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Research Article pubs.acs.org/journal/ascecg

Unraveling Optimal Biomass Processing Routes from Bioconversion Product and Process Networks under Uncertainty: An Adaptive Robust Optimization Approach Jian Gong, Daniel J. Garcia, and Fengqi You* Department of Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States S Supporting Information *

ABSTRACT: A bioconversion product and process network converts different types of biomass to various fuels and chemicals via a plethora of technologies. Reliable bioconversion processing pathways should be designed considering the effect of uncertain parameters, such as biomass feedstock price and biofuel product demand. Given a large-scale bioconversion product and process network of 194 technologies and 139 materials/compounds, we propose a two-stage adaptive robust mixed-integer nonlinear programming problem. The model allows for decisions at the design and operational stages to be made sequentially and considers budgets of uncertainty to control the level of robustness. Nonlinearity in this model appears in the first-stage objective function, and the secondstage problem is a linear program. We efficiently solve the proposed problem with a tailored algorithm. The robust optimal solutions corresponding to various uncertainty budgets show that the minimum total annualized cost is more sensitive to biofuel demand uncertainty compared to biomass feedstock price uncertainty. KEYWORDS: Two-stage adaptive robust optimization, Network optimization, Biomass, Uncertainty, MINLP

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INTRODUCTION Substitution of fossil-based fuels and chemicals for biofuels and bioproducts could be a promising remedy to slow global warming and climate change. The scientific foundation of this opinion lies in the fact that atmospheric carbon dioxide can be captured during biomass growth and accumulation. After decades of development, there are many bioconversion processing pathways available featuring a variety of biomass feedstocks and products, but most comparisons among these pathways remain empirical. Identifying optimal processing pathways of product and process network models is crucial for the competitiveness and sustainability of the resulting processing pathway.1−9 While parameters are treated as deterministic in most existing models, in practice uncertainty can arise due to unpredictable variations in compositions of feed streams, product demands, and material prices.10 An optimal pathway that ignores risks and considers only the nominal parameter values may lead to suboptimal or even infeasible solutions when uncertainties are inevitably realized.11,12 Therefore, it is both practical and crucial to address uncertainty in the bioconversion processing network optimization model in order to obtain robust optimal pathways.13,14 As uncertainty data for internal factors, such as process yields and technology conversion coefficients, are not available for many bioconversion © XXXX American Chemical Society

technologies, we focus on well-documented, external sources of uncertainty. We consider the case where biomass supplies in our network are relatively abundant while their prices are volatile in the market. Specifically, we consider biomass feedstock price and biofuel product demand uncertainty in this work as they are defined with well-documented data. There are several systematic methods for handling uncertainties in process and network design problems. One well-known method focuses on process flexibility.10,15−17 Although a general framework for optimal design under uncertainty has been proposed,10 existing strategies for process flexibility problems require fixed process designs. Another method for handling uncertainty is stochastic programming,18 and several tailored solution methods have been proposed.19−21 Stochastic programming allows simultaneous optimization of design variables under uncertainty, but this method requires both consideration of a moderate number of scenarios and predefined probability distribution functions that are usually difficult to acquire in practice. A trending methodology for handling uncertainty is robust optimization.22−28 Robust optimization approaches do not require probability distribuReceived: January 27, 2016 Revised: April 21, 2016

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DOI: 10.1021/acssuschemeng.6b00188 ACS Sustainable Chem. Eng. XXXX, XXX, XXX−XXX

ACS Sustainable Chemistry & Engineering

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PROBLEM STATEMENT The problems addressed in this work can be formally stated as follows. As depicted in Figure 1, we are given a comprehensive

tions for uncertain parameters and allow tractable solution methods for certain types of uncertainty sets.29 Recent contributions for network systems design either consider only deterministic models30,31 or address uncertainty using stochastic programming approaches.32,33 To the best of our knowledge, no previous investigation applies two-stage adaptive robust optimization to a product and process network design and optimization problem. Moreover, the first-stage optimization problems of existing adaptive robust optimization models in the literature are restricted to mixed-integer linear programming (MILP) problems, which unfortunately are not suitable for a superstructure design problem with separable nonconvex terms for modeling the economies of scale. As a result, it remains challenging to develop a two-stage adaptive robust optimization model to capture sequential decisionmaking processes under uncertainty in a product and process network design problem. A further challenge lies in the development of an efficient solution algorithm for the resulting problem, since two-stage adaptive robust mixed-integer nonlinear programming (MINLP) problems cannot be solved directly by any off-the-shelf optimization solvers. In this work, we focus on a large-scale bioconversion process and product network with biomass feedstock acquisition, processing, upgrading, and production of final products. We develop a deterministic MINLP model for minimizing the processing pathway’s total annualized cost. Two robust counterparts, a conventional static robust MINLP and a twostage adaptive robust MINLP, are proposed based on the deterministic model. In the adaptive robust optimization model, decisions are made at two stages: (1) the design stage pursuing the minimum total annualized cost that includes capital expenditures (CAPEX) and fixed operating expenses (OPEX); and (2) the operational stage pursuing the minimum OPEX after the first-stage decisions are made and uncertainty is revealed. Moreover, budgets of uncertainty are introduced in the uncertainty sets to adjust the level of conservatism. In order to model economies of scale, nonlinear terms are employed in the first-stage objective function of the two-stage adaptive robust optimization model. Moreover, the second-stage recourse problem is a linear program. It is worth noting that the resulting two-stage adaptive robust MINLP (a min−max− min problem) cannot be solved by any general-purpose solvers directly. In order to address this computational challenge, we develop a novel solution strategy that integrates a column-andconstraint generation algorithm and a branch-and-refine algorithm. We show that the proposed two-stage adaptive robust optimization method provides insights and perspectives on how different uncertainty levels affect the optimization results, i.e. how the solutions hedge against uncertainty. Such insights cannot result from deterministic or conventional static robust optimization techniques. The rest of the paper is organized as follows. In the next section, we provide necessary information on the bioconversion processing pathway optimization problem. The formulation of a deterministic MINLP, a conventional static robust MINLP, and a two-stage adaptive robust MINLP as well as the solution strategy for the two-stage adaptive robust MINLP are presented and described in the following section. We present the optimal solutions and computational performance in the Results and Discussion section, followed by Conclusions at the end of the article.

Figure 1. Structure of the product and process network considered in this work.

bioconversion network for converting biomass to biofuels and chemicals with 194 technologies and 139 materials/compounds. The complex bioconversion product and process network contains a plethora of biomass feedstocks, bioconversion technologies, and potential products, providing a large, complex platform from which to choose the optimal pathways given various conditions and restrictions. The network is divided into four components: biomass feedstocks, processing technologies, upgrading technologies, and final products. A condensed depiction of the network with a sampling of technologies and materials/compounds is shown in Figure 2. Considering the entire network allows us to take a comprehensive view of the processing pathways without loss of generality. In contrast, prescreening is essentially a heuristic method that helps reduce the size of the problem, but could eliminate possible solutions. As we will show later, we develop a solution method such that computational times are relatively short for all cases. Thus, considering the whole network does not impede solution of the problem. Much of the data collection strategy and some data corresponding to technologies for this network can be found in previous works.34−36 A list of technologies used in this model as well as their inputs and outputs can be found in the SI. Model parameters include the following: B


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• Variable OPEX are assumed to scale linearly with capacity. • The identified pathways can be adjusted to meet the uncertain demands as long as the demands do not exceed the capacities. The goal of the deterministic problem is to determine the optimal bioconversion processing pathway by minimizing the pathway’s total annualized cost. Additionally, the robust optimization problems consider the same objective function while taking biomass feedstock price and biofuel product demand uncertainties into account. All the problems consider mass balance and economic evaluation constraints. The major decision variables include the following: • Technology pathway selection • Sizing of each technology in the pathway • Operating level of each technology in the pathway • Quantities of feedstocks to purchase and products to sell during operation • CAPEX and OPEX

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METHODOLOGY

Deterministic MINLP Model. A new deterministic MINLP model (DM) is proposed to minimize the total annualized cost of a bioconversion processing pathway. Instead of using technology capacity directly,34,35 a variable for technology operating levels is introduced in the mass balance constraints, and it is bounded above by its corresponding technology capacity. In addition, we introduce integer variables to set lower and upper bounds of technology capacities. The change reflects the fact that real facilities may not always be in full operation, perhaps due to exogenous uncertainties. (DM) is also equipped with a compact objective function to show the dependence of total annualized cost on all decision variables. As a result, (DM) is more suitable for extension to robust optimization problems and permits simpler model reformulation.

Figure 2. Simplified structure of the biomass processing network used in this study. Representative technologies and materials/compounds are shown.

• The upper and lower bounds of the capacity of each technology • The conversion coefficients for each process output • The availability of each biomass feedstock • The uncertain demands for each final product • A base capacity for each technology • An initial capital cost corresponding to the base capacity for each technology • Expected life span in years of the processing pathway • Interest rate • The fixed OPEX for each technology • An initial, variable OPEX corresponding to the base capacity for each technology • The average trucking distance to transport the biomass feedstocks to the processing facility • The variable transportation and fixed transportation costs for biomass • Uncertain biomass feedstock prices Assumptions include the following: • All purchased biomass feedstocks are successfully converted. • Given that several products can be coproduced in the same network, we assume there are other markets to sell extra products into if the demands are exceeded. • All biomass feedstocks are transported to the facility solely by diesel-burning trucks. • Fixed costs and some components of the variable OPEX are set fractions of the capital cost. • Capital costs scale with capacity via concave power functions.

(DM) sf

min ∑ c1, jQ j j +

∑ c2,jWj + ∑ c3,iPi j∈J

j∈J

i∈I

(1)

∀j∈J

(2)

s.t.

a1, jYj ≤ Q j ≤ a 2, jYj , Wj ≤ Q j , Pi − Si +

∀j∈J

(3)

∑ a3, i , jWj = 0,

∀i∈I (4)

j∈J

Pi ≤ bi ,

∀i∈I

(5)

Si ≥ di ,

∀i∈I

(6)

Q j , Wj , Pi , Si ≥ 0,

Yj ∈ {0, 1},

∀ i ∈ I, j ∈ J

(7)

In (DM), the constraints include those for the design stage and for the operational stage. At the design stage, binary variable Yj represents the selection of technology j, and continuous variable Qj represents the capacity of technology j. At the operational stage, Wj denotes the operating level of technology j, and Pi and Si denote the quantities of biomass feedstock i purchased and biofuel product i produced/sold, respectively. Various parameters are employed in the deterministic MINLP model: a1,j and a2,j represent the lower and upper bounds of the capacity of technology j, respectively, a3,i,j represents the conversion coefficient of compound i in technology j, bi is the availability of biomass feedstock i, di is the demand for biofuel product i, and c1,j, c2,j, and c3,i are parameters for economic evaluation and are given in (A1)−(A3) in the Appendix. C


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ACS Sustainable Chemistry & Engineering Constraint (2) quantifies the lower and upper bounds for the capacity of each technology. Constraint (3) sets an upper bound to the operating level of each technology at the operational stage. Constraint (4) describes the mass balance between biomass feedstocks and intermediate/biofuel products at the operational stage. Biomass feedstock availability and biofuel product demand are represented by constraints (5) and (6), respectively. We allow extra biofuel and/or byproducts to be generated in order to expand the feasible region which contributes to a potentially lower total annualized cost. The total annualized cost is compactly calculated in the objective function and consists of three terms. The first term evaluates the CAPEX and fixed OPEX, the second term calculates the variable OPEX, and the third term accounts for the biomass feedstock purchasing and transportation costs. Note that the objective function of this model minimizes the total annualized cost, so revenue need not appear in the objective. This model employs binary variables Yj for pathway selection and nonlinear power functions Qsfj j for technology capital cost evaluation. A nonlinear function for calculating the CAPEX is much more accurate than using linear functions.37 Here we follow the same assumption applied in many papers that consider nonlinear functions for capital cost estimation.38−42 Due to the combinatorial and nonconvex nature of these terms, the resulting nonconvex MINLP can be computationally expensive to solve. Uncertainty Sets. The major parameters that are potentially uncertain in the deterministic model (DM) include conversion coefficients, biomass feedstock availability, biofuel product demands, and biomass feedstock prices. The uncertain data for conversion coefficients are scarce, therefore the introduction of conversion coefficient uncertainty may add extra uncertainty into the model. As mentioned previously, biomass feedstocks are assumed abundant, so in this work, we handle biomass feedstock price and biofuel product demand uncertainty since they are widely addressed, and their data can be retrieved in available literature. We develop two uncertainty sets B1 and B2 for the two types of uncertainty, respectively. |I |

B1 = {c3 ∈  : c3, i = c3,̅ i + c3,̃ iT1, i , ∀ i ∈ I,

nonnegative number for the uncertainty budget, an integer budget has a practical interpretation which demonstrates the number of feedstock prices or products influenced by uncertainty. Moreover, an integer budget is widely applied in adaptive robust optimization problems.44−46 In this work, we consider Γ1 and Γ2 to be nonnegative integers. Robust MINLP Models. Unlike robustness measures that can be integrated in a stochastic programming framework,47−49 robust optimal solutions guarantee feasibility for all realizations of uncertain parameters, including the worst-case realization.50−52 If all variables are determined before the realization of uncertainty, we can formulate a conventional static robust optimization problem. The specific form and the corresponding solution approaches are shown in the Appendix. In conventional static robust optimization, decisions are made once and for all, but in practice decision-makers naturally adjust future operational decisions depending on realization of uncertainties. As a result, the optimal solutions from conventional static robust optimization tend to be overly conservative.53 In order to ensure the success of new bioconversion processing pathways, flexible methods to robustly handle uncertainties must be developed. To cope with these limitations of conventional static robust optimization, adaptive (a.k.a., adjustable) robust optimization problems have been proposed and studied.27,54 Two-stage adaptive robust optimization separates the variables into “here-and-now” decisions, which are made before uncertainty is revealed, and “wait-and-see” decisions, which are made based on complete information on both the here-and-now decisions and uncertainty realization. This approach meets the need for modeling sequential decision-making processes, and more importantly, grants flexibility and reduces conservatism in the resulting robust optimal solutions.53,54 (2SROM) sf

min ∑ c1, jQ j j + Y ,Q

T1, i ∈ [− 1, 1],

max

min

c3∈ B1, d ∈ B2 (W , P , S) ∈ O


i∈I

(8)

∑ |T1, i| ≤ Γ1}

s.t.

i∈I

B2 = {d ∈ |I | : di = di̅ + dĩ T2, i ,

j∈J

T2, i ∈ [− 1, 1],

a1, jYj ≤ Q j ≤ a 2, jYj ,

∀ i ∈ I,

Q j ≥ 0,

∑ |T2, i| ≤ Γ2} i∈I

∀j∈J

(9)

∀j∈J

(10)

Yj ∈ {0, 1},

where

In the uncertainty sets, c3,̅ i and di̅ denote the nominal values of the biomass feedstock cost coefficients and biofuel product demands, respectively. c3,̃ i and dĩ represent the maximum deviations from the corresponding nominal values, resulting in two uncertain regions [ c3,̅ i − c3,̃ i , c3,̅ i + c3,̃ i] and [di̅ − dĩ , di̅ + dĩ ]. In order to allow the uncertain parameters to take any of the values within the corresponding uncertainty regions, we introduce two continuous variables T1,i and T2,i both bounded by [−1, 1]. Finally, we define the uncertain parameters based on the nominal values, the maximum deviations, and the continuous variables, T1,i and T2,i. Γ1 and Γ2 are uncertainty budgets for each uncertain parameters. As inspired by Bertsimas and Sim,29,43 such uncertainty sets are able to adjust the robustness of the optimal solution via introducing uncertainty budgets. These uncertainty budgets set the maximum amount of allowed deviation of the uncertain parameters from nominal values. If a budget is set to 0, no uncertainty is permitted to occur, and the corresponding parameters are restricted to their nominal values. In contrast, if a budget is set to the maximum number of the uncertain parameters, all the uncertain demands are allowed to be realized, and the optimal solution will be the most conservative. On the other hand, selecting any value between 0 and the maximum number can yield a less conservative robust optimal solution. In other words, the optimal solution is more conservative when the budget increases. Therefore, the budget parameter is an important indicator of the level of conservatism for the uncertainty set. In contrast to a noninteger and

O = {(W , P , S) ∈ |J| × |I| × |I | :

(11)

Wj ≤ Q j ,

(12)

Pi − Si +

∀j∈J

∑ a3, i , jWj = 0,

∀i∈I

j∈J

(13)

Pi ≤ bi ,

∀i∈I

(14)

Si ≥ di ,

∀i∈I

(15)

Wj , Pi , Si ≥ 0,

∀ i ∈ I,

j ∈ J}

B1 = {c3 ∈ |I | :

(17)

c3, i = c3,̅ i + c3,̃ iT1, i ,

− 1 ≤ T1, i ≤ 1,

∀i∈I

∀i∈I

∑ |T1, i| ≤ Γ1}

(18) (19)

i∈I

(20)

B2 = {d ∈ |I | :

(21)

di = di̅ + dĩ T2, i , D

(16)

∀i∈I

(22) DOI: 10.1021/acssuschemeng.6b00188 ACS Sustainable Chem. Eng. XXXX, XXX, XXX−XXX

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Figure 3. Flowchart of the proposed algorithm.

− 1 ≤ T2, i ≤ 1,

∑ |T2, i| ≤ Γ2} i∈I

∀i∈I

obtain the exact global optimal solution. The flowchart of the proposed algorithm is shown in Figure 3. The detailed formulation of the master problem (MA) and subproblem (SUB), algorithm as well as global convergence properties are shown in the Appendix.

(23)

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(24)

We develop a two-stage adaptive robust MINLP model for identifying an optimal processing pathway from a product and process network under biomass feedstock price and biofuel product demand uncertainties (2SROM). This model represents the first attempt to employ two-stage adaptive robust optimization to handle uncertainty in bioconversion product and process network optimization problems. In the first stage, the minimization problem determines the selection of processing technologies using binary variables Yj and the capacity of each technology selected using continuous variables Qj. The objective function consists of deterministic and uncertain parts. The deterministic part evaluates the CAPEX and fixed OPEX based on power functions Qsfj j, and the uncertain part is influenced by uncertain biofuel product demands and uncertain biomass feedstock prices. Integer variables are used only in the first-stage optimization problem, and all variables in the second stage are continuous. The model employs nonlinear functions to evaluate technology capital costs in the first-stage objective function, and the objective function in the second stage is linear. The model allows building backup technologies with nonzero capacities and zero operating levels to hedge against uncertainty in biofuel product demand. Solution Strategy for the Two-Stage Adaptive Robust MINLP. Two-stage adaptive robust optimization problems are frequently considerably expensive to solve. There are several methods to approach this problem, such as approximating recourse decisions with functions of uncertain parameters54−60 and master-subproblem methods.43,44,61 We tackle the min-max formulation using a columnand-constraint generation method,62 which dynamically generates constraints with recourse variables in the primal space. In addition, the separable concave functions in the master problem are handled by a branch-and-refine algorithm63 which iteratively solves relaxed problems with successive piecewise linear approximations in order to

RESULTS AND DISCUSSION All computational experiments are performed on a Dell Optiplex 790 desktop with an Intel(R) Core(TM) i5-2400 3.10 GHz CPU, 8GB RAM, and Windows 7 64-bit operating system. All of the models and solution algorithms are coded in GAMS 24.7.1.64 The subproblem and relaxed master problem are solved using CPLEX 12. The relative optimality tolerance of CPLEX 12 is set to 10−6, and the absolute optimality tolerance for the proposed algorithm is set to 10−6. The deterministic problem has 876 continuous variables, 197 discrete variables, and 1018 constraints; the master problem has 1578 continuous variables, 197 discrete variables, and 1774 constraints; the subproblem has 1050 continuous variables, 142 discrete variables, and 910 constraints. The computational performance results are shown in the Appendix. Uncertain Parameters. As shown in Tables 1 and 2, the nominal values of the uncertain biomass feedstock prices and biofuel product demands are taken from previous works.34,35 The ratio of the largest deviation of an uncertain parameter to its nominal value is retrieved from open-resource statistics and official reports.65,66 The maximum deviation of the demand for gasoline is assumed to be 20%, following the same value for biodiesel and ethanol. Since there are eight biomass feedstocks, the budget of biomass feedstock price uncertainty can range from 0 to 8. Similarly, the budget of biofuel product demand uncertainty can range from 0 to 3. These data are used to define the uncertainty sets. E


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(see the Appendix), its optimal solution can only vary with the budget of biomass feedstock price uncertainty. As shown in the right heat map in Figure 4, the minimum total annualized costs of the conventional static robust optimization are equal to those of the two-stage adaptive robust optimization with a biofuel product demand uncertainty budget of 3. It is noted that both conventional static robust optimization and adaptive robust optimization typically provide different results when applied to the same problem.54 In our problem, the uncertain biomass feedstock prices are constraintwise and involve only one constraint after reformulation. Therefore, given the worst case biofuel product demands, our problem becomes a special case where both robust optimization models provide the same results. However, two-stage adaptive robust optimization demonstrates advantages over conventional static robust optimization when recourse decisions are taken into account to hedge against biofuel product demand uncertainty. In the results for the two-stage adaptive robust optimization model, increasing either the biomass feedstock price or biofuel product demand uncertainty budgets increases the minimum total annualized cost. However, increasing the biomass feedstock price uncertainty budget beyond a value of 2 does not significantly change the minimum total annualized cost when the biofuel product demand uncertainty budget is constant. In other words, when moving from the bottom to top of each column in Figure 4, no difference results beyond a biomass feedstock price uncertainty budget of 2. However, when holding the biomass feedstock price uncertainty budget constant, varying the biofuel product demand uncertainty budget produces significant changes in the minimum total annualized cost. Thus, the economic performance of bioconversion processing pathways appears to be more sensitive to changes in biofuel product demand levels than to changes in biomass feedstock prices. In other words, robust, optimal bioconversion processing pathways should be less biomass feedstock specific and focus more on satisfying biofuel demand. This phenomenon could provide useful information for decision makers in the bioconversion space as they design and implement bioconversion processing pathways. Such insight is invaluable when making important decisions such as selection and sizing of technologies in processing pathways. A deterministic solution and a series of robust optimal solutions with varying uncertainty budgets were identified. The deterministic optimal solution is shown in the bottom half of Figure 5 with a minimum total annualized cost of $17.9 M/y. Two biomass feedstocks of soybeans and softwood are used to satisfy the demands of the three fuels (gasoline, biodiesel, and ethanol). Biodiesel demand is satisfied by traditional soybean processing technologies. Byproduct glycerol from this process is later upgraded to PHB (polyhydroxybutyrate). Ethanol demand is satisfied by indirect gasification of softwood, followed by methanol synthesis from the resulting syngas, acetic acid synthesis from the methanol, and finally hydrogenation of the acetic acid. The methanol from this process is also used to make gasoline through the methanol to gasoline synthesis pathway. The processing pathways of the robust optimal solutions differ from that of the deterministic pathway shown in the bottom half of Figure 5, but technology selections do not differ among the robust optimal solutions. We propose the solution where both biomass feedstock price and biofuel product demand uncertainty budgets are 1 as an illustrative solution to compare and contrast with the deterministic and most

Table 1. Biomass Feedstock Price Uncertainty Data feedstocks

nominal prices ($/kg)

largest deviation/ nominal price

ref for the deviations

soybean corn sugar cane corn-stover hardwood softwood switchgrass microalgae

0.109 0.0317 0.0925 0.0881 0.0728 0.0728 0.0878 0.220

30% 40% 20% 30% 20% 20% 30% 30%

67 66 68 69 70 70 71 72

Table 2. Biofuel Product Demand Uncertainty Data products

nominal demands (Mgal/y)

largest deviation/ nominal demand

ref for the deviations

biodiesel gasoline ethanol

1.29 1.58 1.44

20% 20% 20%

73 65 74

Optimal Solutions. Figure 4 shows how varying the biomass feedstock price uncertainty budget and the biofuel

Figure 4. Heat map for the optimal objective function values ($MM) under various uncertainty levels and using different robust optimization models. A budget of 0 represents that the corresponding uncertain parameters can only take nominal values. The dotted box represents the illustrative two-stage adaptive robust solution.

product demand uncertainty budget affects the minimum total annualized cost of both robust optimization models. In the results for two-stage adaptive robust optimization (left heat map in Figure 4), the point with no uncertainty budgets for both uncertain parameters corresponds to the deterministic optimal solution. The minimum total annualized cost of the deterministic model is the lowest ($17.9 M/y), and that of the most conservative two-stage adaptive robust optimal solution (the solution with maximum budgets of uncertainty) is the highest ($22.5 M/y). When uncertainty budgets decrease, the total annualized cost decreases, while the risk of being unable to meet demand or achieve the minimum cost increases. A decision-maker might choose any of the robust optimal solutions in Figure 4, depending on the decision-maker’s willingness to accept higher levels of risk with lower minimum total annualized costs. Since we predetermine the biofuel product demands in the reformulated conventional static robust optimization problem F


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demand in cases where the true demand is larger than the expected nominal demand. Different quantities of PHB are produced among the robust optimal solutions when the biofuel product demand uncertainty varies. For example, 0.62 kt/y of PHB is produced in the illustrative solution, but 0.74 kt/y of PHB is produced at the most conservative two-stage adaptive robust optimal solution, an increase of approximately 19%. Thus, as uncertainty increases in the model, more bioproducts are produced. Therefore, production of bioproducts, or other sellable byproducts of bioconversion processing pathways, might be an attractive strategy to minimize total annualized costs under uncertainty, boosting the economic performance of the processing pathway. Additionally, increasing production of the various products under demand also appears to help hedge the processing pathway against uncertainty. In addition to an increase in the minimum total annualized cost when uncertainty budgets increase, the breakdowns of the minimum total annualized costs change with changes in the uncertainty budgets. Figure 6 displays the breakdown of the

Figure 5. Optimal processing pathway for the deterministic and the illustrative, robust optimal solution. Red text denotes differences between the pathways.

Figure 6. Spider chart showing differences in transportation costs, annualized capital costs, OPEX, and feedstock costs between the deterministic solution, the illustrative solution, and the most conservative two-stage adaptive robust solution (MCTAS).

conservative two-stage adaptive robust optimal solutions. As shown in Figure 4, the illustrative solution has a total annualized cost of $21.2 M/y. The processing pathway of the illustrative solution is shown in the top half of Figure 5, where the red numbers represent different results from those in the deterministic robust optimal solution. Uncertain gasoline demand is identified as the worst case among all possible realizations with a biofuel product demand budget of 1. Ethanol in the illustrative solution is instead primarily produced through an acetic acid synthesis and hydrogenation pathway. The strategies of traditional processing of soybeans to biodiesel and indirect gasification of softwood are selected in all of the robust optimal solutions. However, the capacities of the technologies vary among all robust optimal pathways depending on the biofuel product demand uncertainty budgets. For example, the illustrative solution uses less softwood than other robust optimal solutions with a higher biofuel product demand uncertainty budget. Thus, some robust optimal processing pathways produce more of each biofuel product. Naturally, more biofuel products would be produced in order to satisfy

CAPEX, OPEX, biomass feedstock costs, and biomass transportation costs for the deterministic solution, the illustrative solution, and the most conservative two-stage adaptive robust solution. There are significant increases in most of the four cost categories between the deterministic solution and the robust optimal solutions except for transportation costs. In all three scenarios, transportation costs are relatively low. Annual capital costs are similar for the two robust optimal solutions, as the solutions utilize a similar processing pathway structure. More significant differences between the two robust optimal solutions predictably arise in the OPEX and feedstock costs. The OPEX would be expected to rise when the biofuel product demand uncertainty budget is increased, and feedstock costs could reasonably be expected to increase when the biomass feedstock price uncertainty budget is increased. Worst-case scenarios are selected in robust optimization, so if more uncertainty is allowed in the model, then this will necessarily increase the conservatism of the model, resulting in a worse objective value. G



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The two-stage adaptive robust optimization approach yields less conservative optimal solutions and allows the decision maker to choose an acceptable trade-off between cost decreases and risk. We focus on uncertain parameters in the objective function and right-hand side of second-stage constraints in the proposed two-stage adaptive robust optimization model. These parameters usually correspond to external factors. In addition, uncertain internal factors or model uncertainty, residing in the left-hand side of constraints, are equally important source of uncertainty. The relevant models and solution strategies are worth future investigation. Addressing more complicated uncertainty, such as correlated uncertainty, and multiobjective robust optimization are also among the future research directions to further refine the models and tools introduced in this work.

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APPENDIX

Parameters in the Objective Function of the Deterministic Model

⎡ r(1 + r )ls ⎤ ⎢ ⎥, + fc ⎦ idbase, j(rcj)sfj ⎣ (1 + r )ls − 1 idj

c1, j = iccj

∀j∈J (A1)

c 2, j =

iocj rcj

,

∀j∈J (A2)

c3, i = fpi + ftcci + vtcci· disi ,

∀i∈I

(A3)

iccj is the initial capital cost of technology j; idj and idbase,j are the chemical engineering plant cost index of technology j in the current year and in the reference year, respectively; rcj is the reference capacity of technology j; sf j is the sizing factor of technology j; r is the interest rate; ls is the life span; fc is the fixed OPEX; iocj is the initial OPEX of technology j; fpi is the price of feedstock i; f tcci is the fixed transportation cost coefficient of compound i; vtcci is the variable transportation cost coefficient of compound i; disi is the average distance of compound i for transportation.

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CONCLUSIONS Consideration of uncertainty is a critical undertaking to ensure competitiveness and success of bioconversion enterprises. In previous research, parameters within bioconversion process and product network optimization models were treated as deterministic. However, such treatment is not warranted in most real-world scenarios due to various sources of uncertainty. To that end, we proposed a two-stage adaptive robust MINLP model that minimizes the total annualized cost of bioconversion processing pathways from a product and process network. We focused on biomass feedstock price and biofuel product demand uncertainties due to lack of access to conversion coefficient uncertainties and an assumed abundance of biomass feedstock availability. The two-stage adaptive model considered budgets of uncertainty to control the level of robustness or conservatism in the model, and it allowed for sequential decision-making at the design and operational stages. Thus, this approach allowed the decision maker to set a desired level of conservatism. Since the resulting min−max−min MINLP problem cannot be directly solved by existing commercial solvers, we proposed a solution strategy that integrated a column-and-constraint generation algorithm and a branch-andrefine algorithm. After reformulating a master problem and a subproblem, the optimal solution of the complex problem was obtained using only an MILP solver. The proposed solution method was then utilized to solve a two-stage adaptive robust MINLP for minimization of the total annualized cost of a processing pathway from a process and product network converting biomass into fuels and chemicals. Demand for three different types of biofuels were assumed uncertain and given a maximum uncertainty budget. Furthermore, biomass feedstock prices were assumed to be uncertain and also were given a maximum uncertainty budget. A variety of different solutions with different objective values and levels of conservatism were identified. An overall minimum total annualized cost of $17.9 M/y was found in the deterministic case. The largest minimum total annualized cost of $22.5 M/y was found in the most conservative two-stage adaptive robust optimal solution. An illustrative solution that accounted for the trade-off between risk and cost had an intermediate minimum total annualized cost of $21.2 M/y. Biofuel product demand uncertainty was found to have a significantly larger effect on the total annualized cost than biomass feedstock price uncertainty. Overall, the proposed solution method was found to quickly and efficiently identify robust, optimal processing pathways for product and process networks.

Conventional Static Robust MINLP Model min

max

Y , Q , W , P , S c3∈ B1

sf

∑ c1, jQ j j + ∑ c2, jWj + ∑ c3, iPi j∈J

j∈J

i∈I

s.t. a1, jYj ≤ Q j ≤ a 2, jYj , ∀ j ∈ J Wj ≤ Q j , ∀ j ∈ J Pi − Si +

∑ a3, i , jWj = 0, ∀ i ∈ I j∈J

Pi ≤ bi , ∀ i ∈ I Si ≥ di , ∀ di ∈ B2 , i ∈ I Q j , Wj , Pi , Si ≥ 0, Yj ∈ {0, 1}, ∀ i ∈ I , j ∈ J

where B1 = {c3 ∈ |I | : c3, i = c3,̅ i + c3,̃ iT1, i , ∀ i ∈ I,

T1, i ∈ [−1, 1],

∑ |T1,i| ≤ Γ1} i∈I

B2 = {d ∈ |I | : di = di̅ + dĩ T2, i , ∀ i ∈ I,

T2, i ∈ [−1, 1],

∑ |T2,i| ≤ Γ2} i∈I

Reformulation of the Conventional Static Robust MINLP Model

Only the third term in the original objective function is associated with the biomass feedstock uncertainty. We replace this term with a new variable G and add constraint (A4) to determine the worst case realization of G. max ∑ c3, iPi ≤ G

c3∈ B1

i∈I

(A4)

In order to simplify the uncertainty budget constraint with absolute value functions, we make one observation: the multiplier Pi associated with the uncertain parameter c3,i is nonnegative. Therefore, the optimal solution of c3 in the H


Research Article

ACS Sustainable Chemistry & Engineering maximization problem takes nonnegative values and the corresponding uncertainty set can be simplified as follows. B3 = {c3 ∈ |I | : c3, i = c3,̅ i + c3,̃ iT1, i ,

B1 = {c3 ∈ |I | : c3, i = c3,̅ i + c3,̃ iT1, i , i∈I

B2 = {d ∈ |I | : di = di̅ + dĩ T2, i ,

i∈I

To solve the conventional static robust optimization problem, we employ the duality-based technique for budgetconstrained robust optimization problems.29 R1,i and R2 are dual variables for the constraints in uncertainty set B3. In order to achieve the robustness of each constraint in conventional static robust optimization, uncertain parameters of a constraint must be able to vary in the projection of the correlated uncertainty set on that constraint.75 As a result, each biofuel product demand must be able to vary between [di̅ − dĩ , di̅ + dĩ ] for each i regardless of the uncertainty budget Γ2. Based on such simplification, the worst case biofuel product demand can be predetermined as the largest demand of each biofuel product, i.e. di̅ + dĩ . The reformulated conventional static robust optimization problem is shown as follows. min

T2, i ∈ [−1, 1],

∑ |T2,i| ≤ Γ2}

∀ i ∈ I,

i∈I

We apply the simplified uncertainty set B3 and convert the inner minimization problem into its dual formulation, which is then incorporated into the outer maximization problem. The new subproblem is named (SUBC). (SUBC) max

Z1, Z 2 , Z3 , Z4 , T1, T2

+

−∑ Q jZ1, j − j∈J

∑ biZ3,i + ∑ di̅ Z4,i i∈I

i∈I

∑ dĩ T2,iZ4,i

(A10)

i∈I

s.t.

sf

∑ c1, jQ j j + ∑ c2, jWj + G j∈J

∑ |T1,i| ≤ Γ1}

∀ i ∈ I, T1, i ∈ [0, 1],

∑ T1,i ≤ Γ1}

∀ i ∈ I,

T1, i ∈ [−1, 1],

j∈J

−Z1, j +

s.t.

∑ a3,i ,jZ 2,i ≤ c2,j , ∀ j ∈ J i∈I

∑ c3,̅ iPi + ∑ R1, i + R 2 Γ1 − G ≤ 0 i∈I

Z 2, i − Z3, i ≤ c3,̅ i + c3,̃ iT1, i , ∀ i ∈ I

i∈I

R1, i + R 2 ≥ c3,̃ iPi , ∀ i ∈ I

−Z 2, i + Z4, i ≤ 0, ∀ i ∈ I

a1, jYj ≤ Q j ≤ a 2, jYj , ∀ j ∈ J

0 ≤ T1, i ≤ 1, ∀ i ∈ I

Wj ≤ Q j , ∀ j ∈ J

∑ T1,i ≤ Γ1

Pi − Si +

i∈I

∑ a3, i , jWj = 0, ∀ i ∈ I

−1 ≤ T2, i ≤ 1, ∀ i ∈ I

j∈J

Pi ≤ bi , ∀ i ∈ I

∑ |T2,i| ≤ Γ2

Si ≥ di̅ + dĩ , ∀ i ∈ I

i∈I

Z1, j , Z3, i , Z4, i ≥ 0, ∀ i ∈ I , j ∈ J

Q j , Wj , Pi , Si , R1, i , R 2 ≥ 0, Yj ∈ {0, 1}, ∀ i ∈ I , j ∈ J

where Z1,j, Z2,i, Z3,i, and Z4,i are dual variables for constraints (A6)−(A9), respectively. Note that a set of bilinear terms T2,iZ4,i appears in the objective function (A10) and could be computationally expensive to handle. Since the budget of biofuel product demand uncertainty is an integer value, the optimal solutions for variables T2,i must be either 0 or 1 given that both variables in the bilinear terms are nonnegative. Following this result, we change the continuous variable T2,i to an integer variable ωi and then reformulate the nonlinear subproblem to an equivalent MILP by applying the Glover’s linearization scheme.76 As in the final formulation of the subproblem (SUB), each bilinear term in the objective function is linearized by introducing an auxiliary variable Mi, an upper bounding parameter u, and several auxiliary constraints (A16)−(A18).

Subproblem for Solving the Two-Stage Adaptive Robust Optimization Problem

The original second-stage problem (SUBO) is shown below: (SUBO) max

min

c3∈ B1, d ∈ B2 W , P , S


i∈I

(A5)

s.t. Wj ≤ Q j ,

Pi − Si +

∀j∈J

∑ a3,i ,jWj = 0,

(A6)

∀i∈I

j∈J

(A7)

Pi ≤ bi ,

∀i∈I

(A8)

Si ≥ di ,

∀i∈I

(A9)

(SUB) max

Z1, Z 2 , Z3 , Z4 , Z5 , M , ω

Wj , Pi , Si ≥ 0,

∀ i ∈ I,

j∈J

+

where

∑ dĩ Mi i∈I

I

−∑ Q jZ1, j − j∈J

∑ biZ3,i + ∑ di̅ Z4,i i∈I

i∈I

(A11) DOI: 10.1021/acssuschemeng.6b00188 ACS Sustainable Chem. Eng. XXXX, XXX, XXX−XXX

Research Article

ACS Sustainable Chemistry & Engineering P

s.t. −Z1, j +

∑ a3,i ,jZ 2,i ≤ c2,j ,

∑ PWj ,p = 1,

∀j∈J

∀j∈J (A25)

p=1

(A12)

i∈I

P−1

Z 2, i − Z3, i − c3,̃ iT1, i ≤ c3,̅ i ,

∀i∈I

∑ IWj ,p = 1,

(A13)

∀j∈J (A26)

1

−Z 2, i + Z4, i ≤ 0, 0 ≤ T1, i ≤ 1,

∀i∈I

∀i∈I

PWj ,1 ≤ IWj ,1 ,

(A15)

PWj , p ≤ IWj , p − 1 + IWj , p ,

i∈I

∀i∈I

0 ≤ Mi ≤ ωiu ,

∀i∈I

Mi ≥ Z4, i − (1 − ωi)u ,

∀i∈I

PWj , P ≤ IWj , P − 1 ,

(A17)

a1, jYj ≤ Q j ≤ a 2, jYj ,

(A18)

η≥

(A27)

∀ j ∈ J,

2≤p≤P−1

j∈J

(A20)

ωi ∈ {0, 1},

∀ i ∈ I,

Pi , k − Si , k +

j∈J

Pi , k ≤ bi , Si , k ≥ di , k ,

Based on the framework of a column-and-constraint generation method,62 we formulate the master problem by replacing the second-stage problem with an auxiliary variable η and introducing a set of primal cuts (A30)−(A35). The master problem provides an underestimation to the original problem since only finitely many realizations of the demand uncertainty are considered. The identified realizations of the demand uncertainty are indexed by k, which are also applied to all of the operational variables in constraints (A30)−(A35). Consequently, each realization corresponds to a set of exclusive operational variables. In addition to the primal cuts (A30)−(A35), another important change in the master problem compared to the original first-stage problem involves the linearization of the separable concave functions. It is computationally expensive to solve an MINLP with linear constraints and separable concave terms in the objective function.63 A branch-and-refine algorithm can efficiently tackle this type of problem. The key idea of this algorithm is to approximate the concave power functions with successive piecewise linear approximations. The piecewise linear functions are formulated using SOS1 (specially ordered set of type 1) variables in constraints (A22)− (A28).77,78 (MA)

∀ j ∈ J,

∀k∈K (A31)

k∈K

∑ a3,i ,jWj ,k = 0, ∀ i ∈ I, ∀ i ∈ I,

IWj , p SOS1 variables, k∈K

(A32)

∀ i ∈ I,

k∈K

k∈K

(A34)

k∈K

(A35)

Yj ∈ {0, 1},

∀ i ∈ I,

j ∈ J,

p ∈ P, (A36)

The set for partition points is indexed by p. PWj,p and IWj,p are the weighting factor for the partition points and the position indicator, respectively. IWj,p is defined as the SOS1 variable, and the sum of IWj,p over all intervals is limited to one, meaning one and only one interval is selected for the solution point. On the basis of constraints (A27)−(A29), only the weighting factors for the partition points associated with the selected interval are allowed to be greater than zero. f xj,p and fej,p represent the predefined partition point value and the corresponding power function value, respectively. Constraints (A23)−(A25) determine the solution point on the selected intervals based on the variable weighting factors and predefined partition point and power function values. Algorithm

As shown in the pseudo code (Figure A1), the column-andconstraint generation algorithm serves as the outer loop of the algorithm and begins by solving the master problem with identified uncertain demand scenario set K. In order to efficiently solve the nonlinear and nonconvex master problem, a branch-and-refine algorithm is employed as the inner loop of the algorithm. Before the inner loop starts, the piecewise linear approximations are initialized for each power function. The optimal solution of the relaxed master problem is an underestimation of the optimal objective function value of the original master problem because the piecewise linear approximations lie strictly below the original, nonlinear cost functions. Next, a feasible objective function value can be evaluated using the original, nonlinear cost calculation equations. The lower bound, the upper bound, and the gap for the inner loop are then updated. In the last step of the

(A22)

s.t. P

∀j∈J (A23)

P

∑ fxj ,p PWj ,p = Q j ,

(A30)

i∈I

Q j , Wj , k , Pi , k , Si , k , PWj , p ≥ 0,

min ∑ Ej + η

p=1

∀j∈J

(A33)

Master Problem for Solving the Two-Stage Adaptive Robust Optimization Problem

∑ fej ,p PWj ,p = Ej ,

(A29)

i∈I

(A21)

j∈J

∀j∈J

∑ c2,jWj ,k + ∑ c3,i ,kPi ,k ,

Wj , k ≤ Q j ,

i∈I

Z1, j , Z3, i , Z4, i ≥ 0,

(A16)

(A19)

∑ ωi ≤ Γ2

p=1

∀j∈J

(A28)

∑ T1,i ≤ Γ1 Mi ≤ Z4, i ,

(A14)

∀j∈J (A24) J


Research Article


sets).79 As previously proved, branch-and-refine algorithm in the inner loop of the proposed algorithm guarantees the global optimal solution of the master problem (MA) given a predefined optimality tolerance.80,81 As a result, lower bounds obtained from the inner loop represent the global lower bounds of the original problem (2SROM). The reformulated subproblem (SUB) is an MILP, whose global convergence is guaranteed by the state-of-the-art branch-and-cut methods implemented in solvers such as CPLEX. As proved by Zeng and Zhao,62 the column-and-constraint generation algorithm converge to the optimal solution in finitely many iterations as long as the master problem and subproblem can be solved to their global optimal solutions. Therefore, as the global lower bound and the non-increasing upper bound converge, we obtain the global minimizer of the two-stage adaptive robust optimization problem. Computational Results

Table A1 shows the computation times of solving master problems and subproblems for each combination of uncertainty budgets by the proposed algorithm and the column-andconstraint generation algorithm combined with an MINLP solver. Computation times strongly depend on the size of each uncertainty budget and the type of uncertainty budget. Holding the budget of biomass feedstock price uncertainty constant, biofuel product demand uncertainty budgets of 0 and 3 result in problems that are generally more quickly solved than those with biofuel product demand uncertainty budgets of 1 and 2, and vice versa. This is because there are potentially more candidates for uncertain parameters with uncertainty budgets of 1 and 2. Therefore, more outer loop iterations are usually needed to converge to the optimal solutions. The computation times for optimizing master problems are several orders of magnitude longer than those for solving corresponding subproblems, indicating that the bottleneck of solving this class of two-stage adaptive robust optimization problem lies in achieving the optimal solutions of nonlinear master problems. All optimal solutions can be obtained in less than 1 h by the proposed algorithm, and 12 of the 16 solution points were found in less than 1000 s. In contrast, much longer computation times are consumed if the master problem is handled directly by the MINLP solver BARON 14.4.82 Thus, the solution method proposed in this work is both a practical and highly efficient approach to handle the proposed two-stage adaptive robust optimization problem that cannot be directly solved with off-the-shelf solvers.

Figure A1. Pseudo code of the proposed algorithm.

branch-and-refine algorithm, piecewise linear approximations are updated using the current optimal solution of the relaxed master problem as an additional partition point. The algorithm checks whether the stopping criterion for the inner loop is satisfied. If the stopping criterion is not met, the updated master problem is solved and the aforementioned procedures are repeated; otherwise, the inner loop is terminated. Subsequently, in the outer loop, the optimal objective function value from the inner loop is used to update the current lower bound for the outer loop. The optimal solution of the master problem obtained from the inner loop is employed to solve the subproblem. The optimal solution is used to update the upper bound as well as the gap for the outer loop. In addition, the current iteration count of the outer loop is used to expand the identified uncertain demand scenario set K. Next, a new price, a new demand, and a set of new operational variables for the new master problem are added. The algorithm then checks the stopping criterion of the outer loop and terminates if the gap is sufficiently close to the tolerance. Otherwise, the new master problem will be solved via the branch-and-refine algorithm, and the evaluation procedures outlined above continue until a solution within the optimality tolerance is found. Global Convergence of the Proposed Method

The proposed algorithm guarantees global convergence in O(p) iterations (p is the number of extreme points of the uncertain

Table A1. Computation Times (CPU s) under Various Uncertainty Levels and by Different Solution Strategiesa budget of biofuel product demand uncertainty budget of biomass feedstock price uncertainty b

0 1 2 3 0 1 2 3 a b

0b 607.9 492.8 543.0 284.7 1509.6 8304.3 21242.6 1371.9

1 + + + + + + + +

0.1 0.1 0.1 0.1 0.1 0.2 0.1 0.1

805.0 2091.7 1948.2 1067.7 21715.6 17237.8 48094.2 41542.1

2 + + + + + + + +

0.1 0.2 0.2 0.2 0.3 0.2 0.4 0.3

502.3 1234.8 1522.0 655.8 6551.9 16301.0 10751.8 33994.5

solution strategy 3

+ + + + + + + +

0.2 0.1 0.1 0.1 0.2 0.2 0.1 0.3

199.0 524.6 492.8 337.4 1421.4 4958.3 3596.7 1691.9

inner loop + + + + + + + +

0.1 0.1 0.1 0.1 0.2 0.2 0.1 0.1

branch and refine

outer loop column and constraint generation

BARON 14.482

The first value in each cell of the computation times is for optimizing the master problem, and the second value is for optimizing the subproblem. A budget of 0 represents that the corresponding uncertain parameters can only take nominal values. K


Research Article

ACS Sustainable Chemistry & Engineering P set of partition points, indexed by p

The proposed solution method identifies robust optimal solutions quickly. Figure A2 shows convergence of the upper

Parameters

a1,j a2,j a3,i,j bi c1,j c2,j c3,i c3,̅ i c3,̃ i di di̅ dĩ

Figure A2. Change of bounds for the proposed algorithm. The results correspond to the illustrative solution.

disi fc fej,p

and lower bounds of the outer loop within the proposed solution method for the case of a biomass feedstock price uncertainty budget of 1 and a biofuel product demand uncertainty budget of 1. Convergence of the inner loop (the branch-and-refine algorithm) is shown at each lower bound point of the outer loop. Eight iterations are required each run of the inner loop, but only 4 outer loop iterations are required, resulting in an average time per outer loop iteration of approximately 567 CPUs/iteration. The relative gap between the upper and lower bounds drops relatively quickly, from 61% in the second iteration to 58% in the third iteration and finally to 0% in the fourth and final iteration. Thus, the proposed solution method was shown to be computationally efficient both in time and number of iterations.

■

f pi f tcci f xj,p gapin gapout iccj idbase,j idj iocj iter k lbin lbout ls mo*

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acssuschemeng.6b00188. List of technologies used in this model as well as their inputs and outputs (PDF)

■

moΔ Q* r rcj so*

AUTHOR INFORMATION

Corresponding Author

*Tel.: +1 847 467 2943. Fax: +1 847 491 3728. E-mail: you@ northwestern.edu. Notes

sf j T1* u ubin ubout vtcci εin εout Γ1 Γ2 ω*

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors acknowledge financial support from the Institute for Sustainability and Energy at Northwestern University (ISEN) and the National Science Foundation (NSF) CAREER Award (CBET-1554424).

■

NOTATION

Sets

lower bound of the capacity of technology j upper bound of the capacity of technology j conversion coefficient of compound i in technology j availability of compound i coefficient for economic evaluation coefficient for economic evaluation coefficient for economic evaluation nominal value of c3,i maximum deviation of the uncertain parameter c3,i from c3,̅ i demand of compound i nominal value of di maximum deviation of the uncertain parameter di from di̅ average distance of compound i for transportation fixed OPEX predefined power function value for point p of technology j price of compound i fixed transportation cost coefficient of compound i predefined partition point value for point p of technology j gap for the inner loop gap for the outer loop initial capital cost of technology j chemical engineering plant cost index of technology j in the reference year chemical engineering plant cost index of technology j in the current year initial OPEX of technology j iteration count for the inner loop iteration count for the outer loop lower bound for the inner loop lower bound for the outer loop expected life span in years of the processing pathway optimal objective function value of the relaxed master problem feasible objective function value of the original master problem optimal solution of the relaxed master problem interest rate reference capacity of technology j optimal objective function value of the second-stage problem with respect to Q* sizing factor of technology j optimal solution of T1 in the subproblem upper bound of Mi upper bound for the inner loop upper bound for the outer loop variable transportation cost coefficient of compound i optimality tolerance of the inner loop optimality tolerance of the outer loop budget of uncertain biomass feedstock prices budget of uncertain biofuel product demands optimal solution of ω in the subproblem

Binary Variables

I set of compounds, indexed by i J set of technologies, indexed by j K set of identified uncertain demand scenario, indexed by k

Yj selection of technology j ωi selection of uncertain demand of compound i L


Research Article

ACS Sustainable Chemistry & Engineering SOS1 Variable

(15) Grossmann, I. E.; Sargent, R. W. H. Optimum Design of Chemical-Plants with Uncertain Parameters. AIChE J. 1978, 24, 1021−1028. (16) Swaney, R. E.; Grossmann, I. E. An Index for Operational Flexibility in Chemical Process Design 0.1. Formulation and Theory. AIChE J. 1985, 31, 621−630. (17) Rooney, W. C.; Biegler, L. T. Design for model parameter uncertainty using nonlinear confidence regions. AIChE J. 2001, 47, 1794−1804. (18) Pistikopoulos, E. N.; Ierapetritou, M. G. Novel-approach for optimal process design under uncertainty. Comput. Chem. Eng. 1995, 19, 1089−1110. (19) Gupta, V.; Grossmann, I. E. Multistage stochastic programming approach for offshore oilfield infrastructure planning under production sharing agreements and endogenous uncertainties. J. Pet. Sci. Eng. 2014, 124, 180−197. (20) You, F.; Grossmann, I. E. Multicut Benders decomposition algorithm for process supply chain planning under uncertainty. Ann. Oper. Res. 2013, 210, 191−211. (21) Li, X.; Tomasgard, A.; Barton, P. I. Nonconvex Generalized Benders Decomposition for Stochastic Separable Mixed-Integer Nonlinear Programs. Journal of Optimization Theory and Applications 2011, 151, 425−454. (22) Ben-Tal, A.; Nemirovski, A. Robust convex optimization. Math. Oper. Res. 1998, 23, 769−805. (23) Ben-Tal, A.; El Ghaoui, L.; Nemirovski, A. Robust optimization; Princeton University Press, 2009. (24) Ben-Tal, A.; Nemirovski, A. Robust solutions of Linear Programming problems contaminated with uncertain data. Math. Program. 2000, 88, 411−424. (25) El Ghaoui, L.; Lebret, H. Robust solutions to least-squares problems with uncertain data. SIAM J. Matrix Anal. Appl. 1997, 18, 1035−1064. (26) El Ghaoui, L.; Oustry, F.; Lebret, H. Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 1998, 9, 33−52. (27) Bertsimas, D.; Brown, D. B.; Caramanis, C. Theory and Applications of Robust Optimization. SIAM Rev. 2011, 53, 464−501. (28) Kouvelis, P.; Yu, G. Robust discrete optimization and its applications; Springer Science & Business Media, 2013; Vol. 14. (29) Bertsimas, D.; Sim, M. The price of robustness. Oper. Res. 2004, 52, 35−53. (30) Tovar-Facio, J.; Lira-Barragan, L. F.; Napoles-Rivera, F.; Bamufleh, H. S.; Ponce-Ortega, J. M.; El-Halwagi, M. M. Optimal Synthesis of Refinery Property-Based Water Networks with Electrocoagulation Treatment Systems. ACS Sustainable Chem. Eng. 2016, 4, 147−158. (31) Santibanez-Aguilar, J. E.; Ponce-Ortega, J. M.; GonzalezCampos, J. B.; Serna-Gonzalez, M.; El-Halwagi, M. M. Synthesis of Distributed Biorefining Networks for the Value-Added Processing of Water Hyacinth. ACS Sustainable Chem. Eng. 2013, 1, 284−305. (32) Erbis, S.; Kamarthi, S.; Cullinane, T. P.; Isaacs, J. A. Multistage Stochastic Programming (MSP) Model for Carbon Nanotube Production Capacity Expansion Planning. ACS Sustainable Chem. Eng. 2014, 2, 1633−1641. (33) Tong, K.; Gong, J.; Yue, D.; You, F. Stochastic Programming Approach to Optimal Design and Operations of Integrated Hydrocarbon Biofuel and Petroleum Supply Chains. ACS Sustainable Chem. Eng. 2014, 2, 49−61. (34) Garcia, D. J.; You, F. Multiobjective Optimization of Product and Process Networks: General Modeling Framework, Efficient Global Optimization Algorithm, and Case Studies on Bioconversion. AIChE J. 2015, 61, 530−551. (35) Garcia, D. J.; You, F. Network-Based Life Cycle Optimization of the Net Atmospheric CO2-eq Ratio (NACR) of Fuels and Chemicals Production from Biomass. ACS Sustainable Chem. Eng. 2015, 3, 1732− 1744. (36) Garcia, D. J.; You, F. Life Cycle Network Modeling Framework and Solution Algorithms for Systems Analysis and Optimization of the Water-Energy Nexus. Processes 2015, 3, 514−539.

IWj,p weighting factor for interval p of technology j Continuous Variables

Ej G Mi Pi PWj,p Qj R1,i R2 Si T1,i T2,i Wj Z1,j Z2,i Z3,i Z4,i η

■

approximated nonlinear term of technology j substitution variable in conventional static robust optimization problem substitution variable for bilinear term ωiZ4,i quantity of compound i to purchase weighting factor for partition point p of technology j the capacity of technology j dual variable for constraint T1,i ≤ 1, ∀ i ∈ I dual variable for constraint Σi∈I T1,i ≤ Γ1 quantity of compound i to sell variable for uncertain biomass feedstock price definition variable for uncertain biofuel product demand definition operating level of technology j dual variable for constraint (A5) dual variable for constraint (A6) dual variable for constraint (A7) dual variable for constraint (A8) substitution variable of the second-stage problem

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