or 55  prove that no such solution exists (Bonami & Gonçalves, 2012). The solution methods include branch-and-bound method (Dakin, 1965), Benders decomposition (Geoffrion, 1972) and Outer Approximation (OA) algorithm (Duran & Grossmann, 1986).  In this study, the Outer Approximation (OA) algorithm was used since it was provided in the AIMMS software package (AIMMS. Paragon Decision Technology). The idea in this algorithm is to switch between solving the nonlinear programming sub-problems and the relaxed versions of a mixed integer linear master program for a finite sequence. The main assumption in the Outer Approximation algorithm is the convexity of the nonlinear sub-model.  3.4 Case study  The above model was applied to a case study located in Canada to help maximize profit and manage supply chain efficiently. The supply chain is the same as what was described in section 3.3. Most of the biomass used by the power plant was supplied from residues from local sawmills at a low cost. These inexpensive sources of raw material started to decline in recent years due to the economic downturn that resulted in the closure of some of the mills in the area, and the competition for biomass from pellet mills. Therefore, biomass prices have increased and the power plant was forced to use other sources of biomass such as logging debris. Biomass from different sources has different quality and cost which affect the costs and efficiency of the operation.  Table 3-3 shows some of the characteristics of the power plant.    56  Table 3-3: Characteristics of the case study Number of suppliers with a fixed (long term) contract 4 Number of suppliers without a fixed contract More than 50 suppliers Average ash content 8% Average moisture content (MC) 32.4% (range: 10.2-46.7) Average higher heating value (HHV) 5.53 MWh/tonnes (range: 4.06-5.89) 8565 BTU/lb (range: 6275-9110) Efficiency 30% Range of available biomass from suppliers with a fixed contract  0-12,861 (green tonnes per month) Range of available biomass from suppliers without a fixed contract  0-18,900 (green tonnes per month) Some of the specifications of the case study supply chain are presented below.  Raw material: The feedstock used in the power plant includes bark, sawdust, shavings and roadside logging debris (RLD):  ∈                                                     All types of biomass are transported to the power plant by trucks and then kept in storage and mixed together before combustion. There is a long list of possible suppliers for the power plant with different contracts, terms, conditions and prices. For some suppliers, if they are in the operation, the power plant has to buy their biomass. These mills are located close to the power plant and their biomass is relatively inexpensive because of the short transportation distance and long term contracts. However, if these mills decide not to operate, there will be a raw material shortage for the power plant. Therefore, the amount of biomass that can be purchased from these suppliers is not exactly known in advance. There are other suppliers which can be considered as ad-hoc suppliers which have no contract with the power plant and usually 57  provide more expensive biomass. Therefore, Equation 3.7 can be broken into two sets of constraints:                                                 [For all s in suppliers with a fixed contract, t]          (3.20)                                            [For all s in suppliers without a fixed contract, t]          (3.21) Storage: The capacity of the storage for biomass is limited and known. Storing more than a certain level of biomass forces the power plant to hire an additional operator and use additional material handling equipment. In addition, there is another upper storage limit, above which additional operator and additional handling material equipment are needed, and the risk of fire in the storage facility increases. Therefore, in the model if storage is more than the first upper level in month t, a binary variable called Xt becomes 1 and if it is more than the second upper storage level, another binary variable called Yt becomes also 1. Then, the storage penalty cost will be:                      ∑                                                                                 (3.22) If the level of the storage decreases below a certain level called the minimum storage level, the quality of biomass is decreased by a certain amount (6% in this case) and there is a risk of biomass shortage (Equation 3.12).  Demand: The power plant has a long term contract with a customer to provide a fixed amount of electricity per year, called the firm load. The rest of the production, named the surplus load, can be produced and sold to the same customer whenever it is profitable. The firm load demand has to be met at all times, while the power plant has the option to not produce the surplus load. 58  There is also a total amount for the firm load demand in a year. Usually, the power plant decides whether or not to produce the surplus load in the beginning of the year and informs the customer about its decision. If the power plant decides to produce the surplus load, the first fixed amount of production in each hour is considered as the firm load and the rest is the surplus load. When the production covers the firm demand, the rest of the production is considered as the surplus demand. If the power plant decides to not produce the surplus load, the total production in each hour is considered as the firm load and the power plant will be shut down for the rest of the year after meeting the total firm demand.  To model different demand types, several parameters and decision variables have to be added explained in Table 3-4.  Table 3-4: Variables and parameters of the case study Symbol  Type  Definition     Continuous Variable  Total electricity generated in each month (MWh)    Binary Variable 1 if surplus electricity is produced in a year, 0 otherwise    Parameter Energy price for firm load ($/MWh)    Parameter  Total firm demand in one year (MWh)     Parameter  Amount of firm electricity generated in month t if the surplus is not being produced (MWh)    Parameter  Total surplus demand in one year (MWh)    Parameter Energy price for surplus load ($/MWh)     Parameter Number of working hours in month t Then, the following constraint (Equation 3.23) needs to be added to the model. This constraint implies that the electricity produced in each month has two forms. If the surplus load is being produced, it is equal to the total firm and surplus demand in one year distributed monthly based on the ratio of working hours of that month to the total working hours in one year. If the 59  surplus load is not being produced, the electricity demand in each month is equal to the firm load generated in each month.               ∑                                                                                (3.23)                                                                              The objective function also needs to be modified since the revenue from selling electricity to the customer is equal to the firm load demand multiplied by the power price for the firm load plus the surplus load demand multiplied by the surplus energy price multiplied by the binary variable if the surplus power is being produced.                                                                                                       (3.24)                                                            The optimization model for this case study includes Equations 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.15, 3.17, 3.18, 3.20, 3.21 and 3.23 (and corresponding constraints for definition of Zt) and the objective function is to maximize (3.24 – 3.3 – 3.4 – 3.6 – 3.22).  The MINLP model was solved using the AIMMS software (AIMMS. Paragon Decision Technology) and the Outer Approximation algorithm. The model had 260 variables (73 binary variables and 187 continuous variables) and 333 constraints. It took less than a minute to solve the problem using a 2.80 GHz CPU. The model was implemented with real data and validated by the power plant managers. It is notable that this model can be used for monthly decision making. It means that the model can be used at the beginning of the planning horizon (Jan) to indicate the optimal decision variables. However, if some parameters change later, the model can be updated using the new information, the previous month’s variables have to be fixed based on the real values and the model can be run again for the rest of the months.  60  3.5 Results  Table 3-5 shows the results of the optimization model for the power plant using the 2011 data. The profit from the optimization model is higher than the actual profit of the power plant in 2011. The actual total biomass procurement cost of the power plant in 2011 was $11M which was 15% higher than the optimum total biomass procurement cost resulted from the model. Table 3-5 also shows different costs components of the power plant. The biomass purchase cost accounts for more than 63% of the total cost. Transportation cost encompass 33% of the total cost which is relatively low compared to other studies (Allen et al., 1998) due to the power pla  ’s access  o biomass from local suppliers.  Table 3-5: Results of cost, revenue and profit for the optimization model (in $M)  Biomass Purchase Cost Transportation Cost Ash Handling Cost Firm Revenue  Surplus Revenue Total Profit Model results 6.08 3.18 0.32 21.5 4.72 15.03 Figure 3-2 shows the firm and surplus production profile in different months. The production in May is lower than that in other months due to scheduled maintenance.  Figure 3-3 shows the optimum decision variables for the amount of biomass purchased, stored and consumed in each month based on the 2011 data. The initial storage level was 72,500 green tonnes in 2011. The optimum storage profile declines until April, then increases in May due to the maintenance shut down in the power plant.  61    Figure 3-2: The amount of firm and surplus electricity production in each month  Figure 3-3: Optimum amount of biomass stored, purchased and consumed in each month based on the 2011 data 3.5.1 Scenario analysis In addition to the base case scenario considered in the previous section, three other scenarios are analyzed here. These scenarios are important for the power plant managers.   Scenario 1: One of the suppliers with a fixed contract shuts down its operation which is possible due to the economic situation (pessimistic scenario).  010,00020,00030,00040,00050,00060,000Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMWh FirmSurplus010,00020,00030,00040,00050,00060,00070,00080,000Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecGreen Tonnes StoragePurchaseConsumption62   Scenario 2: One of the previous suppliers with a fixed contract which was closed in the past couple of years resumes its operation (optimistic scenario).   Scenario 3: The power plant makes an investment in a new piece of equipment for ash recovery. Having a new ash recovery system with a capital cost of $2M, the ash generated in the power plant is screened and the unburnt parts are separated and sent back to the operation process. It has been estimated that this system will collect 20% of the ash and use it as a biomass source. It is assumed that the energy value of the unburnt part of the ash is the same as the average energy value of the biomass consumed in that month. Therefore, it has the potential benefit of providing more biomass and less ash for the power plant which reduces the ash disposal costs. Table 3-6 shows the total profit and the total purchase cost of biomass for different scenarios. The results show that Scenario 1 has a lower profit (24.48%) than the base case scenario. Scenario 2 has a higher profit than the base case scenario since the power plant has access to cheaper biomass from a supplier with a fixed contract. The amount of increase in profit is $3.27M (21.76%) which is the result of reduction in the biomass purchase cost. Scenario 3, investment in a new ash recovery system, has a higher profit than the base case scenario ($160,000, 1.06%). This means that investing in a new ash recovery system has economic and environmental benefits for the power plant. The increase in profit is partly from the reduction in the biomass purchase cost (3.46%), and partly from reduction in the ash handling cost (20.75%). The capital cost of the ash recovery equipment was converted to the yearly cost using a 2% interest rate and 10 years’ service life, and then the yearly cost was subtracted from the profit.   63  Table 3-6: Total profit and biomass procurement cost for four different scenarios   Base Case Scenario Scenario 1 Scenario 2 Scenario 3 Total Profit ($M) 15.03 11.35 18.30 15.19 Biomass Procurement Cost ($M) 9.26 12.94 6.00 8.94 Profit difference compared to the base case scenario 0 -24.48% +21.76% +1.06%  3.5.2 Sensitivity analysis  To evaluate the effect of variations in model parameters on the profit, sensitivity analysis was performed. The results of variation in profit are depicted in Figure 3-4. It can be seen that the impact of variation in some of the parameters, such as electricity price, higher heating value and maximum available biomass from suppliers with a fixed contract have a large impact on the profit. When these parameters increase by 20%, the increase in profit is between 20-35%. On the other hand, when they are reduced by 20%, the loss is 23-45%. Other parameters are: moisture content, biomass cost and transportation costs which have the opposite impact on the profit.  Increasing these parameters by 20%, decreases the profit by about 4-15%. The effects of variation in other parameters on the overall profit are low and can be considered negligible.  The sensitivity of the results to changes in initial storage level was also determined. The initial storage level was changed from 0 to 118,000 green tonnes, while the target storage level was set to the initial storage level. As shown in Figure 3-5, an initial storage level of 109,000 green tonnes generates the highest profit.   64   Figure 3-4: Variations in profit with  20% change in different parameters  Figure 3-5: Variations in profit for different initial storage levels -50%-40%-30%-20%-10%0%10%20%30%40%-30% -20% -10% 0% 10% 20% 30%Profit Difference  Change in Parameter Max biomass from supplierwith fixed contractMax biomass from supplierwithout fixed contractAsh contentElectricity priceQuality reduction factorHHVMCTransportation costWood cost02468101214160 20,000 40,000 60,000 80,000 100,000 120,000 140,000Profit ($M) Initial Storage (Green Tonnes) 65  3.6 Discussion and conclusions  In this chapter, a mathematical optimization model was developed to determine the amount of biomass to purchase from each supplier in each month, the amount of mixed biomass to burn and store in each month, and whether or not to produce extra electricity to maximize the total profit. The developed model was a mixed integer non-linear model, which was solved by the Outer Approximation algorithm using the AIMMS software package. The results for the specific case study presented here indicated that the profit from the optimization model was higher than that of the actual profit generated by the power plant in 2011 when  he compa y’s managers made decisions based on their experience. For instance, the optimum biomass procurement cost was $9.26M, which was 15% lower than the actual purchase and transportation costs of the power plant in 2011 ($11M). The model provided the optimum profile for biomass storage, purchase and consumption in each month to achieve maximum profit over a year. The model could be re-optimized and updated if input data varied from what had been used here. The power plant situation in 2011 was considered as the base case scenario and three other scenarios were evaluated in this study. The first one was a pessimistic scenario in which the production shutdown of a supplier with a fixed contract was investigated. The second scenario, an optimistic scenario, was related to the resumption of production at one of the previous suppliers with a fixed contract.  And finally, the third scenario investigated the investment in a new ash recovery system. This system collects and reuses the unburnt parts of the produced ash. Based on the results, the profit of the first scenario was 24% lower than that of the base case scenario, while the profit of second and third scenarios were 22% and 1% higher than that 66  of the base case scenario, respectively. The reason for having lower profit in the first scenario is that the power plant had no longer access to one of its supplier (located in the area and providing inexpensive biomass). Therefore, it needed to purchase more expensive biomass from ad-hoc suppliers. For the second scenario, it was other way around, meaning that an extra inexpensive source of biomass with a fixed long term contract was available for the power plant. Therefore, it resulted in having a higher profit than the base case scenario. In the third scenario, the ash handling cost reduced by 21%. Therefore, investing in a new ash recovery system could have economic benefit as well as environmental advantage for the power plant. This chapter addressed the first, second and part of the third objectives mentioned in 1.2. It provided an integrated framework for optimizing the supply chain of a forest biomass power plant. The developed model was then applied to a real case study. The model and its results were validated by the power plant managers. It was a mixed integer non-linear programming model with multiple time steps and captured seasonal variations in parameters particularity in biomass availability and quality. The quality of the mix of material in the storage was calculated, and the impact of any change in the quality of each biomass type from each supplier in each month on the solution could be examined. Therefore, it included the issue of quality in the modeling and provided optimum decisions on biomass purchase, storage and consumption in order to meet electricity demand in each month. The impact of changes in input parameters on the profit was determined using sensitivity analysis. Parameters that had higher impact on the model solution were also identified. However, uniform changes in input parameters were considered in sensitivity analysis. In other words, it was assumed that uncertainties in all parameters had the same nature and they could 67  have any value within a defined range with the same probability. This is not a realistic assumption since there are different sources of uncertainty in the system and some of the uncertain parameters are correlated with one another. Moreover, not all the possible realizations of uncertain parameters occur with the same probability. To capture these issues, Monte Carlo simulation is used to model the uncertainties in input parameters more realistically, as can be seen in the next chapter.    68  Chapter 4 Monte Carlo simulation 4.1 Synopsis  In this chapter, Monte Carlo simulation is used in combination with the optimization model developed in Chapter 3 to evaluate the impact of uncertainty in different parameters including biomass quality, availability and cost and electricity prices based on historical data. Different types of uncertain parameters are incorporated in the model. Historical data on biomass quality (moisture content and higher heating value) obtained from the power plant is used to generate fitted probability distributions. The Monte Carlo simulation and the optimization model provide a probability distribution for possible optimal solution that can be used for u ders a di g  he sys em’s behaviour i   he prese ce of u cer ai  y, ide  ifyi g  he risks a d opportunities in the supply chain, and making informed decisions.  4.2 Uncertainty and Monte Carlo simulation  Vose (2008) divided the uncertainty based on its origin into two categories: 1) uncertainty resulted from incomplete understanding of a system due to its complexity, and 2) uncertainty related to the random nature of a parameter, which was called “variabili y” by  he au hor. Approaches to be used for dealing with uncertainties in the model and actions to be taken in response to them vary given the source of uncertainty in stochastic parameters. While collecting more information is effective in expanding the knowledge about a random event, understanding a system or trying to modify it might be more useful in managing uncertainties resulting from system complexity. Thus, it is important to evaluate the impacts of these two types of uncertainty separately to realise the impact of each on the solution. This helps decision 69  makers identify the proper steps to take to mitigate the risks and take advantage of possible opportunities associated with variations in the system (Vose, 2008).  The results of the sensitivity analysis performed and presented in Chapter 3 showed that parameters with higher impact on the optimum solution were the monthly available biomass from each supplier (MaxFs,t), biomass MC and HHV, biomass cost and electricity prices. Availability and cost of biomass, as well as electricity price can be grouped together because they are affected by economic and market conditions. Moreover, due to interdependency between different forest sectors and diverge production structure of industries, variations in one part of this supply chain can propagate into the other parts. Therefore, availability and cost of biomass are impacted by variations in other forest product sectors such as lumber and pulp and paper sectors. On the other hand, uncertainty in biomass quality is related to the heterogeneous nature of wood resulting from the variations in its physical and chemical characteristics from different species and different parts of a tree (Bowyer et al., 2012). Biomass quality is also affected by external factors, such as growth conditions and climate, and can change during storage, production, and transportation. It impacts collection, transportation, handling, storage and the efficiency of conversion technology (Rentizelas et al., 2009b).  AIMMS software package (AIMMS. Paragon Decision Technology) was used to model Monte Carlo simulation along with the optimization model. Monte Carlo simulation is first performed for each group of the uncertain parameters separately to study their specific impact on the power pla  ’s profi . The , a  overall u cer ai  y a alysis is co duc ed by allowi g  he i pu  parameters to change over their respective probability distributions/ ranges simultaneously. The 70  probability distributions of the outputs, including total profit, monthly storage level, biomass consumption and biomass purchase, are recorded and analyzed.  4.2.1 Uncertainty in biomass quality  The feedstock for the power plant is provided by different suppliers. Suppliers either have a fixed contract or no contact with the power plant. Suppliers 1 to 4, which include 3 sawmills (Suppliers 1, 2 and 4) and a sawmill and a plywood mill (Supplier 3), have fixed contracts with the power plant. Suppliers 5 to 8 do not have fixed contracts with the power plan. Suppliers with no contract include a sawmill (Supplier 5) which produces bark and sawdust, a harvesting company (Supplier 6) that only provides roadside logging debris (RLD), a number of small suppliers that occasionally provide biomass to the power plant (Supplier 7) and some suppliers that are considered as ad-hoc suppliers and occasionally sell biomass to the power plant (Supplier 8). The biomass cost provided by Suppliers 7 and 8 are different from each other and the types and the amount of biomass from them are not known. Table 4-1 shows the product mix (based on weight percentage) and type of contract for the suppliers.  Table 4-1: Product type of suppliers and their contract type   Product Mix (%) Type of contract Supplier 1 Bark (29%), Sawdust (71%)  Fixed Supplier 2 Bark (100%) Fixed Supplier 3 Bark (41%), Sawdust (29%), Shavings (30%) Fixed Supplier 4 Bark (63%), Sawdust (37%)  Fixed Supplier 5 Bark (70%), Sawdust (30%) Flexible Supplier 6 Roadside logging debris (100%) Flexible Supplier 7 Uncertain  Flexible Supplier 8 Uncertain Flexible 71  Based on the historical data received from the power plant, the best fitted probability distributions are determined for MC and HHV of biomass using the Stat::Fit software.   Moisture Content  Suppliers 1 to 5 produce bark, sawdust and shavings. The operation process of these suppliers takes place in closed areas year-round. Therefore, it is assumed that the seasonality does not impact the distribution of MC of their products, but the MC of different products varies. In contrast, RLD provided by Supplier 6, is collected and stored at roadside before being delivered to the power plant and is affected by seasonality. For Suppliers 7 and 8, the product types are unknown, so, only the seasonality effect is considered for their biomass MC.  Table 4-2 shows average and standard deviation as well as the fitted probability distribution and its parameters for MC of bark, sawdust and shavings from Suppliers 1 to 5. Table 4-2: Average and standard deviation of bark, sawdust and shavings MC for Suppliers 1 to 5   Average MC (%) Standard Deviation (%)  Fitted Probability Distributions Bark 30.0 8.7  Weibull (1.8,13,18.6) Sawdust 34.8 8.4  Beta (4.2,5.9,16.0,61.1) Shavings 11.5 11.3  Triangular (0.01,2.3,45.2) Figure 4-1 illustrates the histogram and probability distribution fitted to the MC of bark, sawdust and shavings.  72  Figure 4-1: Histogram and probability distribution of MC of bark (a), sawdust (b), shavings (c) Table 4-3 presents the average and standard deviation as well as the fitted probability distribution and its parameters for biomass in different months for suppliers 6 (RLD only). It should be noted that harvesting operation is not done in the area in May so there is no RLD available for that month. Figure 4-2 shows the histogram and probability distribution fitted to the MC of RLD in different months.      a b c 73  Table 4-3: Average and standard deviation of biomass MC for Supplier 6   Supplier 6 (RLD) Fitted Probability Distributions Ave MC (%) SD (%) January 26.6 5.6 Triangular (0.9,13.8,33.5) February 36.2 8.4 Gamma (2.8,21,5.41) March 33.7 9.1 Weibull (2.6,10.0,26.4) April 45.5 16.6 Beta (0.9,0.8,15.0,69.7) May - - - June 25.5 10.7 Triangular (0.3,7.3,51.5) July 23.7 5.6 Weibull (1.3,15.0,9.32) August 21.1 4.9 Beta (0.8,0.7,12.0,27.4) September 24.0 5.6 Triangular (0.5,10.3,38.7) October 27.1 10.1 Weibull (1.5,12.0,16.8) November 28.5 6.1 Weibull (1.5,19.0,10.4) December 35.6 7.6 Triangular (0.1,25.3,55.7)  a b 74        c d e f g h i j 75   Figure 4-2: Histogram and probability distribution for MC of RLD in January (a), February (b), March (c), April (d), June (e), July (f), August (g), September (h), October (i), November (j), and December (k)  Higher Heating Value  Table 4-4 shows the average, standard deviation, the fitted probability distribution and its parameters for HHV of different types of biomass received by the power plant from different suppliers. It was not possible to fit a distribution for the HHV of bark and shavings because of the low number of data points, therefore, a range was considered for their variations.  Table 4-4: Average and standard deviation of HHV for different biomass types  HHV Average  Standard Deviation Fitted Probability Distributions  Bark 8534 BTU/lb2       5.51 MWh/tonnes 670 BTU/lb       0.43 MWh/tonnes Uniform (7570.0,9110.0) (BTU/lb) Uniform (4.89,5.89) (MWh/tonnes)   Sawdust 8537 BTU/lb 5.51 MWh/tonnes 325 BTU/lb       0.21 MWh/tonnes Uniform (8050.0,8960.0) (BTU/lb)       Uniform (5.20,5.79) (MWh/tonnes)   Shavings 8517 BTU/lb 5.50 MWh/tonnes 254 BTU/lb       0.16 MWh/tonnes Uniform (8360.0,8810.0) (BTU/lb) Uniform (5.40,5.69) (MWh/tonnes)   RLD 8392 BTU/lb 5.42 MWh/tonnes 347 BTU/lb         0.22 MWh/tonnes Weibull (6.1,6960.0,1580.0) (BTU/lb) Weibull (6.19, 4.41,1.089) (MWh/tonnes)                                                             2 BTU/lb= 0.0006461MWh/tonnes k 76  Figure 4-3 show the histogram and probability distribution fitted for HHV of bark and RLD.     Figure 4-3: Histogram and probability distribution of HHV of sawdust (a) and RLD (b)  4.2.2 Uncertainty in biomass availability and cost and electricity price The amount of by-products generated by  he power pla  ’s suppliers depends on the production of their main products, which are mainly used in housing construction. In the past few years, many sawmills in the area were closed due to the real estate meltdown in the US. Thus, the power plant faced a shortage of raw material. Conversely, when the market is promising, the sawmills produce a large amount of biomass that the power plant is obliged to buy. The situation with a supplier without a fixed contract is less critical since there is no obligation for the power plant to buy their biomass. However, there still exists uncertainty in the available biomass that they can provide. Moreover, biomass procurement and transportation costs vary due to the competition for biomass from other sectors, such as the wood pellets sector, and as a result of variations in fuel cost. Electricity prices are also uncertain and have to be forecasted ahead of the time so that the power plant can decide about whether or not to produce surplus demand in the upcoming year.  a b 77  Uncertainty in electricity prices, and biomass availability and cost are considered together because they are all related to the economic situation and considered to be correlated. Based on the historical data, three electricity price scenarios (high, medium and low) with their corresponding annual available biomass from different suppliers were selected. The annual available biomass was then distributed over a year based on the monthly production profile of 2011. The Monte Carlo simulation model was run for each scenario with ±20% variation in monthly biomass availability and ±10% variation in biomass cost.  4.3 Results  The number of iterations for Monte Carlo simulation depends on the desired accuracy and confidence levels, and is calculated using Equation 4.1 (You et al., 2009):                                                                                                                                                           where N is the number of iterations, Z is the confidence interval of a two tailed normal distribution, σ is  he model’s ou pu  s a dard devia io  a d εμ is the desired marginal error. Based on the marginal error of 6%, confidence level of 95% (Z=1.96), and standard deviation of model results for 100 runs, the required number of iterations was estimated to be 1000.   Table 4-5 shows mean, minimum, maximum and standard deviation of profit and the feasibility rate obtained from the output of the Monte Carlo simulation-optimization model when MC and HHV were generated separately from their probability distributions. It can be seen that incorporating the variability in MC resulted in higher average and standard deviation for the profit compared to that of HHV. Variability in HHV changes the profit within a $280,000 78  interval with an average of $15.1M, while for MC, the variation in the profit is $700,000 with an average of $15.3M as shown in Figure 4-4 and Figure 4-5.  Table 4-5: Minimum, maximum, average and standard deviation of profit for considering uncertainty in different parameters Uncertain Parameter Mean, (Min, Max) of Profit ($M) Standard Deviation ($M) Feasibility Rate MC  15.3 (14.1-16.5) 0.35 (2%) 99.9% HHV  15.1 (14.6-15.6) 0.14 (1%) 99.9%  Figure 4-4: Histogram of profit when MC varies  Figure 4-5: Histogram of profit when HHV varies 0%5%10%15%20%25%30%14.10-14.1914.20-14.2914.30-14.3914.40-14.4914.50-14.5914.60-14.6914.70-14.7914.80-14.8914.90-14.9915.00-15.0915.10-15.1915.20-15.2915.30-15.3915.40-15.4915.50-15.5915.60-15.6915.70-15.7915.80-15.8915.90-15.9916.00-16.0916.10-16.1916.20-16.2916.30-16.3916.40-16.49Frequency Profit ($M) 0%5%10%15%20%25%30%14.60-14.6914.70-14.7914.80-14.8914.90-14.9915.00-15.0915.10-15.1915.20-15.2915.30-15.3915.40-15.4915.50-15.59Frequency Profit ($M) 79  As explained before, three scenarios were considered for electricity price and biomass availability and cost. The results of the Monte Carlo simulation-optimization model for the three scenarios are shown in Table 4-6.   Table 4-6: Results of Monte Carlo simulation-optimization model for scenarios of electricity price and biomass availability and cost Scenario Surplus Electricity Price ($/MWh) Mean Profit ($M) SD ($M) Feasibility rate (%) Low 35.00 14.4 0.1 100 Average 42.50 15.1 0.5 99.9 High 103.87 17.6 0.2 99.3 Figure 4-6 represents the profit histogram when availability and prices vary together for low, average and high electricity price scenarios. It can be seen that despite the low standard deviation of low electricity price scenario, its profit histogram is more bell-shaped, while the histogram of profit for average and high electricity price scenario are more skewed to higher profit ranges.  0%10%20%30%40%50%60%70%14.30-14.3914.40-14.4914.50-14.5914.60-14.6914.70-14.7914.80-14.8914.90-14.9915.00-15.09Frequency Profit ($M) a 0%10%20%30%40%50%60%70%13.80-13.8913.90-13.9914.00-14.0914.10-14.1914.20-14.2914.30-14.3914.40-14.4914.50-14.5914.60-14.6914.70-14.7914.80-14.8914.90-14.9915.00-15.0915.10-15.1915.20-15.2915.30-15.3915.40-15.49Frequency Profit ($M) b 80  Figure 4-6: Histogram of profit when electricity price and biomass availability and cost vary for  a) low, b) average and c) high scenarios  Table 4-7 provides the probability of having profit within a certain range obtained from the Monte Carlo simulation-optimization model. The last row represents the results for the case that uncertainty is included in all parameters. The risk of having profit less than $14M for the low electricity price scenario is 99%. For the average electricity price scenario, it is more likely to have profit between $14M and $15M, while in the high electricity price scenario the profit is always more than $17M. When variation and uncertainty in all parameters are considered, the risk of having a profit less than $14M is 6%, while there is 32% chance of having a profit higher than $17M. Most of the time (62%), the compa y’s profi  is be wee  $14M a d $16M. Because variations in biomass cost and availability and electricity prices affect the profit of the power plant significantly and increase the risk of having low profits when electricity prices are low, the power plant managers should pay special attention to the contract details with suppliers and forecast of electricity prices.  The outputs of the Monte Carlo simulation-optimization model show that in order to have a profit higher than $15.5M, the average MC should be less than 28% for bark, 34% for sawdust, 10% for shavings and 25% for RLD. If the power plant desires a profit higher than $15.0M, the 0%10%20%30%40%50%60%70%17.30-17.3917.40-17.4917.50-17.5917.60-17.69Frequency Profit ($M) c 81  MC level will be 31% for bark, 41% for sawdust, 24% for shavings and 37% for RLD. This indicates that drying of biomass in order to reduce its moisture content level could increase the annual profit by $0.5M. Table 4-7: Probability of having profit within different ranges when considering uncertainty in different model parameters Uncertainty in Pr.(Profit< $14M ) Pr.($14M<Profit <$15M) Pr.($15M<Profit <$16M) Pr.($16M<Profit <$17M) Pr.(Profit >$17M) MC 0 0.17 0.81 0.02 0  HHV 0 0.27 0.73 0 0 Availability and Prices (Low)  0.99 0.1 0 0 0 Availability and Prices (Average) 0.01 0.99 0 0 0 Availability and Prices (High) 0 0 0 0 1 All parameters 0.06 0.51 0.11 0 0.32 Table 4-8 represents ranges of annual biomass purchase from suppliers without a fixed contract, biomass consumption and monthly storage levels from the Monte Carlo simulation-optimization model. It can be seen that when variability in HHV is included, total consumption and total biomass purchase from suppliers without a fixed contract have a higher average and a lower standard deviation compared to the case when variability in MC is considered. It is also indicated that the maximum monthly storage never goes beyond the storage upper level (109,000 green tonnes), therefore, the penalty storage cost is not active in any cases. For the model with uncertainty in availability and prices and also all parameters, there is a risk of having monthly storage lower than the minimum storage level (45,000 green tonnes), and consequently lower biomass quality.  82  Table 4-8: Ranges of biomass purchase from suppliers without contract, biomass consumption and storage levels when considering uncertainty in different parameters (1000 green tonnes) Uncertainty in  Annual Consumption Annual purchase from suppliers without a fixed contract Monthly storage Average SD Average SD Average Min Max MC 483 5 118 5 56 45 89 HHV  488 3 122 3 56 45 94 Availability and prices (Low) 478 1 76 1 57 38 68 Availability and prices (Average) 486 1 119 1 51 33 63 Availability and prices (High) 485 1 140 1 59 45 74 All parameters  489 6 117 25 56 34 102  4.4 Discussion and conclusions In this chapter, uncertainty is studied in the mixed integer non-linear programming model of a forest biomass power plant supply chain through developing a Monte Carlo simulation-optimization model in order to determine the range of variations in profit due to variations in input parameters.  The results showed a lower standard deviation for profit when variation in higher heating value (HHV) was included compared to the case when variation in moisture content (MC) was included. Therefore, it could be concluded that the impact of variations in MC on the optimum profit was more severe than the variation in HHV. Moreover, variations in MC of biomass resulted in the highest variation in the annual consumption and annual purchase of biomass from suppliers without a fixed contract. This was in contrary with the result obtained from sensitivity analysis which showed that variations in HHV change the profit more than 83  variations in MC. The reason for that was that in sensitivity analysis the HHV and MC changed within a same range (±20%), while variations in the historical data showed different ranges with different probability distributions. These results showed that the Monte Carlo simulation provided more realistic results that might differ from those ones obtained from sensitivity analysis.  It was concluded that to have a profit of more than $15.0 M, the MC had to be less than 31% for bark, 41% for sawdust, 24% for shavings and 37% for road logging debris (RLD). Reducing MC to less than 28% for bark, 34% for sawdust, 10% for shavings and 25% for RLD would increase the profit to $15.5 M, if all other parameters of the model remained the same. This information would be helpful for the power plant supply manager when making supply contracts and decisions on the drying of biomass because the amount of profit earned when biomass was dried and had lower MC was shown.  The results indicated that when the electricity price was high (103.87 $/MWh), the profit probability density function was more skewed to higher profit, while when the electricity price was low (35.00 $/MWh), the profit distribution was more bell-shaped. This means that higher electricity prices mitigate the impact of variations in cost and availability of biomass. The results also showed that when all the uncertain parameters fluctuated at the same time, the profit ranged between $13.4M and $17.9M with an average of $15.5M. Therefore, uncertainty in the real system would resul  i  $4.5M varia io  i   he power pla  ’s profi .  Uncertainty in biomass availability and cost and electricity prices, not the variability in biomass quality (MC and HHV), resulted in the risks of having a profit lower than $14M and a storage level below the minimum storage level, as well as the opportunity of having a profit 84  higher than $17M.  Therefore, to mitigate the risks of having low profit and low storage levels, it is needed to control uncertainties in the availability and prices.  This chapter addressed objective 3 of the thesis mentioned in 1.2. Historical data on important uncertain parameters, which were identified previously through sensitivity analysis in 3.5.2 were analysed here. Uncertain parameters with different sources were grouped together and the Monte Carlo simulation was implemented for each group as well as for all of them at the same time. The profit distribution was determined when uncertain parameters varied based on their distribution. Therefore, the output of this model represented the model behaviour when uncertainty occurred and risks associated with uncertainties.  The Monte Carlo simulation method is a post-optimization approach, which means that the decisions have to be made before this model is implemented and it only provides possible changes in the decision variables and the profit in response to uncertainty. Taking uncertainty into account when modeling the supply chain is a challenging task because the model becomes complex and sometimes it may become computationally intractable especially if uncertainty is included in different parameters simultaneously. Therefore, in the next chapter, uncertainty in only one parameter, biomass availability, is included in the optimization model.   85  Chapter 5 Stochastic programming 5.1 Synopsis  So far, the impact of uncertainty on model solution has been evaluated through sensitivity analysis and Monte Carlo simulation which are post-optimization approaches. In this chapter, uncertainty is incorporated in the optimization model during the decision making process. This is done by considering uncertainty in the monthly available biomass from different suppliers during the planning horizon. The mixed integer non-linear programming model is first reformulated into a mixed integer programming model. This enables a direct formulation of a linear stochastic programming model. A two-stage stochastic programming model is developed to maximize the expected profit of possible scenarios of available biomass. Two risk measures are also included in the model to make the results more stable. The trade-off between maximizing the profit and minimizing the risk is then evaluated.  5.2 The mixed integer programming model of the power plant supply chain To facilitate incorporating uncertainty in the model, a new mixed integer linear programming (MILP) model is proposed in this chapter. The main difference between the formulation of MILP model in this chapter and the one developed in Chapter 3 is in the definition of the storage and consumption variables. The notation of these variables is presented in Table 5-1.  Table 5-1: Decision variables of the linear programming model Decision Variables          Amount of biomass stored in month   from supplier   (green tonnes)      Amount of biomass consumed in month   from supplier   (green tonnes)  86  The mixed integer linear model is as follows:  Objective function:                                ∑ (∑                        )          ∑            ∑        ∑                          ∑       (5.1)  Here, Profit equals revenues minus costs as shown in Equation 5.1. Ash handling cost has changed to ∑t AshHC × AshC × ∑s Cs,t.            The changes in the constraints are as follows:  The total storage level of biomass received and stored from all suppliers in month t has to be less than or equal to the maximum storage level. Therefore, Equation 3.8 is changed to Equation 5.2. ∑                                                                                                           [For all t]        (5.2) In the last month of the year, the total storage level of biomass from all suppliers has to be equal to a target storage level. Therefore, Equation 3.9 is changed to Equation 5.3.   ∑                                                                                                                              (5.3)  The storage level of biomass received from supplier   and stored in month t is equal to the storage level of biomass received from supplier s and stored in previous month (t-1) plus biomass purchase minus biomass consumption for the same supplier and month. For the first time step, the initial storage level for each supplier has to be indicated. Therefore, Equation 3.10 is changed to Equation 5.4. 87                                                                                                   [For all s, t]            (5.4) In the MINLP model shown in Chapter 3, the decisions on purchase and consumption amounts were not separated for each supplier because the total initial storage level was provided in the obtained data. The initial energy value of the biomass in the storage was not provided and an initial value for it in the MINLP was assumed. In the MILP model developed here, the initial storage level for each supplier at the beginning of the planning horizon has to be known in the mass balance equation for January (Equation 5.4). Therefore, it is assumed that the share of biomass i   he i i ial s orage level for each supplier is  he same as each supplier’s share i   he total annual available biomass for the power plant in 2011. The initial average energy value in the MILP is then calculated using those ratios. Equation 3.12 is changed to Equation 5.5 to connect the energy demand in month t to the biomass consumption variable. Et equals to the total biomass consumption for all suppliers in month t multiplied by the total ratio of biomass type p for supplier s in month t, the energy value of biomass type p from supplier s in month t, system efficiency, and the ratio of quality reduction factor, if the storage level in month t is lower than SLL.     ∑       ∑                                                [For all t]        (5.5) There is no need for variable AveEVt and Equation 3.13. It can be seen that constraint 5.5 contains the multiplication of a binary variable Zt and a continuous variable Cs,t. This non-linear term can be converted to a linear constraint through replacing the term Zt×∑sCs,t by an additional continuous variable, i.e. Lt≥0 and the following constraints 5.6, 5.7 and 5.8, where M is a sufficiently large number. 88                                                                                                            [For all t]          (5.6)    ∑                                                                                                           [For all t]         (5.7)    ∑                                                                                         [For all t]         (5.8) Having the above changes in the constraints and the objective function of the model presented in Chapter 3 forms a mixed integer linear programming (MILP) model which was solved using the AIMMS software package and the CPLEX solver.   5.3 The stochastic mixed integer programming model of the power plant supply chain A two-stage stochastic programming approach is used here to deal with uncertainty in available monthly biomass and incorporate it into the above multi-period planning model. In this approach, it is assumed that at the beginning of the planning horizon, the available biomass from each supplier for the first quarter of a year is known and it is uncertain for the rest of the year. Therefore, the first stage includes the first three months (each period is a month) of a year and the next nine months are considered as the second stage. The decision variables, including the amount of biomass to be purchased from each supplier, the amount of biomass from each supplier to be stored and the amount of biomass from each supplier to be consumed depend on the available biomass, hence, these decision variables are considered as the first stage decision variables in the first three months and the second stage decision variables in the next nine months. The scenarios are made by varying the monthly available biomass from each supplier by ±20% of the average scenario together with the expected volume, i.e. 3 different values. It is assumed that the change in available biomass is stationary for three months since uncertainty in available biomass is the result of fluctuation in lumber and other wood products market and the 89  market situation is assumed to be constant for that period. Consequently, the total number of scenarios is 3 multiplied by the number of aggregated periods in the second stage, i.e. 33=27.  The general formulation for a two-stage stochastic programming is shown in the literature (e.g. Birge & Louveaux (1997)) and is not repeated here. In this study, an implicit formulation of the stochastic model for the forest biomass power plant supply chain is presented. This means that there is one set of decision variables and the first stage decision variables are distinguished from the second stage decision variables using non-anticipatory constraints.  There is a new set and a new parameter used in the developed two-stage stochastic model defined as below:  Scenarios: List of scenarios (index i) {Scenario1, …, Scenario27}           : Maximum available biomass from supplier s in month t for scenario i.  Table 5-2 represents the definition of the stochastic model decision variables. The objective function is to maximize the expected profit of all scenarios. To calculate the expected value of profit, profit of scenario i (Profiti) is multiplied by Pri, the probability of occurrence of scenario i (∑iPri =1), and the objective function is the sum of expected values as shown in Equation 5.9.                                                              ∑                                                          (5.9) Here, Profiti equal revenues minus costs for scenario i as shown in Equation 5.10.  90  Table 5-2: Stochastic model decision variables        Amount of biomass purchased from supplier s in month t for scenario i (green tonnes), first stage decision variable for t=1-3, and second stage decision variable for t=4-12.        Amount of biomass from supplier s stored in month t for scenario i (green tonnes), first stage decision variable for t=1-3, and second stage decision variable for t=4-12.        Amount of biomass from supplier s consumed in month t for scenario i (green tonnes), first stage decision variable for t=1-3, and second stage decision variable for t=4-12.      Amount of electricity generated in month t for scenario i (MWh), first stage decision variable for t=1-3, and second stage decision variable for t=4-12.     Binary variable, 1 if surplus electricity is produced in a year, 0 otherwise for scenario i.      Binary variable, 1 if Ss,t,i is higher than the desired storage level (SDL) in month   for scenario i, first stage decision variable for t=1-3, second stage decision variable for t=4-12.      Binary variable, 1if Ss,t,i is higher than the upper storage limit (SUL) in month   for scenario i, first stage decision variable for t=1-3, and second stage decision variable for t=4-12.      Binary variable, 1 if Ss,t,i is less than lower storage limit (SLL) in month   for scenario i, first stage decision variable for t=1-3, and second stage decision variable for t=4-12.                          ∑ (∑                        )            ∑            ∑           ∑     (         )     ∑                                   (5.10) Subject to: Monthly available biomass for each supplier s with fixed contract in each month t and each scenario i:                                                      [For s∈{Supplier1, …, Supplier4}, all t and i]         (5.11) 91  Monthly available biomass for each supplier s without fixed contract in each month t and each scenario i:                                                      [For s∈{Supplier5, …, Supplier8}, all t and i]         (5.12) Maximum storage level in each month t and each scenario i: ∑                                                                                              [For all t and i]          (5.13) Last time period target storage level for each scenario i: ∑                                                                                    [For t=Dec and all i]          (5.14)  Mass balance equation for storage, consumption and purchase for each supplier s, each month t and each scenario i:                                                                                        [For all s, t and i]           (5.15) Energy demand in each month t and each scenario i:                 ∑                                                              [For all t, i]          (5.16) Converting biomass to electricity in each month t and each scenario i:      ∑ (∑                    )         (          )              [For all t, i]     (5.17)  92  And non-negativity of all variables for each supplier s, each month t and each scenario i:       ,       ,       ,                                                                               [for all s, t, i]         (5.18) In the stochastic programming model, it is assumed that at the start of each stage, the decision makers have sufficient information on the amount of available biomass in that stage and can adjust the production plan for that scenario. A decision xt is called implementable if it cannot distinguish between different scenarios that are indistinguishable at a stage (Kazemi Zanjani et al., 2010b). The constraints that are implying this are called non-anticipatively constraints as follows.                ,                ,                     [for t ∈{Jan,…,Mar}, all i and i' (i≠i') and s]         (5.19) 5.4 Managing the risk The output of a stochastic programming model is the total expected profit of all scenarios. This model is risk-neutral because only the expected profit is considered in the formulation and profit variations are ignored. The desired solution from a management perspective may include decisions that generate higher than the expected profit or a desired target profit for a scenario. Usually, decision makers are interested in knowing the potential risk and managing it while optimizing the economic objective. To do this, different risk metrics can be defined and included in the optimization model. In this paper, variability index and downside risk are considered as two risk measures (You et al., 2009) and a weighted bi-objective model is developed for the first risk measure. The objective functions are: to maximize the total expected profit and minimize the risk measures.  93  5.4.1 Variability index Uncertainty in input parameters of an optimization model results in uncertainty in the optimal solution. Therefore, the solution, which in our case is profit, has mean and variance. Considering the variance of the optimum solution in addition to its expected value is one of the formula io s  ha  have bee  called “robus  op imi a io ” i   he li era ure (Mulvey et al., 1995). In this method, the variance of the total profit is minimized while the expected total profit is maximized. However, using variance adds a quadratic term to the objective function which makes the optimization model non-linear. Moreover, it treats the profits higher and lower than the expected profit in the same way. Therefore, instead of variance in this study the negative deviation between the scenario profit (Profiti) and the expected profit (Profit) is considered as the risk measure as defined by Ahmed & Sahinidis (1998). The variability index is defined as a non-negative continuous variable (∆i) for each scenario and the following constraint is added to the model:      ∑                                                                    [    , for all i]               (5.20) According to Equation 27, if the profit of scenario i (Profiti) is less than the expected profit (Profit), ∆i is equal to their positive difference; otherwise it is 0. The objective function then is to maximize the weighted sum of the total expected profit minus the expected variability index as follows:      ∑                ∑        )                                                                         (5.21) 94  Where ρ is the weight associated with the second objective function. The output of this bi-objective model provides the trade-off between the expected profit and the profit variability index.  5.4.2 Downside risk  When considering risk, the extremes of the optimum solutions are also important in addition to their variability. It is desirable to have a lower probability of having low profit or a higher probability of having high profit. To address this, another risk metric is used here called downside risk, which is defined as the probability that the real profit is less than a certain threshold or target Ω (You et al., 2009). The difference between the variability index defined earlier and the downside risk is that in the latter the deviation is evaluated by comparing to a target level while in the former it is compared to the expected profit. Here, the variable ψi is defined as the positive deviation between the target Ω and the profit of scenario i (Profiti). If the scenario profit (Profiti) is less than the target Ω, ψi is equal to their difference, otherwise it is 0, as shown by the following constraints:                                                                                           [ψi≥0, for all i]             (5.22) The following equation defines the downside risk metric associated with target Ω:         ∑                                                                                                                  (5.23) In this case, Risk(Ω) is set to be equal to an appropriate predefined level and constraint 5.22 is added to the model.  95  5.5 Results  5.5.1 Result of deterministic models Based on the assumption for the initial storage explained in 5.3, the optimum profit for the MILP model was $16.2M, 5% higher than that of the MINLP mode, which was $15.4M. It has to be noted that there was no guarantee for the MINLP model to provide the exact optimal solution since the model was not convex. 5.5.2 Results of the stochastic model  Table 5-3 shows the expected value of profit for three cases. Column 2 indicates the results if perfect information on the scenario that is going to happen was available. In this case, the deterministic model is solved for each scenario one by one. The average of profit for all scenarios are calculated and shown in column 2 (scenario analysis model). However, in reality, such perfect information is not available, and this is the motivation for developing a stochastic model. Column 3 shows the expected profit for the stochastic model. The deterministic model is usually solved using the average values for uncertain parameters. If the first stage decisions based on average values are implemented, while other scenarios than the average scenario occur in the second stage, the optimization model can be solved with implemented first stage decisions to determine second stage decisions. Column 4 of Table 5-3 shows the expected profit when the deterministic model is solved for each scenario using the first stage decisions of the average scenario.    96  Table 5-3: Expected value of profit for scenario analysis, stochastic and average scenario models ($M)  Deterministic model –scenario analysis Stochastic model Deterministic model  - first stage decisions based on average scenario Expected value of profit 16.2 16.0 15.8 It can be seen that the expected value of profit for the scenario analysis is $16.2M which is $0.2M higher than the expected value of profit for the stochastic model ($16.0M). This difference is also called “Expected Value of Perfect Information” (EPVI) meaning that this is the cost that the power plant is paying due to the existence of uncertainty in available biomass. On the other hand, the expected value of profit of the stochastic model is $0.2M higher than the expected value of profit for the case that the deterministic model result for the average scenario is implemented but other scenarios occur. This difference is named “Value of Stochastic Solution” (VSS) and represents the profit that is associated with inclusion of uncertainty in available biomass and justifies using more sophisticated methods such as stochastic programming in production planning of the power plant (Birge & Louveaux, 1997).  Figure 5-1 shows the histogram of profit for using average scenario results and stochastic model.  It can be seen that the standard deviation of the average scenario is lower than that of the stochastic model. It also indicates that the frequency of having very high profit ($17-18M) and very low profit ($12-13M) is higher in the case of the stochastic model while there is a higher probability of having profit in the range of $16-17M in the deterministic model using average scenario results for the first stage decisions. 97   Figure 5-1: Histogram of profit distribution for the deterministic model with first stage decisions based on average scenario and the stochastic model  Table 5-4 presents the biomass procurement cost, including purchase and transportation costs, from the deterministic model with average scenario and the stochastic model and their difference for each scenario. It can be seen that while the difference between the average procurement costs over all scenarios is $0.1M, this difference could be as high as $500,000 for some scenarios. It means that employing the stochastic programming model results has $0.5M lower costs for the company for some of the scenarios whereas the maximum loss of using stochastic model is only $0.2M for one scenario. This also demonstrates that developing and implementing the stochastic model reduces the biomass purchase cost for the power plant. If the deterministic model is solved for each scenario one by one, then the first stage decision variables are fixed for each scenario, and the model is solved to determine the second stage decisions of each scenario, 9 scenarios will have infeasible solutions as shown in Table 5-5. Therefore, ignoring uncertainty and using the deterministic model results for different scenarios, result in a 33% risk of having an infeasible solution. Moreover, as shown in Table 0%10%20%30%40%50%60%70%12.0-12.9 13.0-13.9 14.0-14.9 15.0-15.9 16.0-16.9 17.0-17.9Frequency  Profit ($M) Stochastic Model Deterministic Model with Average Scenario98  5-5 the average profit for those scenarios with feasible results are always less than that of the stochastic solution, which is $16.0M. Table 5-6 indicates the standard deviation of the amount of biomass consumption, which is one of the decision variables of the model, of 27 scenarios for each month from all suppliers, for both the deterministic scenario analysis and the stochastic model. It can be seen that for the stochastic model the standard deviation is zero for the first stage decisions, as expected from definition of non-anticipatory constraints. The decision variables are therefore independent of the scenario occurring in the future which makes them implementable. The same table can be presented for other decision variables, such as the amount of biomass purchased in each month.  Other remarks from the results are as follows. In the scenario analysis model, the optimization results indicate that the surplus load should not be produced for three scenarios, while in the stochastic model the surplus load is always produced. Moreover, the storage level never go beyond the upper and lower limits (   ,     and    ) neither in stochastic nor in deterministic models. Therefore, there is no risk of having very low or very high storage levels related to the uncertainty in available biomass.  It should be noted that the model and the above comparisons are related to a process where decisions on the first stage are used and implemented. It is possible that after implementing the first three months, the system is re-planned with a new stochastic model over the rest of the year, i.e. 3+6 months as the new first and second stages.  99  Table 5-4: Biomass procurement cost for each scenario of stochastic and deterministic models ($M) Scenario Stochastic Model (SM) Deterministic Model - Average Scenario (DM) Difference (SM-DM) S1 6.7 7.0 -0.5 S2 6.9 7.4 -0.5 S3 7.2 7.6 -0.4 S4 7.0 7.5 -0.5 S5 7.2 7.7 -0.5 S6 8.0 8.0 0.0 S7 7.4 7.8 -0.4 S8 8.2 8.1 0.1 S9 9.1 9.0 0.1 S10 7.0 7.5 -0.5 S11 7.2 7.7 -0.5 S12 7.9 7.9 0.0 S13 7.4 7.8 -0.4 S14 8.1 8.1 0.0 S15 9.0 8.9 0.1 S16 8.4 8.3 0.1 S17 9.2 9.1 0.1 S18 10.1 10.0 0.1 S19 7.4 7.8 -0.4 S20 8.1 8.1 0.0 S21 9.0 8.9 0.1 S22 8.4 8.3 0.1 S23 9.2 9.1 0.1 S24 10.1 10.0 0.1 S25 9.4 9.3 0.1 S26 10.3 10.2 0.1 S27 11.3 11.1 0.2 Average 8.3 8.4 -0.1 100  Table 5-5: Average profit if the first stage decision variables of each scenario is implemented and other scenarios happen ($M) Scenario which its first stage decisions are implemented Expected value if other 26 scenarios happen S1 Infeasible  S2 Infeasible S3 Infeasible  S4 Infeasible  S5 Infeasible  S6 15.8 S7 15.9 S8 15.8 S9 15.7 S10 Infeasible  S11 Infeasible  S12 15.9 S13 15.9 S14 15.8 S15 15.5 S16 15.8 S17 15.4 S18 Infeasible S19 15.9 S20 15.8 S21 15.5 S22 15.8 S23 15.4 S24 15.3 S25 15.3 S26 15.3 S27 Infeasible  101  Table 5-6: Average and standard deviation of the monthly biomass consumption for deterministic model with scenario analysis and stochastic models (1000 green tonnes)  Deterministic  model - Scenario analysis  Stochastic model Average STD Average STD Jan 41 0.3 42 0 Feb 41 0.4 41 0 Mar 40 0.3 40 0 Apr 41 1.3 41 0.4 May 26 0.3 25 0.3 Jun 35 0.4 35 0.7 Jul 43 1.4 43 1.2 Aug 41 1.4 41 0.7 Sep 40 0.8 39 0.3 Oct 42 2.7 43 0.2 Nov 42 0.3 42 0.4 Dec 43 0.2 43 0.2  5.5.3 Results for the variability index Figure 5-2 shows the mean and standard deviation of the total profit for the bi-objective model containing variability index mentioned in 5.4.1. It can be seen that as coefficient ρ increases, lower standard deviation is obtained with lower average profit e.g. to reduce standard deviation from $1.2M to 0, the profit reduces from $16M to $13.5M too. Figure 5-3 presents the histogram of profit distribution before and after managing the variability index. It is observed that after managing the variability index (ρ=2) the expected profit is less spread, while it has more variation when risk is not managed (ρ=0). However, assigning a high weight to the risk in the objective function (e.g. ρ=5) results in lower probability of having higher profit without reducing the probability of having lower profit. It 102  means that although the standard deviation of profits of different scenarios has been reduced, they are distributed around a lower expected value. Therefore, it is not reasonable to assign high weight to the risk in the objective function.   Figure 5-2: Profit mean and standard deviation for different weights (ρ)  Figure 5-3: Histogram of profit distribution for different weights (ρ) associated with variability index   00.20.40.60.811.21.41212.51313.51414.51515.51616.50 1 2 5 7 10 20 27Standard Deviation ($M) Profit ($M) ρ Profit STDV0%10%20%30%40%50%60%70%80%90%12.0-12.9 13.0-13.9 14.0-14.9 15.0-15.9 16.0-16.9 17.0-17.9Frequency Profit ($ M) ρ=0 ρ=2 ρ=5 103  5.5.4 Results for the downside risk Figure 5-4 shows the result after managing the downside risk as mentioned in 5.4.2 with the target profit set at $14M (Ω=$14M). Before managing the downside risk, the expected value of profit was $16.0M and Risk($14M)=$40,669, calculated based on Equations 5.22 and 5.23. In order to reduce the downside risk to $30,000 (Risk($14M)=$30,000), the total the expected profit decreases to $15.8M. It can also be observed that after managing downside risk, the probability of having high profit in the range of $17.0-18.0M is zero.   Figure 5-4: Histogram of profit distribution for before and after managing the downside risk (Ω=$14M) 5.6 Discussion and conclusions  In this chapter, a mixed integer linear programming model is developed to optimize the supply chain of a forest biomass power plant at the tactical level, based on a previously developed mixed integer non-linear programming model. The developed model was then extended to a two-stage stochastic linear programming model to include uncertainty in available biomass 0%10%20%30%40%50%60%70%12.0-12.9 13.0-13.9 14.0-14.9 15.0-15.9 16.0-16.9 17.0-17.9Frequency Profit ($M) Risk=$40669 [max E(profit)] Risk=$30000104  from different suppliers. The results of the case study showed that having the perfect information about monthly available biomass from suppliers,  he power pla  ’s a  ual profi  would be $16.2M. However, in reality, the amount of available biomass from supplier varies due to a number of factors such as market fluctuations resulting in an average profit of $15.8M, $0.4M lower than $16.2M. Part of this loss could be compensated using a stochastic programming model. The profit of the stochastic model was $0.2M higher than the solution when the average scenario was implemented while other scenarios occurred and $0.2M lower than the solution of the deterministic model with the perfect information. The procurement cost of the stochastic model was $0.5M lower than that of the average scenario. The reason that the stochastic model had lower profit than the deterministic model was that in the stochastic optimization, the expected value over all scenarios of monthly available biomass, including the scenarios with 20% reduction in supply, were optimized. In addition, ignoring uncertainty and implementing deterministic model results for each scenario resulted in a 33% chance of having an infeasible solution. Moreover, the stochastic model decision variables were implementable since they did not depend on future uncertainties. This was obtained, however, with a more sophisticated model, higher amount of computational effort and the need for data and assumptions about uncertain parameters.  Two risk measures were also considered in this study and two models were developed and solved in order to explicitly consider the risks included in the supply chain planning process. The considered risk measures were the variability index and downside risk. The results of risk management models indicated that the total expected profit of the power plant would reduce after risk management. Lower profit variance was obtained by managing the variability index, but it resulted in lower probability of having higher profit. When the downside risk is 105  controlled, the probability of having very low and very high profit decreased. It is important to manage risk measures properly in order to prevent low expected values for profit. Basically, this is up to the power plant managers to decide on the risk measures as well as the risk threshold that are acceptable to them, and then implement the optimum solution of using those risk measures and thresholds in the models.  This chapter addressed the thesis fourth objective partially by including uncertainty in biomass availability in a two-stage stochastic optimization model. It is important to incorporate the uncertainty in biomass quality into the modeling to have a comprehensive robust model in response to uncertainty. This is done in the next chapter.    106  Chapter 6 Hybrid stochastic programming-robust optimization model  6.1 Synopsis In the previous chapter, uncertainty in one parameter, biomass availability, was included in the optimization model. In this chapter, uncertainty in several parameters of the power plant supply chain is incorporated simultaneously. Uncertain parameters incorporated here are biomass quality, namely moisture content and higher heating value, in addition to monthly available biomass from suppliers. Uncertainties in electricity prices and biomass cost are not included because the power plant has long term contracts with the client and most of the suppliers. This means the prices and costs do not change during the planning horizon. i.e. a year. There is a small variation in biomass cost over a year for suppliers without a fixed contract, therefore, uncertainty in biomass cost for those suppliers can be ignored as well.  Moisture content and higher heating value change within a range and are considered in a developed robust optimization model. Uncertainty in monthly available biomass is included through a two-stage stochastic programming model as explained in Chapter 5. A hybrid stochastic programming-robust optimization model is used to incorporate uncertainty in all of the mentioned parameters simultaneously, and to have an appropriate balance between computational complexity associated with the stochastic programming formulation and conservatism of the optimal solution (lower profit) associated with the robust optimization approach.  6.2 Robust optimization formulation  Two indicators of biomass quality are considered here: moisture content (MC) in % and higher heating value (HHV) in BTU/lb. These parameters are not correlated since HHV is calculated 107  based on dry biomass. Energy value depends on MC and HHV as was shown in Equation 3.14. Based on the historical data provided by the power plant, the range of variation is considered to be 25-35% for MC with an average of 30%, and 8000-9000 BTU/lb (5.17-5.81 MWh/tonnes) for HHV with an average of 8500 BTU/lb (5.49 MWh/tonnes). To be able to include uncertainty in MC and HHV in a stochastic programming model, different scenarios can be obtained for them by discretizing their ranges. However, if these scenarios are combined with the 27 scenarios of monthly available biomass, the total number of scenarios is going to be huge even with a small number of discretized scenarios. For instance, considering only 3 scenarios for each of MC and HHV, within the scenario structure described above for monthly available biomass, the number of scenarios will be 93×27=19683. This is even without considering the combination of different MC and HHV for different suppliers. Therefore, robust optimization is used to include uncertainty in biomass quality into the decision making process. The formulation for the robust optimization is derived from Ben-Tal et al. (2009) which is available in the AIMMS software package. It helps to come up with a tractable model providing a solution that is feasible for all MC and HHV ranges with a price of having lower profit since the worst case is optimized. Here, a brief explanation of the idea of robust optimization is provided.  Consider a general linear programming (LP) model as shown in Equation 6.1:                                                                                                                                (6.1) 108  Where x represent model decision variables, c is the objective function coefficients vector, b is the right hand side values vector and A is the constraints coefficient matrix. If the parameters of the model, c, A and b, are uncertain and range within a convex set of U, then the robust counterpart of this uncertain LP model can be defined as shown in Equation 6.2:                    ∈     ∈        ∈                                                          (6.2) It should be noted that in robust optimization, the objective is to optimize the worst case. Equation 6.2 can be reformulated as follows:                          ∈     ∈        ∈                                                (6.3) The robust counterpart model shown in Equation 6.3 has a single objective function and continuous constraints. Any solution         for Equation 6.3 satisfies the constraints of Equation 6.2 for  c, A and b∈U. These solutions are robust feasible and among such robust feasible solutions, the robust optimal value, namely  ̅, is the one providing the best possible objective function for Equation 6.3. Ben-Tal & Nemirovski (2002) proved that problem shown in Equation 6.3 is computationally tractable (or in the other words polynomially solvable) for a wide choice of the uncertainty sets U. Particularly, problem in Equation 6.3 is an LP model if U is a box or polyhedral set (Li & Floudas). Without loss of generality, only uncertainty in parameters of matrix A, named as  ̃ is considered. If  ̃ belongs to a box set, the uncertainty in  ̃ is defined as:  ̃       ̂                                                                                                                           (6.4) 109  where   is the nominal value of parameters and  ̂ is the positive constant perturbation and   is independent random variables which are subject to uncertainty. Then, constraint    –        can be written as:  ∑   ∑   ̂                                                                                                             (6.5) The solution has to be feasible for any   in a given uncertainty set U or alternatively:  ∑        ∈   ∑    ̂                                                                                            (6.6) In particular for the box uncertainty set the random vector   is defined as:                                                                                                                                  (6.7) Where   is an adjustable parameter controlling the size of uncertainty set. Then the corresponding robust counterpart in Equation 6.3 is equivalent to the following constraints: ∑        ∑    ̂  –                                                                                                           (6.8)                                                                                                                                    (6.9) Where u is a decision variable (u≥0). Equation 6.9 is still linear (Li & Floudas). In the present model, uncertain parameters belong to certain ranges and therefore can be considered as box. This formulation is being used to include uncertainty in biomass quality in the optimization model presented in this thesis. The MILP model presented in Chapter 5 is first extended to a robust optimization model. The formulation for robust optimization model, in case of uncertainty only in MCs,p,t is included is as follows:                                                                                                                                           (6.10) 110                 ∑ (∑                         )          ∑             ∑       ∑                            ∑                              (6.11) Subject to:                                                           [For s∈{Supplier1, …, Supplier4} and all t]       (6.12)                                                          [For ∈{Supplier5, …,  upplier8} and all t]       (6.13) ∑                                                                                                          [For all t]      (6.14) ∑                                                                                                                            (6.15)                                                                                              [For all s and t]       (6.16)                ∑                                                                    [For all t]       (6.17)      ∑  ∑                          ̃                 –                                                                         [For    ̃     ∈[25,30] and all t]       (6.18)   All continuous variables have to be non-negative as shown in Equation 6.19.     ,     ,     ,                                                                                           [for all s, t]     (6.19) Notice that constraint (6.18) is converted to inequality in order to make sure it can be met for all realization of  ̃     .  111  The robust counterpart of Equation 6.18 is as follows (according to Equations 6.8 and 6.9):    ∑ (∑                      (         ))         –                      ∑ (∑                           ̂      )         –                      [for all t]     (6.20)                  ,                                                                                                       (6.21) Where   ̂s,p,t is the positive constant perturbation in MCs,p,t and ψ is the adjustable parameter controlling the size of uncertainty set. The same formulation can be written for uncertainty in HHVs,p,t. When uncertainty in both parameters is included, the term HHVs,p,t × (1 – MCs,p,t) is replaced by EVs,p,t and the formulation is written for this parameter with the range derived from ranges in both HHVs,p,t and MCs,p,t. 6.3 Hybrid stochastic programming-robust optimization model  Here, the formulation for hybrid stochastic programming-robust optimization model, in case of uncertainty in MCs,p,t only, is presented:                                                                                                                                           (6.22)                 ∑ (∑                         )            ∑             ∑         ∑     (           )              ∑                         (6.23) 112  Subject to:                                                         [For s∈{Supplier1, …,Supplier4}, all t and i]      (6.24)                                                        [For s∈{Supplier5, …,Supplier8}, all t and i]       (6.25) ∑                                                                                                  [For all t and i]      (6.26) ∑                                                                                       [For t=Dec and all i]       (6.27)                                                                                            [For all s, t and i]       (6.28)                  ∑                                                                      [For all t, i]       (6.29)        ∑  ∑                          ̃                   –                                                                               [For    ̃     ∈[25,30] and all t and i]       (6.30)   All continuous variables have to be non-negative as shown in Equation 6.19.       ,       ,       ,                                                                                  [for all s, t, i]     (6.31) Again, constraint (6.30) is converted to inequality in order to make sure it can be met for all realization of  ̃     .  The robust counterpart of Equation 6.30 is as follows: 113        ∑ (∑                                 )         ( –        )                 ∑ (∑                          ̂       )         ( –        )                                                                                                                                         (6.32)                                                                                                                                (6.33) The same formulation can be written for uncertainty in HHVs,p,t. When uncertainty in both parameters is included, the term HHVs,p,t×(1-MCs,p,t) is replaced by EVs,p,t and the formulation is written for this parameter with the range derived from ranges in both          and       . 6.4 Results  First, the results of the robust optimization model are presented where variations in biomass quality are considered. A solution that is feasible for all possible instances of uncertain parameters will likely be a conservative solution. The model is solved with different ranges for MCs,p,t and HHVs,p,t values to assess how the solution changes when the uncertainty set is widened, the results can be seen in Table 6-1. The profit of the robust optimization model using the average MCs,p,t (30%) and average HHVs,p,t (8500 BTU/lb) is $15.64M. Using the average HHVs,p,t in the model, as the uncertainty set of MC widens from 2% to 10%, the optimum profit drops from $15.07M to $13.13M, which is also shown in Figure 6-1. Alternatively, using the average MCs,p,t of 30% in the model, as the uncertainty set of HHVs,p,t is expanded from 200 BTU/lb (0.13 MWh/tonnes) to 2000 BTU/lb (1.29 MWh/tonnes), the optimum profit reduces from $15.15M to $13.58M, as shown in Figure 6-2. When both parameters are considered uncertain at the same time, the result becomes more conservative with a more severe reduction in profit to $11.97M for HHVs,p,t∈[8100,8900], MCs,p,t∈[26,34]. In the extreme ranges, the 114  model is infeasible. Figure 6-3 shows the results when uncertainty is included in both MCs,p,t and HHVs,p,t.  Table 6-1: Profit ($M) for different ranges of MCs,p,t  and HHVs,p,t used in the robust optimization model   MCs,p,t (%)   30 (Average) 29-31 28-32 27-33 26-34 25-35 HHVs,p,t  8500 (BTU/lb) (Average) 5.49 (MWh/tonnes) 15.64 15.07 14.61 14.13 13.64 13.13 8400-8600 (BTU/lb) 5.42-5.56 (MWh/tonnes) 15.15 14.72     8300-8700 (BTU/lb) 5.36-5.62 (MWh/tonnes) 14.78  13.76    8200-8800 (BTU/lb) 5.30-5.69 (MWh/tonnes) 14.39   12.94   8100-8900 (BTU/lb) 5.23-5.75 (MWh/tonnes) 13.99    11.97  8000-9000 (BTU/lb) 5.17-5.81 (MWh/tonnes) 13.58     Infeasible   Figure 6-1: Solution of the robust optimization model for different ranges of moisture content 11121314151629-31% 28-32% 27-33% 26-34% 25-35%Profit ($M) Moisture Content Range 115    Figure 6-2: Solution of the robust optimization model for different ranges of higher heating value  Figure 6-3: Solution of the robust optimization model for different ranges of energy value  Despite having a more conservative solution with a lower profit, the decisions provided by robust optimization are feasible for ranges of MC and HHV shown in Figure 6-3. The model was run for the case where the decision variables of biomass purchase, storage and consumption in the first three months are made and implemented based on the results of the model using average MC and HHV (MCs,p,t=30% and HHVs,p,t=8500 BTU/lb). Then, it was 1112131415168400-8600 8300-8700 8200-8800 8100-8900 8000-9000Profit ($M) Higher Heating Value Range (BTU/lb) 11121314151629-31%8400-860028-32%8300-870027-33%8200-880026-34%8100-8900Profit ($M) Moisture Content Range Higher Heating Value Range (BTU/lb) 116  assumed that for the rest of the months, MC and HHV take other values (any of the ranges showed in Table 6-1, for instance HHVs,p,t∈[8100,8900] and MCs,p,t∈[26,34]). It was observed that the model becomes infeasible when MCs,p,t and HHVs,p,t vary from their average values, which in reality is very probable.  Figure 6-4 and Figure 6-5 show the optimum biomass storage and consumption level in different months based on the results of the robust optimization model with HHVs,p,t∈[8100,8900], MCs,p,t∈[26,34] and the deterministic model. It can be seen that robust optimization provides more storage levels compared to the deterministic model for most of the months and more consumption levels in all months. The consumption level is higher due to a lower than average energy value of biomass used in the robust optimization model.   Figure 6-4: The optimum storage level in different months from the robust optimization model with HHVs,p,t∈[8100,8900] and MCs,p,t∈[26,34] and the deterministic model  0102030405060708090Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecStorage Level (1000 Green Tonnes) Robust OptimizationDeterministic117   Figure 6-5: The optimum biomass consumption level in different months from the robust optimization model with HHVs,p,t∈[8100,8900] and MCs,p,t∈[26,34] and the deterministic model   The result of the hybrid stochastic programming-robust optimization model is provided in Table 6-2. The first two columns show MC and HHV ranges. The next columns show the robust optimization solutions, the hybrid stochastic programming-robust optimization solutions and their differences, respectively. It can be seen that the hybrid stochastic programming-robust optimization model provides a more conservative solution compared to the robust optimization model. However, the reduction in profit is small (0.2-0.7%). The reduction in profit increases as the range of uncertainty decreased. For the average values of HHV and MC, the profit for the deterministic model is $15.64M and it is $15.45M for the stochastic programming model. Despite having a lower profit from the hybrid stochastic programming-robust optimization model, it provides feasible solutions for all scenarios of monthly available biomass and all ranges of variation in MC and HHV. Figure 6-6 shows the optimum storage level over 27 scenarios for the robust optimization model and the hybrid stochastic programming-robust optimization model when HHVs,p,t∈[8100,8900] and MCs,p,t∈[26,34]. To make the graphs 01020304050Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecBiomass Consumption (1000 Green Tonnes) Robust OptimizationDeterministic118  readable, only the storage levels from January to May are depicted. It can be seen that the first stage decision variables are unique and independent when stochastic programming is used. Table 6-2: Profit for different ranges of MCs,p,t and HHVs,p,t used in the robust optimization and hybrid models MCs,p,t range (%) HHVs,p,t range (BTU/lb) Robust optimization solution ($M) Hybrid model solution ($M) Difference ($M) 26-34 8100-8900 11.97 11.93 0.03 27-33 8200-8800 12.94 12.87 0.07 28-32 8300-8700 13.76 13.66 0.10 29-31 8400-8600 14.72 14.61 0.11    Figure 6-6: Optimum storage level of 27 scenarios for the first five months  for a) robust optimization model, and b) hybrid stochastic programming-robust optimization model when HHVs,p,t∈[8100,8900] and MCs,p,t∈[26,34] 6.5 Discussion and conclusions  In this chapter, first uncertainty in biomass quality is included in the optimization model of a forest bioenergy supply chain through developing a robust optimization model. This model provides a solution that is feasible for any value of uncertain parameters within a defined range. Different profit can be generated depending on the variation range for MC and HHV. For the average MC (30%) and average HHV (8500 BTU/lb), the profit was $15.64M. For 405060708090100Jan Feb Mar Apr MayStorage (1000 Green Tonnes) 405060708090100110120Jan Feb Mar Apr MayStorage (1000 Green Tonnes) 119  average HHV and MC range of 29-31%, the profit decreased to $15.07M, while it dropped to $13.13M for MC range of 25-35%. When HHV changed within 200 BTU/lb range (8400-8600 BTU/lb) with the average MC, the optimum profit reduced to $15.15M. It also decreased to $13.58M, as the HHV range was widened to (8000-9000 BTU/lb). When MC and HHV changed simultaneously the solution became more conservative. For instance, the profit for the widest range with a feasible solution, which was (26-34%) for MC and (8100-8900 BTU/lb) for HHV, was $11.97M. From these results, it was concluded that as the range of uncertain parameters widened, the profit obtained from the robust optimization model decreased. The reason was that in the robust optimization model, the worst case scenario was optimized. Expanding the range of the uncertain parameters made the worst case scenario worse.  A hybrid stochastic programming-robust optimization model was then proposed to incorporate uncertainties in different parameters simultaneously. Stochastic programming was used to include uncertainty in monthly available biomass using 27 scenarios over a one-year time horizon. Robust optimization was used to model the uncertainty in biomass quality within a polyhedral set. Comparing the results of robust optimization and stochastic programming models, it can be concluded that the conservatism in the robust optimization model was more severe because it optimized the worst cases while the stochastic programming model optimized the expected value of all scenarios. The degree of conservatism can be controlled by selecting the appropriate range of uncertain parameters. The conservative solution was compensated by having more stable solution which was free of infeasibility risk. No matter what happens in terms of uncertain parameter values, one can be sure that the model remains feasible and the decisions can be adjusted if any perturbation occurs within the defined sets or scenarios. To balance the conservatism of optimizing the worst case, stochastic programming was used to 120  include uncertainty in available biomass. When the hybrid model was implemented, the profit reduced even more. For instance, for the case with (26-34%) range for MC and (8100-8900 BTU/lb) range for HHV, the profit was $11.93M, $0.03M lower than that of the robust optimization model with the same ranges, $11.97M. However, this model has the advantage of providing a solution which is implementable, e.g. unique decision variables for the first stage. The stochastic programming results have the advantage of being unique for the first stage, or in other words, being implementable.  This chapter completed on the fourth objective of the thesis. Making decisions in supply chain planning at the tactical level considering uncertainty in biomass quality was unavailable before. The hybrid model is a unique approach to include uncertainty in several parameters simultaneously and balance the advantages and disadvantages of both stochastic programming and robust optimization approaches.    121  Chapter 7 Conclusions, strength points, limitations and future research  7.1 Conclusions  The optimization models developed and presented in Chapters 3-6 achieved all the objectives of this study. The deterministic model presented in Chapter 3 achieved objectives 1 and 2 and part of objective 3, the Monte Carlo simulation model presented in Chapter 4 fully achieved objective 3 and the stochastic programming, robust optimization and the hybrid model developed and presented in Chapters 5 and 6 achieved the 4th objective.  From the deterministic model presented in Chapter 3 it was concluded that optimization models were useful tools that could help managers of forest biomass power plants to make better decisions on how much biomass to purchase, store and consume in each month to meet the electricity demand and maximize their profit. Different parts of the supply chain from biomass supply to storage and electricity production, were integrated in the deterministic model, therefore, the optimum solution prescribed by the model was based on the impact of parameters in all parts of the supply chain and the interactions between different supply chain parts. This would be helpful for decision makers because it would not be possible for them (without using a decision support tool) to consider all these factors and interactions simultaneously when making decisions. In fact, the decisions in the power plant are made by different people (fibre manager, production manager, and some by finance manager). The managers make their own decisions and then have quarterly meetings to discuss important issues without being able to know how each decision would affect other decisions or the profit as a whole. The model developed here is useful since it integrates all the supply chain parts.  The impact of uncertainty in input parameters on the profit was shown from sensitivity analysis results. These results showed that variations in electricity price, higher heating value, maximum available biomass from suppliers with fixed contracts, moisture content and biomass 122  cost had high impacts on the optimum solution. The results of the Monte Carlo simulation model showed that variation in moisture content had a higher impact on profit than variation in higher heating value.  Including uncertainty in the decision making process may seem to provide lower profit for the power plant. However, it provided a robust solution against perturbations in input parameters. The solution was feasible for all considered instances of uncertain parameters and was implementable. It was also concluded that there was a trade-off between expected value of profit and probability of having low profit. Moreover, it was shown that developing a hybrid model that was a combination of stochastic programming and robust optimization models provided a flexible framework for considering uncertainty in different parameters at the same time. The hybrid model took advantage of both stochastic programming and robust optimization models; it is computationally tractable and not too conservative. If the hybrid model is implemented, the power plant managers would make sure even with the changes in the parameters (biomass availability and quality), that are probable in reality, the power plant still would be able to generate electricity and meet the demand, have biomass for production and do not have shortage or an excess storage.  7.2 Strengths points The main strength and contribution of this thesis is that the issue of uncertainty which is a critical factor in the forest biomass supply chain is incorporated in the modeling and analysis comprehensively. This can be used as a guideline for dealing with uncertainty in forest biomass/ bioenergy supply chains, or even other supply chains that have a similar structure.  The case study presented here, is a real forest biomass power plant. Having access to real data and being able to communicate with people working directly in this industry is a strength point 123  of this research. The model structure and solutions were validated by the power plant managers. Moreover, data analysis, particularly probability distribution of biomass quality for different biomass types in different months, provided useful information that could be used in other relevant studies.  The only study similar to this work was conducted by Alam et al. (2012b). They developed an optimization model with monthly time steps for supply chain management of a forest biomass power plant. The main difference between this deterministic optimization model and the one developed by Alam et al. (2012b) is that in the former, biomass quality for each biomass types, from each suppliers and in each months is considered to be different. This allows variations in biomass availability and quality during the year to be included in the optimization model. Alam et al. (2012b) used an average value for energy value and moisture content in all months. Moreover, they focused more on the harvesting side of the supply chain while the model developed here included the whole supply chain from procurement to storage, production and ash management. It also included the effects of mixing biomass and storage on biomass quality and production efficiency which were not done before, even in studies of other forest bioenergy plants such as district heating systems or biofuel plants.  Most of the time, uncertainty is ignored due to the lack of data, complexity of models with uncertainty, and difficulty to solve them. All the studies in forest biomass power plants supply chains ignored uncertainty. Alam et al. (2012b) only evaluated the impact of uncertainty in the parameters of an optimization model of a forest biomass power plant only through sensitivity analysis. They did not include uncertainty in the modeling and decision making process. In this thesis, historical data were used for identifying and quantifying uncertainty in different parameters. Ignoring uncertainty in parameters gave infeasible solutions (33% of the cases), 124  which means the constraints of the model (supply, demand, storage, etc.) were not met. The approaches for dealing with uncertainties were selected based on the model structure, characteristics of uncertain parameter, the quality of the solution they provided, and the computational effort needed to solve them. In the literature, there were studies that developed stochastic programming models to include uncertainty in similar industries, such as biofuel supply chains. However, all of them were at the strategic level with one time step. In the present study, the stochastic programming model was developed and solved for a multiple time step problem, which is more complex. Developing a hybrid stochastic programming–robust optimization model to incorporate uncertainty in different parameters of the supply chain was also a novel idea in field of supply chain management and helped to balance the model complication and conservatism. Those who work in this field should use models that can incorporate uncertainties in the decision making such as stochastic modeling, robust optimization or the proposed hybrid model to make sure the solutions are feasible for all realizations of uncertain parameters.  7.3 Limitations  One limitation of this study is related to the calculation of the mix of biomass, where it is assumed that all the biomass in the storage yard is mixed together and an average energy value is obtained. In reality a portion of biomass may be mixed completely. Moreover, assumptions were made about the initial storage level and initial quality of biomass provided by each supplier in storage at the beginning of the planning horizon because such information was not available.  The structure of the stochastic model was based on the assumption that biomass availability remains the same for three months and changes by  20% in the following quarters which may 125  not be the case in reality. In the robust optimization model and the hybrid model, the same ranges were taken for MC and HHV of all biomass types, from each supplier in each month. This is contrary to the fact that they could be different for each of the indices as was discussed in Chapter 4. However, in order to address these two limitations a much more complex model is needed that may still be challenging to solve. Another limitation of this study is that it does not consider any environmental or social objectives of the system. Instead, it only focuses on the economic objective and cost/profit in the system. It should be noted that the main incentive in building and operating a forest biomass power plant is to provide sustainable energy and local jobs in remote communities, as explained in the introduction chapter. However, these objectives should be considered more at the strategic level, e.g. when a forest biomass power plant is being designed. 7.4 Future research  As environmental and social impacts of products and processes are becoming more important than the pure economic value, future studies using deterministic modeling should incorporate environmental and social objectives in addition to the economic objective into the models and perform multi-objective optimization. Considering non-economic objectives into the models would help deal with important issues, such as emissions, land use, communities' interests, job creation, governmental policies, ecological impacts of removing residues from forest areas, and recreational aspects of forests, and provide possible trade-offs between different objectives. Moreover, to make the model usable by people in the industry, designing a user interface will be helpful. It is worthy to integrate different forest supply chains due to their dependency and model the whole value chain considering harvest areas, sawmills, pulp mills, and bioenergy conversion 126  facilities, and evaluate the effect of uncertainty in one chain on the supply, production and demand of the other chains. Moreover, integration of planning decisions at strategic, tactical and operational levels could also be done. 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