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a b s t r a c t 摘要
生物燃料是可再生液体燃料的有希望的候选。一个为新兴工业的挑战之一是在供应量，市场需求，市场价格，和加工技术的不确定性的较高水平。这些不确定性的复杂投资决策的评估。不确定性表现为大量的可能影响整体盈利能力和设计随机模型参数。我们研究了供应链网络覆盖美国东南部地区，包括生物质能源的位置和数量，候选场址和2种燃料转化加工能力和运输的林业资源的位置，物流的转换点和然后对最终的市场。为了减少设计问题可管理的大小目标函数的每个参数不确定性的影响，计算该参数的范围的两端。然后导致在利润超过其范围最变化的参数被组合成用于查找通过两阶段混合整数随机节目的设计情形。第一阶段决定是资本投资决策包括加工厂的大小和位置。第二阶段追索权的决定是生物质和产品在各种情况下流动。我们的目标是在预期收益在不同的场景的最大化。鲁棒性和主场迎战强大的设计（适用于多个场景）标称设计（对于单名义方案）的全球灵敏度分析使用蒙特卡罗模拟在从参数范围内形成的超立方体分析。 。保留2011爱思唯尔有限公司保留所有权利。 1.引言木质纤维素生物质是生物燃料最有前途的资源之一。Bio-fuels represent promising candidates for renewable liquid fuels. One of the challenges for the emerging industry is the high level of uncertainty in supply amounts, market demands, market prices, and processing technologies. These uncertainties complicate the assessment of investment decisions. This paper presents a model for the optimal design of biomass supply chain networks under uncertainty. The uncertainties manifest themselves as a large number of stochastic model parameters that could impact the overall profitability and design. The supply chain network we study covers the Southeastern region of the United States and includes biomass supply locations and amounts, candidate sites and capacities for two kinds of fuel conversion processing, and the logistics of transportation from the locations of forestry resources to the conversion sites and then to the final markets. To reduce the design problem to a manageable size the impact of each uncertain parameter on the objective function is computed for each end of the parameter’s range. The parameters that cause the most change in the profit over their range are then combined into scenarios that are used to find a design through a two stage mixed integer stochastic program. The first stage decisions are the capital investment decisions including the size and location of the processing plants. The second stage recourse decisions are the biomass and product flows in each scenario. The objective is the maximization of the expected profit over the different scenarios. The robustness and global sensitivity analysis of the nominal design (for a single nominal scenario) vs. the robust design (for multiple scenarios) are analyzed using Monte Carlo simulation over the hypercube formed from the parameter ranges. . 2011 Elsevier Ltd. All rights reserved. 1. Introduction Lignocellulosic biomass is one of the most promising resources for biofuels. Among potential biofuels feedstocks, forestry sources have some of the best attributes in terms of feasibility and environmental sustainability, but the economic supply, transportation and processing of woody biomass source feedstocks to new biofuels processing infrastructure depends on many assumptions. Forecasts are currently very preliminary and understanding how uncertainties in price, technology, supply, and demand impact design decisions is important to reducing risk and uncertainty that limit investment. Estimating supply and inventory of biomass feedstocks for biofuels is complex, and the quality of estimates varies by feedstock type. Inventory, technical and environmental constraints are spatially very variable and not consistently available at spatial scales finer than the county level. Estimating feedstock price is more complicated than gross potential supply and cost. This is because, . Corresponding author. E-mail address: [email protected] (M.J. Realff). although production and harvest cost components can be estimated, the interactions of a competitive market for fiber and other products from forestry with increased demands for feedstock from biofuels are opaque. This uncertainty mirrors how food prices react to increased demand for starch feedstocks for first generation ethanol production. Price and demand estimates for final fuel markets are uncertain because little current market data exists, significant policy intervention has been undertaken but future regulation is uncertain, and competing technologies such as increased fuel efficiency, hybrid and electric vehicles may reduce the overall demand for liquid fuels. Overall, this creates a dauntingly complex landscape for decision-makers that wish to invest in emerging technologies in lignocellulosic biofuels. The complexity creates a need for tools to help assess which uncertainties will have the biggest impacts on technology and system design, and to help synthesize solutions that are robust to them. This study presents possible future directions of biofuel production, transportation, and logistics of the infrastructure for the lignocellulosic biofuel industry in the SE region of the US. To do this, we identify the key parameters which impact the infrastructure decisions, find optimized supply chain network designs under the key uncertainties, and verify that the 0098-1354/$ – see front matter . 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2011.02.008
J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751 1739 uncertainties that are not considered in the design do not have a significant influence on the overall outcome. Supply chain modeling and optimization for biomass and biofuel systems has been studied by companies and academic research groups alike in recent years. Supply chain design decisions influence the overall structure of the biofuel production network through capital investment, production technology and location choices. Logistics management involves medium to short-term decisions on the procurement of biomass and distribution of products. Decision models of increasing scope and sophistication have been devised, including an emphasis on the environmental performance and incorporating uncertainty. One type of model focuses on the supply element of the biomass processing chain. For decision support, a dynamic integrated biomass supply analysis and logistics model was developed to simulate the collection, storage, and transport operations of supplying agricultural biomass in Sokhansanj, Kumar, & Turhollow (2006). The issue of combining multiple biomass supply chains, aimed at reducing the storage space requirements, is introduced in Rentizelas, Tolis, & Tatsiopoulos (2009). Geographical information systems (GIS) have been introduced to biomass supply chain studies in order to compute more accurately the expected supply of biomass in a given region and the transportation distances and related costs as well as to assess the impacts of spatial feedstock subtraction for different chain designs. A GIS-based decision support system for selecting least-cost bio energy locations when there is a significant variability in biomass farm gate price and when more than one bio energy plant with a fixed capacity has to be placed in the region was presented in Panichelli and Edgard (2008). A methodology using a GIS focused on logistics and transport strategies that can be used to locate a network of bioenergy plants around the region was developed to contribute by outlining a procedure for achieving an optimal use of agricultural and forest residue biomass (Perpina et al., 2009). Beyond the configuration of the supply component of the chain is the production component that converts the biomass into fuel products. These models often include specific geographic regions in their analysis to test the model formulations. A combined production and logistics model was presented in Dunnet, Adjiman, & Shah (2008) to investigate cost-optimal system configurations for a range of technological, system scale, biomass supply and ethanol demand distribution scenarios specific to European agricultural land and population densities. A mathematical programming model was proposed to design the supply chain and manage the logistics of a biorefinery, Eksioglu, Acharya, Leightley, & Arora (2009). The model determines the number, size and location of biorefineries needed to produce biofuel using the available biomass and the model was used for the State of Mississippi as a test case. A mixed integer linear programming (MILP) model was presented to determine the optimal geographic locations and sizes of methanol plants with heat recovery and gas stations in Austria, Leduc, Schwab, Dotzauer, Schmid, & Obersteiner (2008). These static models can be extended to consider planning over multiple periods. A mathematical model that integrates spatial and temporal dimensions was developed by Huang, Chen, & Fan (2010) for strategic planning of future bioethanol supply chain systems. The planning objective is to minimize the cost of the entire supply chain of biofuel from biowaste feed-stock fields to end users over the entire planning horizon, simultaneously satisfying demand, resource, and technology constraints. This model is tested to evaluate the economic potential and infrastructure requirements for bioethanol production from eight waste bio-mass resources in California as a case study. The environmental aspects of the supply chain were integrated into a general modeling framework conceived to drive the decision-making process for the strategic design of biofuel supply networks as presented in Zamboni, Shah, & Bezzo (2009a), Zamboni, Bezzo, & Shah (2009b). The design task is formulated as a mixed integer linear program (MILP) that accounts for the simultaneous minimization of the supply chain operating costs (Zamboni et al., 2009a) as well a the environmental impact in terms of greenhouse gas (GHG) emissions (Zamboni et al., 2009b). The model is devised for the integrated management of the key issues affecting a general biofuel supply chain, such as agricultural practice, biomass supplier allocation, production site locations and capacity assignment, logistics distribution, and transport system optimization. A number of studies have considered the uncertainties associated with supply chain network problems. Grossmann and Guillen-Gosalbez (2010) reviewed major contributions in process synthesis and supply chain management, highlighting the main optimization approaches that are available, including the handling of uncertainty and the multi-objective optimization of economic and environmental objectives. This paper emphasized there is a clear need to develop sophisticated optimization and decisionsupport tools to help in exploring and analyzing diverse process alternatives under uncertainty, and to determine optimal tradeoffs between environmental performance and profit maximization. A dynamic spatially explicit mixed integer linear programming (MILP) modeling framework was devised to optimize the design and planning of biomass-based fuel supply networks according to financial criteria and accounting for uncertainty on market conditions (Mas et al., 2010). The model capabilities for steering strategic decisions are assessed through a real-world case study related to the emerging corn-based bioethanol production system in Northern Italy. Aside from the biofuel network problem, there have also been efforts to address other alternative fuel network problems. The design problem for a hydrogen supply chain was addressed considering various activities such as production, storage and transportation (Kim et al., 2008)Kim, Lee, & Moon, 2008). The purpose of this study was to develop a stochastic model to take into account the effect of the uncertainty in the hydrogen activities and examine the total network costs of various configurations of a hydrogen supply chain in an uncertain environment for hydrogen demand. Another model was developed to consider the availability of energy sources (i.e. raw materials) and their logistics, as well as the variation of hydrogen demand over a long-term planning horizon leading to phased infrastructure development (Almansoori and Shah, 2009). Given that many mathematical or computational models are being developed to design supply chain for biofuels, it is important to develop approaches to identify and incorporate a wide range of sources of uncertainty that can be coupled to these types of models. Good modeling practice requires sensitivity analysis (SA) to ensure the model quality by analyzing the model structure, selecting the best type of model and effectively identifying the important model parameters. Global sensitivity analysis (Sobol, 2001) can be used to identify the most important input parameters, and to understand the contributions of various parameter subsets to the overall objective variation. In this paper, we formulate a general MILP model for a simple biorefinery network structure for single and multiple design scenarios. We start with a single nominal scenario that enables the selection of fuel conversion technologies, capacities, biomass locations, and the logistics of transportation from forestry resources to conversion, and from conversion to final markets. We use the optimization model to design and analyze optimal network systems that process biomass into crude bio-oil and then to biodiesel, using a realistic data set covering the Southeastern region of the United States. We compute ± deviations of the profit of the optimized single scenario design using individual ranges on problem 1740 J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751 Fig. 1. The overall structure of supply chain network system. parameters, and identify the set of parameters that make the dominant contributions to the overall variation of the profit. Based on this, we construct multiple uncertainty scenarios from combinations of the dominant parameters. We optimize the supply chain network through a multi-period formulation that contains the multiple scenarios. Robustness and global sensitivity of the optimized multiple scenario design vs. the single scenario design are analyzed via Monte Carlo simulation. The simulation of a broader set of scenarios incorporates all of the parameters, and sensitivity measures are used to check to see if any of the parameters should be included in the design algorithm. The MILP model is implemented on the commercial software GAMS (version 21.3) and uses the CPLEX (version 9.0) solver. #p#分页标题#e#
2. Problem statement 问题陈述
We consider a two step processing supply chain network depicted in Fig. 1, it includes the following elements: - A set of biomass sites where five biomass types (logging residuals, thinnings, prunings, grasses, and chips/shavings) are harvested to be used as a feedstock to conversion 1 plant. - A set of candidate sites for conversion 1 plants of four capacity options where three kinds of intermediate products (bio-oil, char and fuel gas) are manufactured to be used as feedstock or utility at conversion 2 plants or as a utility locally. - A set of candidate sites for conversion 2 plants of four capacity options where final products (gasoline and biodiesel) are manufactured and transported to the final markets. - A set of markets, where the final products are sold, with certain maximum demands. The objective is to determine the number, location, and size of the two types of processing units and the amount of materials to be transported between the various nodes of the designed network so that the overall profit is maximized while respecting the constraints associated with product demands. This optimization needs to account for a wide range of uncertainty, and the size of each scenario is substantial. This means that only a limited number of scenarios can be considered simultaneously in the mathematical optimization problem. Fig. 2 depicts a flowchart of our overall approach to, first identifying which parametric variations seem important to the objective, and then verifying that a design which accounts for them is overall robust to the full set of parameters’ variations. In the first step, optimization to determine nominal fixed design is performed based on the fourteen key parameters for the problem listed in Table 1.
   These parameters are fixed at values that are considered the most likely for the design problem. In the next step, a set of m dominant parameters among the 14 parameters will be identified using the nominal fixed design. This is achieved by considering each parameter individually at the end points of a range of values centered on the most likely values for the parameters. The parameters with the highest contributions to the positive and negative deviations of the objective function from the nominal case are selected. This selection balances the magnitude of the variation with the additional computational burden of including more scenarios. Next, the m dominant parameters are combined into 2m scenarios where the parameters are taken at the high and low ends of their expected range of variation. A new optimal design is then constructed, called the multiple scenario design, whose objective is to maximize the expected profit over all these scenarios, plus the nominal scenario. Finally, the robustness of nominal design vs. scenario design is analyzed using Monte Carlo simulation over a broader set of scenarios, selected from the entire parameter space defined by the fourteen dimensional hypercube of the original parameter ranges. Global sensitivity analysis (GSA) is performed between the m selected parameters vs. the non-selected parameters to check whether the majority of the profit variability is captured by the Fig. 2. The overall steps of optimal design with multiple scenarios, robustness analysis, and GSA.
J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738–1751 Table 1 Key parametric uncertainties. CSl. mh. Cost of transporting intermediate product h. from conversion 2 plant location l. to final market location m for scenario S CSpm Sale price of each final product p at each market location m for scenario S CShl Value of each intermediate product h at each conversion 1 plant location l for scenario S .Sikh Yield of intermediate product h from biomass type k to for the processing type i of conversion 1 plant for scenario S .Si. hh. Yield of final product p from intermediate product h for the processing type i. of conversion 2 plant for scenario S ik Relative consumption of the capacity for the processing type i of conversion 1 plant by biomass type k i. h Relative consumption of the capacity for the processing i. of conversion 2 plant by intermediate product type h ProS Probability of scenario Decision variables for the problem Xilc Binary variable indicating whether to place a conversion 1 plant of processing type i and capacity of c at location l Xi. l. c. Binary variable indicating whether to place a conversion 2 plant of processing type i. and capacity of c at location l. fSicrkl Flow rate of biomass material of type k from the biomass site of r to the conversion 1 plant of type i and capacity c at location l for scenario S fSi. c. lhl. Flow rate of intermediate product of type h from the conversion 1 site at location l to the conversion 2 plant of type i. and capacity c. at location l. for scenario S fSpl. h. m Flow rate of final product of type p from the conversion 2 site at location l. to the final market location m for scenario S fSilhl Amount of intermediate product of type h to be consumed for utility energy at conversion 1 plant of type i and location l for scenario S
3.1. Design optimization formulation: mixed integer linear program (MILP) model
3.1.1. Mass balance constraints The mass balance must be satisfied at each node of the supply chain system for each scenario. Eqs. (1) and (2) are the flow balances at the nodes of locations l and of nodes at locations l. for each scenario S, respectively. The subscript S would be dropped for the nominal design case in all the equations below. Eq. (1) states that at each conversion 1 plant location l and for each intermediate product type h, the sum of the inward flows of all biomass types (indexed by k) multiplied by their corresponding yield factor .sikh must be equal to the sum of all the outward flows of h plus the total amount of h consumed locally for utility energy. Here i is the index for processing type. Also, r is the index for the biomass locations and c is for capacity options. Eq. (2) states that, at each conversion 2 plant location l. and for each product type h , the sum of the inward flows of all the intermediate product types multiplied by the corresponding yield factor .si hh. must be equal to the sum of all the outward flows of product h . As before, i. here is the index for the processing type and c. is for capacity options in the second conversion step. Also, m is the index for the final market location and p is for the type of final products sold. For this single period model, we assumed no inventory for either the intermediates or the final products. = .S, l, h .Sikh fSicrkl fSi c lhl + fSilhl ikrc i. c. l. i (1) .Si hh. fSi c lhl = fSpl h. m .S, l ,h. (2) ih lc. pm
3.1.2. Availability/capacity constraints
The sum of the flows of each biomass type k from each biomass site r to all the conversion 1 plants for each scenario cannot exceed the total amount of kth type of biomass that can be harvested from Biomass availability for each biomass type Acquisition cost for each biomass type Cost of transporting biomass Yield of final product from intermediate product at conversion 2 processing Yield of intermediate product from biomass at conversion 1 processing Operating costs for conversion 1 & conversion 2 processing Annualized capital cost of conversion 1 & conversion 2 processing Cost of transporting intermediate products Value of each intermediate product at conversion 1 processing site Cost of transporting final products Maximum demands Sale price of each final product variation in the m inputs rather than those that were excluded from the design scenarios.
3. Mathematical models 数学模型
In this section the basic mathematical model that is used in the network design is described. Before describing the mathematical model, the input parameters, the decision variables, and the indices are listed below; Subscript indices used in the model r Biomass locations l Possible locations for conversion 1 processing l. Possible locations for conversion 2 processing m Final market locations k Biomass types h Intermediate types produced at conversion 1 h. Intermediate types produced at conversion 2 p Final product types i Conversion 1 processing types i. Conversion 2 processing types c Processing capacity options for conversion 1 c. Processing capacity options for conversion 2 S Scenarios Input variables for the problem Drl Road-travel distance between the biomass site r and conversion 1 plant site l Dll. Road-travel distance between conversion 1 plant sites l and conversion 2 plant sites l. Dl. Road-travel distance between conversion 2 plant site l. andm final market m rmSkr Available biomass for each biomass type k at each biomass site r for scenario S fdSpm,max Maximum demand for each final product p at each market location m for scenario S fdSpm,min Minimum demand for each final product p at each market location m for scenario S ci Capacity for each size option c, processing type i of conversion 1 plant c. i. Capacity for each size option c , processing type i. of conversion 2 plant CSrk Acquisition cost for each biomass type k at biomass site r for scenario S OSlick Operating cost for conversion 1 type i and size c of conversion 1 plant at location l on biomass type k as the feed for scenario S OSl. i. c. h Operating cost for processing of type i. and size c. of conversion 2 plant at location l , on intermediate product type h as the feed for scenario S Cic Annualized capital cost of conversion 1 plant of processing type i with capacity of choice c Ci. c. Annualized capital cost of conversion 2 plant of processing type i. with capacity of choice c. CSrlk Cost of transporting biomass type k from biomass site r to conversion 1 plant at location l for scenario S CSll. h Cost of transporting intermediate product of type h from conversion 1 plant location l to conversion 2 plant location l. for scenario S #p#分页标题#e#
J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738–1751 the site r. This biomass availability constraint is expressed by fSicrkl ≤ rmskr .S, r, k (3) icl We must also ensure that the sum of the biomass types coming from the different sources to each conversion 1 plant location does not exceed the chosen processing capacity at that location for each scenario case. This is also true for each conversion 2 plant location. These constraints are represented by ˉ ikfSicrkl ≤ icXilc .S, c, l (4) ikr i ˉ i hfSi c. lhl ≤ i c. Xi l c .S, c ,l. (5) i. hl i We assumed that there could exist both lower and upper bounds on the demand for each scenario case (the minimum demand level that must be satisfied and the maximum supply level that can be sold). These are expressed as constraints on the production quantity of each final product at each sink location. fSpl h m ≥ fdSpm,min .S, p, m (6) i. h. fSpl h m ≤ fdSpm,max .S, p, m (7) i. h. Finally, we assume that we are allowed to build only a single plant of each processing type at any given location, although one can choose from multiple capacity options. This constraint is given by Xilc ≤ 1 .i, l (8) c Xi l c ≤ 1 .i ,l. (9) c. 3.1.3. Objective function The objective function to be maximized is the overall Profit, which is the sum of each profit of each scenario with its probability. Each profit for each scenario can be represented by Revenue(S) – Cost(S): Overall profit = ProS(Revenue(S) . Cost(S)) (10) S Revenue(S) = SCSpm + SCShl .S (11) pm pm Cost(S) = (OSil + Cil) + (OSi l + Ci l. ) + CTSk li l. i. k + CTSh + CTSh + OSrk .S (12) hh rk The Pro(S) is the weight (0 < Pro(S) < 1) placed on each scenario in the design process, that may reflect the concern or subjective probability that user has for that scenario. The Revenue(S) includes those from selling various products in the final market plus the credits for the utility energy produced. It is the sum of the following two terms: SCSpm = CSpmfSpl. h m .S (13) l. h. SCShl = CShlfSilhl .S (14) i The Cost(S) has four main components. First is the operating cost for each scenario. Eqs. (15) and (16) are the operating costs for processing types i and i. at locations l and l , respectively. OSil = OSlickfSicrkl .S (15) ck r OSi. l = OSl i c hfSi c lhl .S (16) c. kr The second component is the annualized capital cost. The variables are not indexed by the scenario because the installed capital is a first stage decision. Eqs. (17) and (18) are the total annualized fixed cost of the chosen capacity options for processing types i and i. at locations l and l , respectively. Cil = CicXilc (17) c Ci l = Ci c. Xi l. c. (18) c. The third component is the transportation cost for each scenario. Eqs. (19)–(21) describe the three transportation cost elements related to the flows from all raw material sites to the conversion 1 sites for each feed type k, between all conversion 1 sites to conversion 2 sites for each intermediate type h, and between all conversion 2 sites to final market locations for each product type h. respectively: CTSk = CSrlkDrlfSicrkl .S (19) lr ic CTSh = CSll h. Dll. fSi c. lhl .S (20) ll. i. c. CTSh = CSl mh. Dl mfSpl h m .S (21) l. mp Finally, Eq. (23) represents the biomass acquisition cost for biomass type l at raw material site k for each scenario: OSrk = CSrkfSicrkl .S (22) cil The maximization of the objective function subject to the previously discussed constraints is a mixed integer linear program (MILP). There are efficient commercial solvers for MILPs such as GAMSTM with CPLEX solver, which we used in our case study.
3.2. Global sensitivity analysis: a Monte Carlo method The Monte Carlo method is sampling based and the objective here is to identify regions in the space of the input factors corresponding to particular values (e.g. high or low) of the output (Saltelli et al., 2008). The Monte Carlo method is based on performing multiple evaluations with randomly selected model inputs, and then using the results of these evaluations to determine both uncertainty in model predictions and apportioning to the input factors their contribution to this uncertainty (Pannell, 1997). The Monte Carlo method involves; . selection of ranges and distributions for each input factor (Step 1), . generation of a sample from the ranges and distributions specified in the first step (Step 2), . evaluation of the model for each element of the sample (Step 3), . uncertainty analysis and sensitivity analysis (Step 4).
J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738–1751 The definitions for global sensitivity indices, and the required Monte Carlo calculations, are repeated here from (Sobol, 2001): The ratios Dj1...jsisij1...jsi = (23)D are called global sensitivity indices, where D denotes the variance of the objective function. The integer si is often called the order or the dimension of the index (23). Consider an arbitrary set of u variables and v, total number of variables, 1 ≤ u ≤ v . 1, that will be denoted by one letter y =(xq1,..., xqu), 1 ≤ q1< ... < qu ≤ v, where q is a subset of variable indices among the set of u variables and let z be the set of v . u complementary variables. Thus x =(y,z). Let Q = (q1,...,qu). The variance corresponding to the subset y can be defined as u Dy . Dj1,...,jSi (24) si= 1(j1<...<jsi) ∈ Q The sum in (24) is extended over all groups (j1,...,jsi) where all the j1,...,jsi belong to Q. Similarly, the variance corresponding to the subset z, Dz can be introduced. Then the total variance corresponding to the subset y is Dtot y = D . Dz One can notice that Dtot is also a sum of sij1...jsi; but it is y extended over all groups (j1,...,jsi) where at least one jl ∈ Q. Here 1 ≤ si ≤ v. Two global sensitivity indices for the subset y are introduced. yDy sitot Dtot siy = ,y = . DD sitot = 1 . siz and always 0 ≤ siy ≤ sitot ≤ 1. The most informative are the extreme situations: yy sitot. siy == 0 means that f(x) does not depend on y,y sitot. siy == 1 means that f(x) depends on y only.y Subset’s variance Dy is equal to Dy = f (x)f (y, z. )dxdz . f 2 (25)0 A formula similar to (25) can be written for Dz: Dz = f (x)f (y,z)dxdy . f 2 (26)0 Thus, for computing siy and sitot = 1 . siz one has to estimate y four integrals: f (x)dx, f 2(x)dx, f (x)f (y, z. )dxdz. , and f (x)f (y,z)dxdy. Now a Monte Carlo method can be constructed (Sobol, 2001). Consider two independent random points and . uniformly distributed in Iv and let =( , ), . =( . , . ). implies the random sets of the y parameters and . is another randomly chosen set of the y parameters. and . implies the random values for variables in the complimentary sets of z parameters in both cases. Each Monte Carlo trial requires three computations of the model: f( ) f( , ), f( , . ), and f( . , ). After N trials, Monte Carlo estimates are obtained: N 1 Pf ( j).→ f0N j= 1 N 1 P f ( j)f ( j, j. ).→ Dy + f02 N j= 1 N 1 Pf 2( j).→ D + f 2 0N j= 1 N 1 Pf ( j)f ( . j, j).→ Dz + f02 (27)N j= 1 Sobol (2001) states that the stochastic convergence above is implied by the square integrability of f(x). We can estimate an average of optimal value with the random parameter sets, f0, the variance of the objective function, D, and two subset variances Dy and Dz using Eq. (27).
4. Case study 案例分析
In this case study, we examine a supply chain network design problem for thermo-chemical conversion of biomass into biofuels, gasoline and diesel, for the Southeastern part of the United States. The data used were developed with an industrial collaborator and are representative estimates of various parameters that define the problem. We examine a particular processing option where the first step (‘conversion 1’) is Fast Pyrolysis to produce a bio-oil, and the second step (‘conversion 2’) is a Fischer Tropsch (FT) synthesis process that converts the bio-oil to a liquid fuel through a syngas intermediate. This process can utilize a wide array of biomass types as its feedstock. The advantage of the bio-oil intermediate is that it is cheaper to transport because it concentrates the energy from the biomass into a smaller volume. Hence there could be some advantage to having a decentralized processing of the biomass to bio-oil followed by shipping the bio-oil to larger scale FT plants. The region of interest in this case study consists of ten states (Oklahoma, Arkansas, Louisiana, Mississippi, Alabama, Tennessee, Georgia, Florida, South Carolina, and North Carolina). Five different biomass types (logging residuals, thinnings, prunings, grasses and chips/shavings) are harvested at 30 biomass source locations. These biomass materials can be converted into three intermediate products (bio-oil, char, and fuel gas) at any number of 29 possible locations for conversion 1 plants. Only bio-oil is transported to and converted into two final products (gasoline and biodiesel) at any number of 10 possible conversion 2 plant locations. Char and fuel gas are to be consumed locally as utility energy sources at conversion 1 plants locations. The final products are transferred to 10 final markets for sale. Each plant, for both conversions, has four different capacity options. Latitude and longitude of each location were provided to us and we converted them into distance matrices using geographic information systems. Key parameter values are summarized in Tables 2–11.
5. Results 结果
5.1. Single nominal scenario supply network design
The proposed optimization MILP model is tested for designing nominal single scenario network systems. In the optimized nominal single scenario network biomass resources are transferred to 15 conversion 1 processing locations selected from the 27 biomass sites. The 15 selected conversion 1 sites include one of the conversion 2 processing locations. Bio-oil converted at conversion 1 processing plants is fed to three conversion 2 processing units. Two final products, gasoline and biodiesel, are delivered to 10 final #p#分页标题#e#
1744 J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751
  Table 2 Biomass material volume at biomass locations (rmkr) [tons]. BM1 BM2 BM3 BM4 BM5 BM6 BM7 BM8 BM9 BM10 Logging residuals 0 0 103600 155400 184800 0 168000 0 319000 139320 Thinnings 116550 116550 0 0 0 50400 0 100050 0 0 Prunings 38850 38850 25900 38850 46200 16800 42000 33350 79750 34830 Grasses 0 0 621600 932400 0 0 0 0 0 835920 Chips/shavings 305000 297000 188000 202000 291000 198000 186000 198000 221000 246000 BM11 BM12 BM13 BM14 BM15 BM16 BM17 BM18 BM19 BM20 Logging residuals 77400 0 77400 0 0 0 561600 0 0 0 Thinnings 0 0 0 0 189000 231000 0 105300 155400 77700 Prunings 19350 0 19350 0 63000 77000 140400 35100 51800 25900 Grasses 464400 0 0 0 0 0 0 0 1243200 621600 Chips/shavings 228000 301000 297000 285000 192000 0 0 0 0 0 BM21 BM22 BM23 BM24 BM25 BM26 BM27 BM28 BM29 BM30 Logging residuals 0 116100 38700 92880 0 131580 0 100620 0 0 Thinnings 116550 0 0 0 105000 0 95700 0 0 0 Prunings 38850 29025 9675 23220 35000 32895 31900 25155 0 0 Grasses 932400 696600 0 557280 0 789480 0 603720 0 0 Chips/shavings 0 0 0 0 0 0 0 0 0 0 Table 3 Maximum final demand at final market locations (fdpm) [tons]. FM1 FM2 FM3 FM4 FM5 FM6 FM7 FM8 FM9 FM10 Gasoline 222,100 813,800 786,500 918,700 590,500 331,200 400,200 307,700 343,900 319,100 Biodiesel 111,100 406,900 393,200 459,300 295,300 165,600 200,100 153,900 171,900 159,600 Table 4 Capacity volume on a raw material basis and fixed cost of processing types of conversions. Fast pyrolysis (conversion 1) process ( ci) (C1(1)–C1(19)) Plantation Reference plant Large coop Scale Capacity [tons] 1,225,000 2,450,000 3,675,000 4,900,000 Fixed cost [$/year] 7,728,000 12,768,000 17,808,000 22,848,000 Fast pyrolysis (conversion 1) process ( ci) (C1(20)–C1(29)) Plantation Reference plant Large coop Scale Capacity [tons] 1,750,000 3,500,000 5,250,000 7,000,000 Fixed cost [$/year] 9,119,000 15,046,000 20,974,000 26,992,000 Fischer Tropsch (conversion 2) process ( c i ) (C2(1)–C2(10)) Small Medium Large Scale Capacity [tons] 1,750,000 3,500,000 5,250,000 7,000,000 Fixed cost [$/year] 52,618,000 79,228,000 99,671,000 117,183,000 Table 5 Yield parameters. Biomass types to intermediate products at the processing types of conversion 1 (.ikh) [dimensionless] Bio oil Char Fuel gas Fast pyrolysis Logging residuals 0.70 0.15 0.15 Thinnings 0.65 0.25 0.10 Prunings 0.65 0.25 0.10 Grasses 0.80 0.10 0.10 Chips/shavings 0.75 0.20 0.05 Intermediate products to final products at the processing types of conversion 2 (.i hh ) [dimensionless] Gasoline Biodiesel Fischer Tropsch Bio oil 0.40 0.20 Char 0.00 0.00 Fuel gas 0.00 0.00
J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751 1745
   Table 6 Transportation cost [$ per ton per mile]. Biomass from biomass site to conversion 1 location (Crlk) Logging Residuals Thinnings Prunings Grasses Chips/shavings Transportation cost 0.25 0.25 0.28 0.31 0.19 Intermediate product from conversion 1 location to conversion 2 location (Cll h) Transportation cost Bio oil Char Fuel gas 0.10 0.10 0.10 Final product from conversion 2 location to final market (Cl mh ) Gasoline Biodiesel Transportation cost 0.10 0.10 Table 7 Value of final product at final market (Cpm) [$/tons]. FM1 FM2 FM3 FM4 FM5 FM6 FM7 FM8 FM9 FM10 Gasoline 469.00 457.00 479.00 425.00 489.00 475.00 505.00 441.00 462.00 436.00 Biodiesel 568.00 562.00 532.00 514.00 548.00 542.00 550.00 523.00 547.00 539.00 market locations. Supply chain networks from forestry resources to conversion 1, from conversion 1 to conversion 2, and from conversion 2 to final markets, are shown in Fig. 3. We analyzed the change in the objective function value using discrete values of the parameters at the nominal plus or minus given percentage changes (.50%, .30%, .10%, 10%, 30%, and 50%) for the optimized nominal single design. In this preliminary screening, one parameter at a time is varied, while the rest have their values at the mid-point of the range. The 14 parameters are listed in Table 1. Changes of the sale price in the final market affect the overall profit the most, as graphed in Fig. 4. The next dominant parameters are the conversion yield ratios of both conversion 1 Table 8 Acquisition cost of biomass materials at biomass locations (Crk) [$ per ton]. Logging residuals Thinnings Prunings Grasses Chips/shavings BM1 25.0 25.0 20.0 35.0 50.0 BM2 25.0 25.0 20.0 35.0 50.0 BM3 25.0 25.0 20.0 35.0 50.0 BM4 25.0 25.0 20.0 35.0 50.0 BM5 30.0 30.0 24.0 42.0 60.0 BM6 30.0 30.0 24.0 42.0 60.0 BM7 30.0 30.0 24.0 42.0 60.0 BM8 27.5 27.5 22.0 38.5 55.0 BM9 27.5 27.5 22.0 38.5 55.0 BM10 25.0 25.0 20.0 35.0 50.0 BM11 25.0 25.0 20.0 35.0 50.0 BM12 25.0 25.0 20.0 35.0 50.0 BM13 25.0 25.0 20.0 35.0 50.0 BM14 25.0 25.0 20.0 35.0 50.0 BM15 25.0 25.0 20.0 35.0 50.0 BM16 25.0 25.0 20.0 35.0 50.0 BM17 25.0 25.0 20.0 35.0 50.0 BM18 25.0 25.0 20.0 35.0 50.0 BM19 25.0 25.0 20.0 35.0 50.0 BM20 25.0 25.0 20.0 35.0 50.0 BM21 25.0 25.0 20.0 35.0 50.0 BM22 25.0 25.0 20.0 35.0 50.0 BM23 25.0 25.0 20.0 35.0 50.0 BM24 25.0 25.0 20.0 35.0 50.0 BM25 30.0 30.0 24.0 42.0 60.0 BM26 25.0 25.0 20.0 35.0 50.0 BM27 27.5 27.5 22.0 38.5 55.0 BM28 25.0 25.0 20.0 35.0 50.0 BM29 27.5 27.5 22.0 38.5 55.0 BM30 27.5 27.5 22.0 38.5 55.0 and conversion 2 processes. Maximum demand and biomass availability have an effect on overall profit when at the lower ends of the ranges explored. The other 9 parameters have relatively less influence over the nominal design profit, thus five parameters are chosen to construct multiple scenarios for design purposes, and for sensitivity studies evaluated through Monte Carlo simulation. 5.2. Multiple scenario supply network design The five dominant parameters form the basis for the scenario set: S = [yield of final product, yield of intermediate product, sale price of final product, maximum demand and biomass availability]. Each scenario is created by varying the parameters by ±20%. We assume that availabilities of five biomass types in 30 biomass sites are varied simultaneously not independently. Sale prices and maximum demands of the two final products from 10 markets are also changed simultaneously. The scenarios are numbered using a binary encoding scheme where the number is given by i(Bi * 2i) + 1, where i refers to the position of the parameter in the list S above in order, and Bi = 1 if the parameter is at +20% and zero if at .20%. For example, the 16th scenario is [0, 1, 1, 1, 1] + 1, which means all the parameters except for biomass availability are at +20%. All 33 scenarios are created based on the 32 combinations and a nominal scenario. A multi-period MILP model was constructed to optimize another flow network design considering the 33 scenarios simultaneously as shown in Fig. 5. The scenarios couple the first stage decisions on the size and location of the processing infrastructure, but the actual flows can be different in each scenario. It was assumed that the weight of each scenario was the same. In the optimized multiple scenarios network, biomass resources are transferred to 14 conversion 1 processing locations selected from the 27 biomass sites. The 14 selected conversion 1 sites include two of the conversion 2 processing locations. Bio-oil converted at conversion 1 processing plants is fed to two conversion 2 processing units. Gasoline and biodiesel are delivered to 10 final product demand locations from the two converion2 locations. Three conversion 1 locations (C1(7), C1(16), and C1(17)) in the single scenario design are not present in the multi-period solution, instead, C1(21) is newly sited in the multiple scenarios design. The location is same site for conversion 2 (C2(2)), which is also selected in the multiple scenarios design instead of C2(5) and C2(8) of the single scenario design.
1746 J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751 Fig. 3. Optimized flow networks from biomass to biofuels with nominal single scenario. a. Louisiana/Arkansas/Oklahoma. b. Mississippi/Alabama/Louisiana/Georgia/North Carolina.
J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751 1747
   Table 9 Operating cost of processing type of conversion 1 on biomass materials as feed (Olick) [$ per ton]. Logging residuals Thinnings Prunings Grasses Chips/shavings C1(1)–C1(19) Fast pyrolysis Plantation 13.38 13.70 14.04 16.12 15.62 Reference plant 11.37 11.65 11.93 13.70 13.28 Large coop 10.40 10.65 10.91 12.53 12.14 Scale 9.73 9.97 10.21 11.72 11.36 C1(20) Fast pyrolysis Plantation 10.22 15.09 15.35 7.02 9.29 Reference plant 10.22 13.97 14.22 6.37 8.96 Large coop 10.22 13.44 13.69 5.31 8.43 Scale 10.22 13.11 13.36 3.05 7.30 C1(21) Fast pyrolysis Plantation 9.73 14.38 14.62 6.69 8.84 Reference plant 9.73 13.30 13.54 6.07 8.53 Large coop 9.73 12.80 13.04 5.06 8.03 Scale 9.73 12.49 12.73 2.91 6.95 C1(22) Fast pyrolysis Plantation 9.97 14.74 14.98 6.86 9.07 Reference plant 9.97 13.63 13.88 6.22 8.75 Large coop 9.97 13.12 13.36 5.19 8.23 Scale 9.97 12.80 13.04 2.98 7.13 C1(23) Fast pyrolysis Plantation 9.97 14.74 14.98 6.86 9.07 Reference plant 9.97 13.63 13.88 6.22 8.75 Large coop 9.97 13.12 13.36 5.19 8.23 Scale 9.97 12.80 13.04 2.98 7.13 C1(24) Fast pyrolysis Plantation 10.22 15.09 15.35 7.02 9.29 Reference plant 10.22 13.97 14.22 6.37 8.96 Large coop 10.22 13.44 13.69 5.31 8.43 Scale 10.22 13.11 13.36 3.05 7.30 C1(25) Fast pyrolysis Plantation 9.97 14.74 14.98 6.86 9.07 Reference plant 9.97 13.63 13.88 6.22 8.75 Large coop 9.97 13.12 13.36 5.19 8.23 Scale 9.97 12.80 13.04 2.98 7.13 C1(26) Fast pyrolysis Plantation 9.97 14.74 14.98 6.86 9.07 Reference plant 9.97 13.63 13.88 6.22 8.75 Large coop 9.97 13.12 13.36 5.19 8.23 Scale 9.97 12.80 13.04 2.98 7.13 C1(27) Fast pyrolysis Plantation 9.73 14.38 14.62 6.69 8.84 Reference plant 9.73 13.30 13.54 6.07 8.53 Large coop 9.73 12.80 13.04 5.06 8.03 scale 9.73 12.49 12.73 2.91 6.95 C1(28) Fast pyrolysis Plantation 9.73 14.38 14.62 6.69 8.84 Reference plant 9.73 13.30 13.54 6.07 8.53 Large coop 9.73 12.80 13.04 5.06 8.03 Scale 9.73 12.49 12.73 2.91 6.95 C1(29) Fast pyrolysis Plantation 9.73 14.38 14.62 6.69 8.84 Reference plant 9.73 13.30 13.54 6.07 8.53 Large coop 9.73 12.80 13.04 5.06 8.03 Scale 9.73 12.49 12.73 2.91 6.95 Table 10 Operating cost of processing type of conversion 2 on intermediate products as feed (Ol i c h) [$ per ton]. Fischer Tropsch Small Medium Large Scale C2(1) Bio-oil 92.53 69.94 59.36 52.83 C2(2) Bio-oil 88.12 66.61 56.53 50.31 C2(3) Bio-oil 90.32 68.28 57.95 51.57 C2(4) Bio-oil 90.32 68.28 57.95 51.57 C2(5) Bio-oil 92.53 69.94 59.36 52.83 C2(6) Bio-oil 90.32 68.28 57.95 51.57 C2(7) Bio-oil 90.32 68.28 57.95 51.57 C2(8) Bio-oil 88.12 66.61 56.53 50.31 C2(9) Bio-oil 88.12 66.61 56.53 50.31 C2(10) Bio-oil 88.12 66.61 56.53 50.31 Table 11 Value of intermediate product at conversion locations (Chl and Chl ) [$/tons]. C1(1) C1(2) C1(3) C1(4) C1(5) C1(6) C1(7) C1(8) C1(9) C1(10) Char 40.0 40.0 40.0 40.0 48.0 48.0 48.0 44.0 44.0 40.0 Fuel gas 20.0 20.0 20.0 20.0 24.0 24.0 24.0 22.0 22.0 20.0 C1(11) C1(12) C1(13) C1(14) C1(15) C1(16) C1(17) C1(18) C1(19) Char 40.0 40.0 40.0 40.0 40.0 40.0 44.0 64.0 40.0 Fuel gas 20.0 20.0 20.0 20.0 20.0 20.0 22.0 32.0 20.0 C1(20) C1(21) C1(22) C1(23) C1(24) C1(25) C1(26) C1(27) C1(28) C1(29) Char 48.0 40.0 44.0 48.0 48.0 44.0 44.0 40.0 40.0 40.0 Fuel gas 24.0 20.0 22.0 24.0 24.0 22.0 22.0 20.0 20.0 20.0 #p#分页标题#e#
1748 J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751 Fig. 4. Analysis of 14 possible influential parameters in the optimized nominal single scenario design, change in profit with percentage change in the parameter value from the nominal case. 5.3. Robustness and global sensitivity analysis: single scenario design vs. multiple scenarios design We used the single scenario design model for each 33 scenario to maximize the overall profits and examined the robustness of the single nominal scenario design and multiple scenarios design. In
   Table 12, the “Optimal Design” column is the best values for each scenario, when all decisions of location, plant size, and flow are made simultaneously for that scenario. The “Nominal Single Scenario Fixed Design” column is the best values when the optimized locations and plant sizes (Fig. 3) are found for the nominal scenario (S0) and only the flows are allowed to change in each scenario. The Table 12 Robustness analysis. Scenario typeb Optimal design (OD) [$]a Nominal single scenario fixed design (NFD) [$]a Percentage of (OD-NFD)/OD (%) Multiple Scenario fixed design (SFD) [$]a Percentage of (OD-SFD)/OD (%) S0 – 1505.9 M 1505.9 M 0.0 1490.1 M 1.0 S1 00000 89.6 M .47.9 M 153.4 .11.4 M 112.7 S2 00001 654.7 M 588.3 M 10.1 593.6 M 9.3 S3 00010 422.3 M 383.5 M 9.2 396.2 M 6.2 S4 00011 1025.3 M 930.8 M 9.2 974.2 M 5.0 S5 00100 754.8 M 681.5 M 9.7 698.1 M 7.5 S6 00101 1732.6 M 1673.9 M 3.4 1680.1 M 3.0 S7 00110 1485.2 M 1469.1 M 1.1 1465.8 M 1.3 S8 00111 2204.0 M 2109.5 M 4.3 2153.0 M 2.3 S9 01000 112.5 M .19.6 M 117.5 15.3 M 86.4 S10 01001 716.1 M 646.3 M 9.7 655.5 M 8.5 S11 01010 475.7 M 441.2 M 7.2 449.3 M 5.5 S12 01011 1412.5 M 1398.5 M 1.0 1384.8 M 2.0 S13 01100 788.7 M 720.8 M 8.6 734.9 M 6.8 S14 01101 1817.5 M 1744.0 M 4.0 1756.9 M 3.3 S15 01110 1560.9 M 1538.3 M 1.4 1541.6 M 1.2 S16 01111 3042.9 M 3026.9 M 0.5 3015.0 M 0.9 S17 10000 154.2 M 47.4 M 69.3 100.9 M 34.6 S18 10001 782.9 M 714.0 M 8.8 759.2 M 3.0 S19 10010 576.4 M 492.8 M 14.5 547.7 M 5.0 S20 10011 1064.6 M 962.5 M 9.6 1010.0 M 5.1 S21 10100 1131.4 M 1028.2 M 9.1 1116.7 M 1.3 S22 10101 1961.6 M 1892.7 M 3.5 1937.9 M 1.2 S23 10110 1755.1 M 1671.5 M 4.8 1726.4 M 1.6 S24 10111 2243.3 M 2141.2 M 4.6 2188.7 M 2.4 S25 11000 179.9 M 95.3 M 47.1 143.8 M 20.1 S26 11001 1058.8 M 965.2 M 8.8 1038.5 M 1.9 S27 11010 708.7 M 652.6 M 7.9 686.3 M 3.2 S28 11011 1664.7 M 1591.6 M 4.4 1636.1 M 1.7 S29 11100 1207.1 M 1083.9 M 10.2 1187.9 M 1.6 S30 11101 2689.1 M 2459.3 M 8.5 2634.5 M 2.0 S31 11110 2333.1 M 2126.2 M 8.9 2014.9 M 13.6 S32 11111 3432.8 M 3359.6 M 2.1 3404.1 M 0.8 Average 1295.3 M 1214.4 M 6.2 1246.3 M 3.8 a M = millions of dollars. b (0 = .20%, 1 = +20%, – no change from base case, biomass availability, maximum demand, sale price, intermediate yield, final yield from highest to lowest digit in binary code.).
J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751 1749 Fig. 5. Optimized flow networks from biomass to biofuels by considering 33 scenarios. (a) Louisiana/Arkansas/Oklahoma. (b) Mississippi/Alabama/Louisiana/Georgia/North Carolina.
1750 J. Kim et al. / Computers and Chemical Engineering 35 (2011) 1738– 1751
  Table 13 Global sensitivity analysis for both designs: the 5 parameters vs. the 9 parameters. Nominal single scenario fixed design Multiple scenario fixed design f0 1.56E+09 1.58E+09 D 1.12E+18 1.13E+18 siy 0.96 0.97 siz 0.02 0.01 “Multiple Scenario Fixed Design” column is the best values when the optimized design (Fig. 5) is found considering all 33 scenarios, equally weighted, to fix the locations and plant size, and then the optimal flows found for each scenario with this design. The deviations from the optimal design are also calculated for both designs. The multiple scenarios design can match the optimal designs more closely in profit, as seen by lower percentage changes, than the single scenario design. It does this with less than 10% variation in twenty eight of the scenarios, but those scenarios in which the biomass flow is particularly low neither design can overcome their higher fixed costs to match the optimal designs which have smaller networks (S0, S7, S12, S16, and S31). To further explore the quality of the two solutions, we performed the global sensitivity analysis using the dominant 5 parameters vs. the others 9 parameters. In Eq. (28), is a set of the 5 parameters chosen randomly and is another randomly chosen set of the same 5 parameters. and are the random values for variables in the complimentary sets of the 9 parameters in both cases. We created 5,000 random cases of the four different sets using the GAMS random number generator. The results of the sensitivity analysis are given in Table 13. The multiple scenarios design has higher values of the sum of 5000 random parameter sets. siy can be interpreted as how much of the variance in the design objective function depends on the 5 parameters and siz represents this for the 9 parameters. Both designs demonstrate that most of the sensitivity of the objective is explained by the five main parameters, with the multiple scenario design capturing slightly more of that variation than the single scenario design. This approach can be used to confirm that designs are not sensitive to parameters that have been left out of the design and higher level interactions, caused by covariation of the parameters, can also be screened this way (Sobol, 2001). However, this requires substantial additional computational effort, as the various subsets of parameters must be explored alone, and in combination with each other, to be able to extract these effects.
6. Conclusions 总结
The objective of this study was to explore the designs of a biofuels network in the SE region of the US considering Fast Pyrolysis and Fischer Tropsch bio-diesel conversion processes with uncertainty represented through scenarios. An optimization model was developed that enables decision making for the infrastructure of biofuel conversion processing including processing locations, volumes, supply networks, and the logistics of transportation from forestry resources to conversion and from conversion to final markets. First, we used our optimization model to design an optimal network with single nominal scenario attaining maximum profit based on a realistic data set provided by our industrial partner. Second, we analyzed the value of the objective function for the extreme points of the range of values of the main 14 parameters for the optimized single scenario design. Based on the range analysis, we selected the 5 dominant parameters (biomass availability, maximum demands, sale price of each final product, yield of intermediate product, and yield of final product). For the next step, we optimized a new supply chain networks by considering the 33 scenarios, which are created by the combinations of the high and low values of the 5 parameters and the nominal scenario. In order to estimate the performance of the multiple scenario design for the 5 parameters, we performed the robustness analysis and Monte Carlo global sensitivity analysis vs the single scenario design. Based on the robustness and GSA, we concluded that the optimized multiple scenario design was able to mitigate the impact of the variation and did not “miss” any major contributor to the overall profit variation. For future research, we have identified several promising directions for the biorefinery network design problem. The issues to be examined include: . Improving the formulation to include the possibility of a more complex structure in the network, specifically more options for processing and the inclusion of mobile processing infrastructure. . Extending the single period model to one that can account for the development of the infrastructure over time. . Integrating the new infrastructure with existing facilities to improve the joint mass and energy efficiency of the network, specifically the heat and mass integration of biofuels and chemicals plants with existing wood and pulp processing plants. . Extending the sensitivity analysis to include correlated uncertainty in the parameters and detecting important interactions in the parameters.
Acknowledgements 致谢
The Authors gratefully acknowledge the partial financial support of the National Science Foundation through grant CBET #093392. Dr. Ludwig Furtner and Dr. Craig Whittaker, of Weyerhaeuser NR, provided significant input into the paper through discussions and feedback on results.
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