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ERROR: type should be string, got "https://doi.org/10.1007/s11367-019-01650-6\nThe International Journal of Life Cycle Assessment (2019) 24:2191–2205 https://doi.org/10.1007/s11367-019-01650-6\nThe International Journal of Life Cycle Assessment (2019) 24:2191–2205 LCI METHODOLOGY AND DATABASES Abstract Purpose This paper aims to demonstrate how LCA can be improved by the use of linear programming (LP) (i) to determine the\noptimal choice between new technologies, (ii) to identify the optimal region for supplying the feedstock, and (iii) to deal with\nmultifunctional processes without specifying a certain main product. Furthermore, the contribution of LP in the context of\nconsequential LCA and LCC is illustrated. Methods We create a mixed integer linear program (MILP) for the environmental and economic assessment of new technologies. The model is applied in order to analyze two residual beech wood-based biorefinery concepts in Germany. In terms of the optimal\nconsequences for the system under study, the principle of the program is to find a scaling vector that minimizes the life cycle\nimpact indicator results of the system. We further transform the original linear program to extend the assessment by life cycle\ncosting (LCC). Thereby, two multi-objective programming methods are used, weighted goal programming and epsilon constraint\nmethod. Results and discussion The consequential case studies demonstrate the possibility to determine optimal locations of newly\ndeveloped technologies. A high number of potential system modifications can be studied simultaneously without matrix inver-\nsion. The criteria for optimal choices are represented by the objective functions and the additional constraints such as the available\nfeedstock in a region. By combining LCA and LCC targets within a multi-objective programming approach, it is possible to\naddress environmental and economic trade-offs in consequential decision-making. Conclusions This article shows that linear programming can be used to extend standard LCA in the field of technological choices. Additional consequential research questions can be addressed such as the determination of the optimal number of new production\nplants and the optimal regions for supplying the resources. The modifications of the program by additional profit requirements\n(LCC) into a goal program and Pareto optimization problem have been identified as promising steps toward a comprehensive\nmulti-objective LCSA. Keywords Eco-efficiency . Epsilon constraint . Goal programming . Mixed integer . Multi-objective optimization . Spatial LCA Keywords Eco-efficiency . Epsilon constraint . Goal programming . Mixed integer . Multi-objective opt Consequential LCA and LCC using linear programming: an illustrative\nexample of biorefineries Maik Budzinski1\n& Mattia Sisca1,2 & Daniela Thrän1,3 Received: 20 November 2017 /Accepted: 11 June 2019\n# The Author(s) 2019\n/Published online: 24 June 2019 * Maik Budzinski\nmaik.budzinski@ufz.de 1 Introduction In contrast,\nattributional LCA focuses on describing the life cycle of a\nproduct by attributing the environmental flows to the product\nunder study. The choice for either determining consequences or attribu-\ntional impacts also affects the way of modeling in terms of the\nmultifunctional problem. This problem arises when a process\nprovides more than one product that is not used within the\nsystem under study (Heijungs and Frischknecht 1998;\nHeijungs and Guinée 2007). In this case, the environmental\nburdens associated with the multifunctional process need to be\nallocated to different products. Several procedures have been\ndeveloped to solve the multifunctional problem (Heijungs and\nSuh 2002). Consequential LCA approaches prefer to use sub-\nstitution method to deal with the problem, whereas attribution-\nal LCA is a synonym for using the partitioning method\n(Ekvall and Weidema 2004; Thomassen et al. 2008; Schmidt\n2010). The ISO standard suggests the expansion of the system\nas one method to deal with multifunctional unit processes\n(ISO 2006). Also, partitioning method is recommended,\nwhere dividing unit processes and expanding the product sys-\ntem cannot be avoided. Several authors have identified that\nsystem expansion is conceptually equivalent to substitution\n(Heijungs and Guinée 2007; Tillman et al. 1994). Thereby,\nthe expansion of additional functions related to the co-\nproducts can either be carried out by subtracting the functions\nfrom the according process (substitution aka avoided burden\nmethod) or including the functions into the final demand vec-\ntor (system expansion by expanding the functional unit). In\nany way, dealing with multifunctional processes by using\npartitioning and/or substitution method depends on a model\nchoice (Majeau-Bettez et al. 2015). The choice of a certain\nmodel to deal with multifunctional processes is usually argued\nby the invertibility of the technology matrix (avoid rectangular\nmatrix) and the argumentation that the model represents real\nworld more closely than others would do (Suh et al. 2010). However, to solve systems with more products than processes\n(rectangular technology matrix), linear programming is a suit-\nable way (Heijungs and Suh 2002). The aim of this paper is, hence, to demonstrate how LCA\ncan be extended by the use of linear programming (i) to de-\ntermine the optimal choice between new technologies, (ii) to\nidentify the optimal region for supplying the feedstock, and\n(iii) to deal with multifunctional processes without specifying\na certain main product. 1 Introduction Responsible editor: Sangwon Suh Electronic supplementary material The online version of this article\n(https://doi.org/10.1007/s11367-019-01650-6) contains supplementary\nmaterial, which is available to authorized users. Electronic supplementary material The online version of this article\n(https://doi.org/10.1007/s11367-019-01650-6) contains supplementary\nmaterial, which is available to authorized users. Life cycle assessment (LCA), and in particular consequential\nLCA, is regarded as an appropriate tool for the assessment of\nenvironmental impacts of new bio-based technologies (e.g.,\nPawelzik et al. 2013). Despite the publication of various con-\nsequential modeling approaches, consequential LCA is still\nfar from a proper systematization (Zamagni et al. 2012). Although the use of consequential and attributional LCA for\ndecision-making is still being discussed (Ekvall et al. 2016;\nHertwich 2014; Plevin et al. 2014), all approaches are based\non the definition of consequential LCA that “is designed to\ngenerate information on the consequences of decisions”\n(Ekvall and Weidema 2004). Consequential LCA attempts to * Maik Budzinski\nmaik.budzinski@ufz.de 1\nHelmholtz Centre for Environmental Research – UFZ,\nPermoserstraße 15, 04318 Leipzig, Germany 2\nPolitecnico di Milano, Piazza Leonardo da Vinci 32,\n20133 Milan, Italy 3\nDeutsches Biomasseforschungszentrum (DBFZ) gemeinnützige\nGmbH, Torgauer Straße 116, 04347 Leipzig, Germany 2192 Int J Life Cycle Assess (2019) 24:2191–2205 of LCA with linear programming (LP) has a longer tradition. Azapagic and Clift (1995, 1998) introduced LP in the field of\nprocess engineering. Thereby, LP is used to allocate environ-\nmental burdens to multiple co-products by means of the mar-\nginal values at the solution of the LP model. Recent studies\nexist in the field of biomass utilization. Vadenbo et al. (2017)\nused a multi-objective optimization model to determine the\noptimal activity levels of a set of biomass process options. Kostin et al. (2012) carried out an assessment with a mixed-\ninteger linear programming approach for ethanol production\nchains. Based on a rectangular choice-of-technology model\n(Duchin and Levine 2011), Kätelhön et al. (2016) built a tech-\nnology choice model (TCM) for consequential LCA of rice\nproduction. Despite the attempts to incorporate LCA into a\nconsistent multi-objective framework combining linear pro-\ngramming and life cycle sustainability assessment (LCSA)\n(e.g., Gong and You 2017; Steubing et al. 2011; Liu et al. 2010), a conceptual discussion on how consequential LCA\nand LCC can be extended by LP is still scarce. determine how environmental flows will change due to the\nconsequences of a decision (Curran et al. 2005). 1 Introduction In doing so, we use the matrix-based\nnotation of Heijungs and Suh (2002) and create a rectangular\nmixed integer linear program and carry out a case study to\nillustrate its applicability. Thereby, two wood-based\nbiorefinery concepts are analyzed in terms of the optimal con-\nsequences for the total system under study. Furthermore, we\nmodify the original linear program into a weighted goal pro-\ngramming problem and a Pareto optimization model those\nextend the environmental focus on sustainability by adding\nthe economic perspective through a life cycle costing (LCC)\napproach. 3.1 Goal and scope Goal and scope\nstudy concerns two exemplar\nepts and data are based on\n6). The first biorefinery co\nnosolv lignin, and biomethan\nuces the same products as the\ns replaced by ethanol. The an\nmultifunctional biorefinery p\ne 1. The goal of the exempl\nly, we want to determine th\nopriate to reduce environmen\nnt production system. Second\nh site the biorefineries shoul\nly, to identify districts for\ntocks. 1\nAnnual input and output of b\nBi\nethane (m3)\n5.0\nene (kg)\n4. olysis lignin (MJ)\n1.9\nol (MJ)\n0 The study concerns two exemplary biorefinery concepts. The\nconcepts and data are based on Budzinski and Nitzsche\n(2016). The first biorefinery concept produces ethylene,\norganosolv lignin, and biomethane. The second configuration\nproduces the same products as the first one except for ethylene\nthat is replaced by ethanol. The annually produced amounts of\nthe multifunctional biorefinery processes are illustrated in\nTable 1. The goal of the exemplary case study is threefold. Firstly, we want to determine the biorefinery that is more\nappropriate to reduce environmental impacts in regard to the\ncurrent production system. Secondly, we want to determine at\nwhich site the biorefineries should be located (Fig. 1). And\nfinally, to identify districts for supplying woody biomass\nfeedstocks. An interesting but not often recognized characteristic of\nLCA is the existence of a price model beside the quantity\nmodel (As = f), which is similar to input-output models. This\nbecomes obvious when a price vector α is introduced (cf. Heijungs et al. 2013). Using this price vector, it is possible\nto compute the value added v per process. Table 1\nAnnual input and output of biorefinery processes\nBiorefinery 1\nBiorefinery 2\nBiomethane (m3)\n5.05·107\n5.05·107\nEthylene (kg)\n4.14·107\n0\nHydrolysis lignin (MJ)\n1.96·109\n1.96·109\nEthanol (MJ)\n0\n1.87·109\nOrganosolv lignin (kg)\n6.22·107\n6.22·107\nSodium hydroxide (kg)\n−5.04·105\n−5.04·105\nMagnesium sulfate (kg)\n−3.57·105\n−3.57·105\nPhosphorus (kg)\n−3.57·105\n−3.57·105\nSulfur (kg)\n−3.57·105\n−3.57·105\nSulfuric acid (kg)\n−3.76·106\n−3.76·106\nFodder yeast (kg)\n−2.32·106\n−2.32·106\nLime, packed (kg)\n−3.84·104\n−3.84·104\nRefrigerant (kg)\n−4.94·104\n0\nHeat, natural gas (MJ)\n−5.86·107\n0\nElectricity, natural gas (kWh)\n−1.08·109\n−1.35·109\nChemical factory (unit)\n−1.05·108\n−9.52·107\nTreated wastewater (m3)\n−3.33·10−2\n−3.33·10−2\nWater (kg)\n−2.84·106\n−3.16·106\nBeech wood (kg dm)\n−4.00·108\n−4.00·108 Table 1\nAnnual input and output of biorefinery processes v ¼ A0α\nð4Þ ð4Þ h ¼ Qg The so-called multifunctional problem arises if a process\ngenerates more than one product and if and only if the deliv-\nered functions are not used in the same proportion in the sys-\ntem (Heijungs and Frischknecht 1998). The most common\napproaches to solve the multifunction problem can be classi-\nfied into partitioning method and substitution method\n(Heijungs and Guinée 2007). By applying these procedures,\nthe technology matrix A becomes square, since an equal\namount of rows and columns exists. The partitioning method\ndivides the multifunctional process into processes that are\nmono-functional by the use of allocation factors. Several ways\nto gain these factors are possible, e.g., due to physical or\nmonetary characteristics of the products. The substitution\nmethod requires the definition of a mono-functional process,\nwhich provides the avoided product to the multi-functional\nprocess. Adding this process to the technology matrix A\nmakes its inversion possible. 2 Computational structure of LCA and LCC The computational structure of LCA is comprehensively de-\nscribed by Heijungs and Suh (2002). Mathematically, the first\nstep of life cycle inventory (LCI) analysis is the determination\nof the scaling vector s. The components of this vector scale\nprocesses of a system up or down such that the output of unit\nprocesses exactly matches the final demand. The final demand\nvector f includes the reference flows r, which are defined\nwithin the goal and scope of a LCA study. The scaling vector\ncan be determined by the multiplication of the inverse of tech-\nnology matrix A with the final demand vector. A major issue for standard life cycle assessment, which\naddresses one state of a particular product system, might be\nthe fact that no optimal production mixes can be determined. Usually, in LCA, this point is addressed by creating a small\nnumber of scenarios analyzing different product system alter-\nnatives iteratively (e.g., Budzinski and Nitzsche 2016; Renouf\net al. 2018). However, such procedure can be very time-con-\nsuming. Particularly in consequential LCA, this could be an\nissue. In many cases, not only the consequences of a potential\ndecision need to be described, but rather the optimal decision\nitself shall be identified. To overcome limitations in LCA,\namong other methods such as fuzzy programming (Tan et al. 2008) or graph methods (Vance et al. 2015), the combined use s ¼ A−1 f; with f i ¼\n> 0if i∈r\n0otherwise\n\u0001\nð1Þ s ¼ A−1 f; with f i ¼\n> 0if i∈r\n0otherwise\n\u0001\nð1Þ ð1Þ 2193 Int J Life Cycle Assess (2019) 24:2191–2205 g ¼ Bs ð3Þ 3 Exemplary case study The next step of the LCI analysis is to specify the inventory\nvector g according to the reference flows. This is done by\nmultiplying the intervention matrix B with the scaling vector. In the following sections, we demonstrate how the previously\nmentioned equations can be modified to create a rectangular\nchoice-of-technology model in regard to the goal and scope of\nthe study (Sect. 3.1). Therefore, we define a mixed integer\nlinear program (Sect. 3.2) and determine the results of an\nexemplary case study (Sect. 3.3). To include LCC aspects,\nwe transform the original problem into a weighted goal pro-\ngramming problem and a Pareto optimization model in Sects. 3.4 and 3.5, respectively. ð2Þ 1 http://www.cechemnet.de/Netzwerk/cechemnet_en v ¼ A0α To determine the value added in terms of a reference flow,\nthe elements of the scaling vector s can be multiplied by the\nvalue added of the corresponding processes, where 1 is the\nsummation operator (vector of ones). v0\nscaled ¼ 1 diag v\n0\n\u0003 \u0004\ns\n\u0003\n\u0004\nð5Þ ð5Þ ð5Þ This value added vector can then be used to determine life\ncycle costs (LCC). Heijungs et al. (2013) define life cycle cost\nof a composite final demand as l ¼ ∑j−vscaled\nj\n, where j runs\nover all case such that technical coefficient aij > 0, where ref-\nerence flows i is such that fi ≠0. In other words, the life cycle\ncosts of reference flows, which are specified in the final de-\nmand vector, are the negative sum of the corresponding entries\nin the scaled value added vector. 2194 Int J Life Cycle Assess (2019) 24:2191–2205 For simplicity reasons, we focus on four chemical parks of\nSchwarzheide. We assume that all chemical parks can provide\nFig. 1 Potential location for the two biorefinery types and annual availability of residual beech wood from forests within German districts (scenario\n100%) the two biorefinery types and annual availability of residual beech wood from forests within German districts (scenario Fig. 1 Potential location for the two biorefinery types and annual availability of residual beech wood from forests within German districts (scenario\n100%) Schwarzheide. We assume that all chemical parks can provide\nthe required infrastructure and utilities in the same manner. Furthermore, we suppose that either one or none of the\nbiorefinery concepts can be built at a location. Data on resid-\nual beech wood are estimated at the level of 37 administrative\ndistricts in Germany according to Polley and Kroiher (2006). For simplicity reasons, we focus on four chemical parks of\nthe Central European Chemical Network1 as potential plant\nlocations, which would in principal allow the building of the\ntwo biorefinery concepts: Leuna, Böhlen, Zeitz, and 2195 Int J Life Cycle Assess (2019) 24:2191–2205 This residual wood is currently not used and remains in forests\nafter chopping, which might be possibly used as feedstock for\nbiorefineries without inducing feedstock rivalry (Michels\n2009). Since the 100% use of available beech wood residuals\nis quite optimistic, we introduce a second scenario in which\nthe availability in each region is reduced by 50% (cf. Michels\n2009). v ¼ A0α The transport distances from the districts to the four\npotential plant locations can be covered by train and/or truck. In this study, we assume that up to 200 km, the wood is\ntransported only by truck. For distances greater than\n200 km, the wood is additionally transported by train. Thereby, we further assume that a minimal distance (between\n10 and 30 km depending on the availability of train network)\nis covered by truck. Varying transport distances for other pre-\nproducts of the biorefineries are neglected. multi-output processes are considered in the upstream system\nand, hence, is more general. The overall program can be summarized by the formulas 8\nto 11 considering that the scaling factors of biorefinery pro-\ncesses s1, s2, s3, s4, s5, s6, s7, and s8 are integers. Thereby, Eq. 8\nrepresents the objective function of minimizing environmental\nimpacts. Equation 9 illustrates the system under study includ-\ning all potential biorefineries, feedstock regions, and transpor-\ntation alternatives. Equation 10 includes the capacity con-\nstraints for feedstock availability in the regions. Equation 11\nconsiders the fact that only one of the two biorefinery options\ncan be built at a potential location. By using the linear pro-\ngramming model, we are interested in the reduction of the\ntotal environmental impacts of the current system. Hence,\nthe objective function results from Eqs. 2 and 3, and the target\nis to find a vector s that minimizes the total impacts h of the\nsystem, which is expressed by matrix A. Avisualization of the\nmodeling principle including the sub-matrices of A is given in\nFig. 2. The fundamental principle of the model is to check\nwhether environmental impacts can be reduced by biorefinery\n1 or 2 considering 4 potential locations (ABRS) and 37 regions\nof wood supply (AS). If not, the model would only choose\ncurrent processes (ARef) to provide the components of the final\ndemand. We assume that the new biorefinery products compete with\nexisting products of the current system. Hence, we define\nsubstitutable reference products for each biorefinery product:\nbiomethane vs. natural gas, bio-based ethylene vs. fossil eth-\nylene, hydrolysis lignin vs. lignite briquettes, ethanol vs. pet-\nrol, and organosolv lignin vs. polyol. The definition of these\navoided products is a long-discussed issue in LCA. Since the\ngeneral societal aim is to move towards a bio-based economy,\nwe use fossil-based references that provide the same\nfunctions. v ¼ A0α The amounts of reference flows are specified in regard to\nthe German demand of these products within the current sys-\ntem. Thereby, the reference flows need to be specified in the\nmanner that the products of current processes must provide the\nsame functions as of the biorefinery products. In the example,\nwe assume that this is guaranteed, e.g., by referring to MJ in\nthe case of ethanol and gasoline. Minimize h ¼ QBs\nð8Þ\nsubject to Minimize h ¼ QBs\nð8Þ ð8Þ e h\nQ s\nð8Þ\nsubject to subject to subject to subject to As\n≥f i > 0\n¼ f i ¼ 0\n≥f i ¼ 0\n8\n<\n:\nif i∈r\nif i∈T; S\notherwise\nð9Þ\n0≤s≤c\nð10Þ\ns1 þ s2 ≤1\ns3 þ s4 ≤1\ns5 þ s6 ≤1\ns7 þ s8 ≤1\nð11Þ As\n≥f i > 0\n¼ f i ¼ 0\n≥f i ¼ 0\n8\n<\n:\nif i∈r\nif i∈T; S\notherwise\nð9Þ As\n≥f i > 0\n¼ f i ¼ 0\n≥f i ¼ 0\n8\n<\n: ð9Þ For LCIA, we chose the category climate change taking\ninto account the three substances CO2, CH4, and N2O. ð10Þ ð11Þ As ¼ f A ¼\nABRS\n0\n0\n0\nARef\n−A*\nup\nAup\n−A**\nup\n−A***\nup\n0\n0\n0\nAS\n−A*\nS\n0\n−A*\nT\n0\n0\nAT\n0\n2\n664\n3\n775\nð12Þ by the inequation ð12Þ ð7Þ 3.2 The linear programming model According to Heijungs and Suh (2002), the principle of ex-\ntending LCA by LP is to relax the balance equations into\ninequations. Hence, we can replace the equation with s1, s2, s3, s4, s5, s6, s7, s8 ∈ℕ with s1, s2, s3, s4, s5, s6, s7, s8 ∈ℕ with s1, s2, s3, s4, s5, s6, s7, s8 ∈ℕ The rectangular technology matrix Awith the dimension of\nproducts-by-processes consists of several sub-matrices. As ¼ f\nð6Þ\nby the inequation\nAs≥f:\nð7Þ ð6Þ As≥f: By using this formulation, the solution of such program\nwould allow flows that are greater than those specified in the\nfinal demand vector. Regarding the special case in which pro-\ncesses are defined to be strictly mono-functional, the formu-\nlation as As = f would also work for solving the LP. The for-\nmulation in Eq. (7) also provides solutions for cases in which These sub-matrices have the following dimensions: These sub-matrices have the following dimensions: ABRS\nir × jBRS (5 × 8)\nARef\nir × jRef (5 × 5)\nAup\niup × jup (17 ×\n17) ABRS\nir × jBRS (5 × 8)\nARef\nir × jRef (5 × 5)\nAup\niup × jup (17 ×\n17) −A*\nup\niup × jBRS (17 ×\n8)\n−A**\nup\niup × jS (17 × 37)\n−A***\nup\niup × jT (17 ×\n296)\n−A*\nT\niT × jBRS (8 × 8)\nAT\niT × jT. (8 × 296)\n−A*\nS\niS × jT. (37 × 296)\nAS\niS × jS.(37 × 37). with,\nir\nbiorefinery/reference product\niup\nproduct of upstream process\niT\ntransported wood\niS\nsupplied wood in a district\njBRS\nbiorefinery process\njRef\nsubstitutable reference process\njup\nupstream process\njS\nregion for supplying wood\njT\ntransport option\nThe columns of matrix ABRS represent the biorefinery op-\ntions jBRS of each type at each potential location. The rows of\nthis matrix represent the products ir which can also be pro-\nduced by current reference processes jRef. Matrix ARef\nincludes current available processes jRef that produce the same\nfunctions ir as the biorefinery processes. To address the choice for a certain district, which deliver\nthe residual beech wood, the model is extended by the matri-\nces AT, −A*\nT, AS and −A*\nS. Matrix AS is an identity matrix to\nconsider the different alternatives of beech wood supply from\nthe 37 districts. Matrices −A*\nS and ATare needed to include the\nalternatives of beech wood transport from each district to each\none of the 8 biorefinery alternatives. The linkage of the these\nalternatives to the biorefinery options is represented by matrix\n−A*\nT. Upstream processes in matrix Aup, which are needed to\nprovide the pre-products of the biorefineries, the cultivation,\nand transportation of wood and the reference products, com-\nplete the rectangular A matrix. As≥f: The program is designed as an LCA for a final demand fr\non reference flows r, which are characterized by the products\nof the biorefinery system and the corresponding reference\nproducts (biomethane/natural gas, bio-based ethylene/fossil\nethylene, hydrolysis lignin/lignite briquettes, ethanol/petrol,\norganosolv lignin/polyol). The amounts for the multiple com-\nponents of the functional unit are taken from statistical sources\n(e.g., VCI 2018; SDK 2017) and represent those of a current\nGerman demand. Other components of the final demand\nFig. 2 Simplified flowchart\nInt J Life Cycle Assess (2019) 24:2191–2205\n2196 Int J Life Cycle Assess (2019) 24:2191–2205 2196 Fig. 2 Simplified flowchart Fig. 2 Simplified flowchart includes current available processes jRef that produce the same\nfunctions ir as the biorefinery processes. To address the choice for a certain district, which deliver\nthe residual beech wood, the model is extended by the matri-\nces AT, −A*\nT, AS and −A*\nS. Matrix AS is an identity matrix to\nconsider the different alternatives of beech wood supply from\nthe 37 districts. Matrices −A*\nS and ATare needed to include the\nalternatives of beech wood transport from each district to each\none of the 8 biorefinery alternatives. The linkage of the these\nalternatives to the biorefinery options is represented by matrix\n−A*\nT. ir\nbiorefinery/reference product\niup\nproduct of upstream process\niT\ntransported wood\niS\nsupplied wood in a district\njBRS\nbiorefinery process\njRef\nsubstitutable reference process\njup\nupstream process\njS\nregion for supplying wood\njT\ntransport option Upstream processes in matrix Aup, which are needed to\nprovide the pre-products of the biorefineries, the cultivation,\nand transportation of wood and the reference products, com-\nplete the rectangular A matrix. The program is designed as an LCA for a final demand fr\non reference flows r, which are characterized by the products\nof the biorefinery system and the corresponding reference\nproducts (biomethane/natural gas, bio-based ethylene/fossil\nethylene, hydrolysis lignin/lignite briquettes, ethanol/petrol,\norganosolv lignin/polyol). The amounts for the multiple com-\nponents of the functional unit are taken from statistical sources\n(e.g., VCI 2018; SDK 2017) and represent those of a current\nGerman demand. Other components of the final demand The columns of matrix ABRS represent the biorefinery op-\ntions jBRS of each type at each potential location. The rows of\nthis matrix represent the products ir which can also be pro-\nduced by current reference processes jRef. f S;T ¼ 0 is introduced. We further define the first eight scaling factors for the\nbiorefinery processes to be integers. This choice is neces-\nsary, since the data collection for the biorefineries is based\non process simulation for a specific capacity. The inter-\nmediate flows and environmental interventions associated\nwith this capacity are not up- or downscalable in a linear\nmanner. The program is completed by introducing lower\nbounds (set to 0) and a vector of upper bounds c. In\nregard to the availability of residual beech wood in each\ndistrict, the data of Fig. 1 is considered as upper bounds\nfor sjS. The upper bounds c for the scaling factors of\nbiorefinery processes (sjBRS ) were set to 1, since we as-\nsume that only one biorefinery can be built at a chemical\npark. To study the origin of wood supply it is possible to analyze\nthe corresponding scaling factors of the transport processes jT. These factors represent the contribution to the overall wood\ndemand of the biorefineries. Figure 3a illustrates the solutions\nfor the 100% scenario. Thereby, the biorefinery in Zeitz is\ndelivered from regions in the middle west of Germany and\nfrom regions in the middle south. In contrast, the biorefinery\nin Schwarzheide is delivered by regions in the northeastern\npart of Germany. The LCI data for upstream biorefienry processes and\nfor the substitutable reference system is taken from the\necoinvent 3.2 database (cut-off model). The whole prob-\nlem and the corresponding data can be studied in more\ndetail in the Electronic Supplementary Material\n(MILP_LCA.xlsx). Due to the reduced wood availability in the 50% scenario,\nregions supplying the feedstock to biorefinery 1 in Leuna are\ndistributed throughout Germany (Fig. 3b). Hence, a large\ncatchment area is required to fulfill the demand of the\nbiorefinery in the scenario with reduced residual beech wood\navailability. 3.3 Results To solve the mixed integer linear program, the intlinprog\nsolver in Matlab was used. An additional scenario is an-\nalyzed in which the available amount of beech wood is\nreduced by 50%. The results for 100% scenario (1.14 Mt\ndm beech wood in Germany) indicate that two\nbiorefineries with a capacity of 0.4 Mt dm are built at\nthe considered chemical parks. Since the scaling factors\nfor biorefineries are defined as integers, not the total\navailable amount of beech wood is exhausted. The opti-\nmal locations to build the biorefineries are Zeitz and\nSchwarzheide. It is visible that for both scenarios, the\nethylene-producing biorefinery 1 is more preferable to\nreduce greenhouse gas emissions than the ethanol-\nproducing biorefinery 2 (Table 2). When reducing the\navailable residual wood for all districts by 50%, another\nlocation becomes more preferable than those of the 100%\nscenario. This is an interesting result, since one might\nassume that one of the locations identified in the 50% As≥f: Matrix ARef Int J Life Cycle Assess (2019) 24:2191–2205 2197 vector f are set to zero. The design of an LCA for a demand on\nthe functional unit does not take into account the linkage of\ndownstream processing with the modeled upstream process-\ning so that economy-wide impacts are cut off (Suh 2004). However, the modeling in regard to the functional unit is a\ngeneral principle of LCA (Peters and Hertwich 2006). To en-\nsure the balance between supply and demand of transported\nbeech wood, the constraint\n0\n0\nAS\n−A*\nS\n0\n−A*\nT\n0\n0\nAT\n0\n\u0005\n\u0006\ns ¼\nf S;T ¼ 0 is introduced. would be chosen. The total reduction of the impacts on\nclimate change compared to the impacts of the current\nsystem (hRef = QRef ∙BRef ∙ARef\n−1 ∙fr) is 4.76 Mt/a CO2\neq. in the 100% scenario and 2.38 Mt/a CO2 eq. in the\n50% scenario. The contribution analysis for the 100% scenario shows\nthat current substitutable processes still provide the ma-\njor part of the overall GHG emissions (lignite production\n88.48%, natural gas production 8.24%, petrol production\n1.95%, ethylene production 1.05%, polyol production\n0.25%) whereas the biorefinery system plays a minor\npart (biorefineries 0.12%, wood production −0.20%, oth-\ner upstream processes including transport 0.12%). Due to\nthe feedstock availability of German beech wood resid-\nuals, only a small ratio of current GHG emissions could\nbe reduced. Detailed information on contribution analy-\nses is provided in the Electronic Supplementary Material\n(MILP_LCA.xlsx). f S;T ¼ 0 is introduced. 3.4 Inclusion of LCC by multi-objective optimization For example, it might also reasonable to explicitly take\ninto account the profit for reference processes competing with\nbiorefineries, or to use discounted cash flow analysis taking\ninto account the time value of money (e.g., net present value). However, additional data must be collected. For simplicity\nreasons, we chose the standard approach of LCC in this study. Minimize whdþ\nh þ wzd−\nz\nð15Þ\nsubject to Minimize whdþ\nh þ wzd−\nz\nð15Þ ð15Þ subject to subject to subject to As\n≥f i > 0\n¼ f i ¼ 0\n≥f i ¼ 0\n8\n<\n:\nif i∈r\nif i∈t\notherwise\nð16Þ As\n≥f i > 0\n¼ f i ¼ 0\n≥f i ¼ 0\n8\n<\n: ð16Þ QBs þ d−−dþ ¼ h\nð17Þ\nps þ d−−dþ ¼ z\nð18Þ\n0≤s≤c\nð19Þ\ns1 þ s2 ≤1\ns3 þ s4 ≤1\ns5 þ s6 ≤1\ns7 þ s8 ≤1\nð20Þ QBs þ d−−dþ ¼ h\nð17Þ\nps þ d−−dþ ¼ z\nð18Þ\n0≤s≤c\nð19Þ\ns1 þ s2 ≤1\ns3 þ s4 ≤1\ns5 þ s6 ≤1\ns7 þ s8 ≤1\nð20Þ QBs þ d−−dþ ¼ h\nð17Þ\nps þ d−−dþ ¼ z\nð18Þ ð17Þ ð19Þ 0≤s≤c To deal with multiple objectives in linear programming,\nbasically, three groups can be distinguished: a priori, interac-\ntive, and a posteriori methods (Hwang and Masud 1979). The\nfirst class requires the definition of preferences between goals\nat the beginning of the solution process. The latter class re-\nquires the prioritization at the end after the presentation of all\nPareto optimal solutions. To illustrate the pros and cons of the\nbroad range of applications, we modify the linear program in\nthe next sections choosing weighted goal programming for an\na priori approach and epsilon-constraint method for an a\nposteriori approach. ð20Þ with s1, s2, s3, s4, s5, s6, s7, s8 ∈ℤ with s1, s2, s3, s4, s5, s6, s7, s8 ∈ℤ Table 3 shows the result of the GP model for both scenar-\nios. Contrary to the LP model, the optimal solution is the\nethanol concept (biorefinery 2) for both scenarios. The opti-\nmal chemical parks of the chosen biorefineries are identical to\nthose of the LP model. The regions that deliver the wood\nfeedstock to the plants are also identical. Therefore we omit\nto illustrate them again in the map. 3.4 Inclusion of LCC by multi-objective optimization For h,\nwe chose 0, since the aim of a sustainable economy can be\nregarded to be carbon neutral. For the profit target z, we\nchoose an arbitrarily large value (1.00E + 15 €) which is\nequivalent to maximize the objective function. The weighting\nfactors w result in a normalization and prioritization of the\ntarget deviations. For greenhouse gas emissions, we use the\nweighting factor wh = 0.025 which is the value of expected\ndamage costs in terms of €/kg CO2-eq (De Bruyn et al. 2010). The choice of this factor is concerned by uncertainty,\ncomparable to those related to the monetization of environ-\nmental impacts. Due to the illustrative character of this study,\nwe do not assess this in detail. However, robust optimization\nmay take uncertainty of parameters explicitly into consider-\nation (Wang and Work 2014). The weighting factor wz is set to\n1, since the unit of the profit target is €. Equation 13 consists of two major terms. The first term\nconsiders the cash-flows related directly to the biorefinery\nprocesses jBRS including revenues for biorefinery products processes jBRS including revenues for biorefinery products\n( ∑\niBRS¼1\nIBRS\nABRS\niBRS;jBRS \u0002 αiBRS ), payments for pre-products\n( ∑\niup¼1\nIup\nA*\niup;jBRS \u0002 αiup ), payments for wood ( ∑\niT¼1\nIT\nA*\niT;jBRS \u0002 αiT ),\nand other payments such as for personal and insurance\n( ∑\nK\nk¼1\nFk jBRS ). The second term represents the payments related\nto the transportation options jT of wood. This expression can\nbe simplified in matrix notation as Maximize z ¼ pËCs;\nð14Þ ð14Þ where the components of vector p specify the profit of pro-\ncesses j. This LCC approach is congruent to the proposed one by\nHeijungs et al. (2013), but differs in the way that only those\ncosts are taken into account, which are relevant for the invest-\nment decision for the biorefinery options. This choice is con-\ncerned with taking the view of potential investors maximizing\nthe profit of a biorefinery. Thereby, only the costs are consid-\nered that are relevant for the decision on the profitability of the\nbiorefinery (including costs for transportation and additional\ninputs). Besides to this simple LCC approach, however, there\nare further possibilities to consider cost information by the\nmodel. 3.4 Inclusion of LCC by multi-objective optimization Since we have not considered transportation and other costs so\nfar in this case study, we adopt the model by introducing an\nadditional objective for life cycle costing. In doing so, we\nintroduce for the specific biorefineries an additional input fac-\ntor matrix F with subcategories k (expressed in monetary\nunits): personal costs, taxes, insurance, and administration. As Heijungs et al. (2013) showed, the consideration of costs\nfor the inputs and outputs of the processes of the technology\nmatrix A can be carried out by using a price vector α. Since\nwe focus on the investor’s perspective, price information is\nonly required for products that are relevant for the profitability\nof the biorefineries. The additional objective function, which\nincludes costs for processed and produced commodities of the\nbiorefinery processes as well as the costs for transported beech\nwood, can be derived as Int J Life Cycle Assess (2019) 24:2191–2205 2198 Table 2\nResults of the mixed integer linear program, per annum Table 2\nResults of the mixed integer linear program, per annum\n100% scenario\n50% scenario\nBiorefinery 1, Leuna (pc.)\n0\n1\nBiorefinery 2, Leuna (pc.)\n0\n0\nBiorefinery 1, Zeitz (pc.)\n1\n0\nBiorefinery 2, Zeitz (pc.)\n0\n0\nBiorefinery 1, Böhlen (pc.)\n0\n0\nBiorefinery 2, Böhlen (pc.)\n0\n0\nBiorefinery 1, Schwarzheide (pc.)\n1\n0\nBiorefinery 2, Schwarzheide (pc.)\n0\n0\nNatural gas (m3)\n9.580053·1010\n9.585027·1010\nFossil ethylene (kg)\n5.017197·109\n5.058598e·109\nLignite (MJ)\n3.214008·1013\n3.214204·1013\nPetrol (MJ)\n7.340000·1011\n7.340000·1011\nPolyol (kg)\n4.342704·108\n4.964352·108\nh (kg CO2-eq.)\n7.152422·1011\n7.156019·1011\nPotential savings h −hRef (kg CO2-eq.)\n−7.193181·108\n−3.596181·108 Maximize z ¼\n∑\njBRS¼1\nJ BRS\n∑\niBRS¼1\nIBRS\nABRS\niBRS;jBRS \u0002 αiBRS\n \n! −\n∑\niup¼1\nIup\nA*\niup;jBRS \u0002 αiup\n \n! −\n∑\niT ¼1\nIT\nA*\niT ;jBRS \u0002 αiT\n \n! −∑\nK\nk¼1\nFk jBRS\n \n! \u0002 sjBRS−∑\njT¼1\nJ T\n∑\niup¼1\nIup\nA***\niup;jT \u0002 αiup\n \n! \u0002 sjT\nð13Þ ð13Þ Fig. 3 Optimal regions to supply beech wood to the biorefineries Fig. 3 Optimal regions to supply beech wood to the biorefineries Fig. 3 Optimal regions to supply beech wood to the biorefineries Fig. 3 Optimal regions to supply beech wood to the biorefineries 2199 Int J Life Cycle Assess (2019) 24:2191–2205 respectively. The target function of the goal program mini-\nmizes the deviations in terms of the desired values. 3.4 Inclusion of LCC by multi-objective optimization The reason for choosing\nbiorefinery 2 is the higher priority to minimize the negative\ndistance to the economic target value (d−\nz ) compared to the\npriority of minimizing the positive distance to the environ-\nmental target value (dþ\nh ). Thus, the total annual amount of\nenvironmental savings is reduced from 0.72 Mt/a and 0.36 Mt/\na CO2-eq to 0.37 Mt/a and 0.18 Mt/a CO2-eq in the 100% 3.4.1 Goal programming In weighted goal programming, the objectives are taken ex-\nplicitly into account as constraints (Miller and Blair 2009). The principle of goal programming (GP) is to use slack vari-\nables dh and dz that measure the deviation from the desired\nenvironmental and economic target values h and z, Int J Life Cycle Assess (2019) 24:2191–2205 2200 Table 3\nResults of the goal program, per annum\n100% scenario\n50% scenario\nBiorefinery 1, Leuna (pc.)\n0\n0\nBiorefinery 2, Leuna (pc.)\n0\n1\nBiorefinery 1, Zeitz (pc.)\n0\n0\nBiorefinery 2, Zeitz (pc.)\n1\n0\nBiorefinery 1, Böhlen (pc.)\n0\n0\nBiorefinery 2, Böhlen (pc.)\n0\n0\nBiorefinery 1, Schwarzheide (pc.)\n0\n0\nBiorefinery 2, Schwarzheide (pc.)\n1\n0\ndþ\nh (kg CO2-eq.)\n7.155941·1011\n7.157785·1011\nd−\nz (€)\n9.999991·1013\n9.999995·1013\nProfit z−d−\nz (€)\n9.174736·107\n4.509840·107\nPotential savings hRefS−dþ\nh (kg CO2-eq.)\n−3.674181·108\n−1.830181·108 Table 3\nResults of the goal program, per annum process-based LCC. However, a possible way is to implement\nan additional constraint in the LP model that ensures the ex-\nceeding of a minimal positive profit value. scenario and the 50% scenario, respectively. The annual\nprofits are 92 Mio € and 45 Mio € in the 100% scenario and\nthe 50% scenario, respectively. When introducing economic\naspects, this result is similar to the one of Budzinski and\nNitzsche (2016) who also concluded that the ethanol-\nproducing concept has a better economic performance than\nthe ethylene-producing concept. However, in Budzinski and\nNitzsche (2016), only the ethanol biorefinery is profitable,\nwhich is contrary to the results of this study. The reason for\nthat is the authors’ use of a dynamic cost calculation approach\nthat takes the time value of money into account. Thereby, the\ninternal rate of return on the investment must be at least equal\nthe minimum rate of return a decision-maker is willing to\naccept. The time value of money usually is not considered in 3.4.2 Epsilon-constraint method Comparing this difference with the\noverall GHG emissions of the system, one might assume that\nthe results are not reliable in terms of data inaccuracies. To\naddress this point, we have a look at the reference flows of the\nfinal demand vector. The chosen amounts represent an esti-\nmated total demand for Germany and, hence, are quite high. However, lower values would not lead to different results and\nwould not reduce uncertainty. In fact, the choice of values for\nthe considered reference flows is arbitrary. The determination\nof optimal locations for biorefineries is, hence, not affected by\nthis choice. Since we here assume identical conditions at the\npotential chemical parks, the identification of optimal loca-\ntions is only determined by the transport distances and the\ncorresponding impacts and costs. On the contrary, the choice\nfor the type of biorefinery in terms of environmental impacts\nis mainly dominated by its emissions and those of the refer-\nence processes. In terms of profit, the optimal solution is\nmainly dominated by the corresponding costs for biorefinery\n(pre)-products. In conclusion, the results for the optimal\nchoices of locations for biorefineries may be considered being\nrobust in the face of data uncertainties in other parts of the\nmodel. A problem that might result is numerical issues when\nsolving a problem with large discrepancies in order of Pareto optimal solutions are solutions that cannot be im-\nproved in any of the two objectives without degrading the\nother objective. The Pareto optimal solutions of the 100%\nscenario are illustrated in Fig. 4. In this example, three\nPareto optimal solutions are identified. The dominated solu-\ntions, which are not Pareto optimal, are below the blue line. However, more comprehensive problems can result in various\nnon-dominated solutions. Solution P1 suggests biorefinery 1\n(ethylene) in Zeitz and in Schwarzheide. In contrast to this\nenvironmentally most preferable solution, P3 contains the\nmost preferable solution in terms of profit (Table 4). Thereby, biorefinery 2 (ethanol) is located in Zeitz and\nSchwarzheide. The solution P2 can be interpreted as a com-\npromise between these extreme solutions of maximal profit\nand minimal impacts on climate change. Furthermore, the\nsolution may become worthwhile due to the fact that both\nbiorefinery concepts (ethylene and ethanol) would be built\nin Zeitz and Schwarzheide. This mixture of technologies\nmight be also interesting for decision-making. 3.4.2 Epsilon-constraint method A disadvantage of goal programming might be the require-\nment of defining of preferences at the beginning of the solu-\ntion process. Alternatively, a posteriori methods are available\nto prioritize between goals after the generation of all Pareto\nefficient solution. In doing so, epsilon-constraint method has\nbeen already used in the field of LCA (e.g., Azapagic and Clift\n1999). Here, in this study, we apply the epsilon-constraint\nmethod in GAMS (Mavrotas 2009a, b). Thereby, the target Fig. 4 Pareto front (100%\nscenario). h: environmental\nimpacts; z: profit Fig. 4 Pareto front (100%\nscenario). h: environmental\nimpacts; z: profit Fig. 4 Pareto front (100%\nscenario). h: environmental\nimpacts; z: profit 2201 Int J Life Cycle Assess (2019) 24:2191–2205 Table 4\nResults of the epsilon-constraint method (100% scenario), per annum\nP1\nP2\nP3\nh (kg CO2-eq.)\n7.154561·1011\n7.155251·1011\n7.155941·1011\nz (€)\n5.183191·107\n7.178963·107\n9.174736·107\na)\nb)\nBiorefinery 1, Leuna (pc.)\n0\n0\n0\n0\nBiorefinery 2, Leuna (pc.)\n0\n0\n0\n0\nBiorefinery 1, Zeitz (pc.)\n1\n0\n1\n0\nBiorefinery 2, Zeitz (pc.)\n0\n1\n0\n1\nBiorefinery 1, Böhlen (pc.)\n0\n0\n0\n0\nBiorefinery 2, Böhlen (pc.)\n0\n0\n0\n0\nBiorefinery 1, Schwarzheide (pc.)\n1\n1\n0\n0\nBiorefinery 2, Schwarzheide (pc.)\n0\n0\n1\n1 Table 4\nResults of the epsilon-constraint method (100% scenario), per annum function of profit (Eq. 14) is additionally involved in the op-\ntimization problem as solution is nearly located to another such as P2 in the 50%\nscenario, clearly is an advantage of a posteriori approaches\nover a priori methods. Minimize h sð Þ; −z sð Þ\nð\nÞ:\nð21Þ ð21Þ Minimize h sð Þ; −z sð Þ\nð\nÞ: A special look requires the influence of transportation dis-\ntances on the overall result (profit and environmental im-\npacts). In this study the transportation of feedstock only has\na limited influence on the overall profit of the biorefinery. For\ninstance, the difference between biorefinery type 2 in Leuna\ncompared to the location in Böhlen is 177,280 € per year\n(Table 5). In regard to the potential reduction of GHG emis-\nsions, the difference is even less significant. Here, the\nbiorefinery 2 in Leuna has a reduction potential at 2000 kg\nCO2 eq. per year higher than in Böhlen (Pareto point 2 and 3\nof 50% scenario, Table 5). 3.4.2 Epsilon-constraint method In contrast to\ngoal programming in which only one solution is determined in\naccordance with the a priori defined priority order, Pareto\noptimization allows the decision-maker to deal with trade-\noffs after investigating the non-dominated solutions. The non-dominated solutions of the 50% scenario are illus-\ntrated in Fig. 5. The most beneficial solution P1 in terms of\nimpacts on climate change is given by the biorefinery 1 locat-\ned in Leuna. Biorefinery 2 in Leuna is the most profitable\nsolution P3. Contrary to the 100% scenario, the compromise\nsolution P2 is near to P3 suggesting biorefinery 2 in Böhlen. An interesting fact is that P3 (in both scenarios) is the solution\nof the goal programming approach (Sect. 3.4.1). However, the\nidentification of all non-dominated solutions, even if the Int J Life Cycle Assess (2019) 24:2191–2205 2202 extension of LCA is to provide a broader and systematic as-\nsessment of consequences. The implicit environmental com-\nparison of new bio-based technologies with fossil reference\ntechnologies can be regarded as a feature that has not been\nprovided by other optimization models within the field of\nLCA. The LCC formulation used four our purpose is congru-\nent to the suggestion of (Heijungs et al. 2013), since we only\nfocus on the processes that provide the reference flows. However, it differs in the way that we only need to collect\ndata for costs which are relevant for the decision-maker as a\nbiorefinery investor and neglecting the costs for current tech-\nnologies. This and the consideration of environmental targets\ncan be interpreted as an eco-efficiency approach. However,\ninstead of simply creating the fraction between an economic\nvalue and an environmental value (ISO 2012), the approach in\nthis study allow to assess a specific target achievement, i.e. being less pollutant than current available technologies. magnitude. Back substituting the solution vectors s into Eqs. 9\nand 16, however, identified the results´ reliability in terms of\naccuracy. magnitude. Back substituting the solution vectors s into Eqs. 9\nand 16, however, identified the results´ reliability in terms of\naccuracy. Fig. 5 Pareto front (50%\nscenario). h: environmental\nimpacts; z: profit 4 Discussion LCA has been developed for the assessment of environmental\nimpacts of a product. To broaden the scope of LCA, Udo de\nHaes et al. (2004) propose three general strategies: the use of\nLCA in a toolbox, hybrid analysis, and the extension of LCA. Using LCA in a toolbox, the limitations shall be overcome by\nadditional separate models that are used without a data link. The extension of LCA is considered with one consistent linear\nmodel. Thereby, LCA and the other tool are fully compatible. Hybrid modeling as a mixture of both approaches that com-\nbines LCAwith other models and linking these by data flows. The extension with linear programming (LP) leads to a con-\nsistent linear model that determines the optimal choice among\nothers for the total system under study. Interdependent choices\nin different regions can be studied simultaneously without\nmatrix inversion, since with LP even rectangular systems\ncan be solved. The criteria for choices are represented by the\nobjective function (minimizing impacts on climate change)\nand the additional constraints (e.g., available feedstock in a\nregion). It is shown that the modification of the program by\nadditional profit requirements (LCC) into a goal program and\na Pareto optimization approach also enables to incorporate\nmultiple objectives within the decision-making process. Thereby, regional biomass availability and transport logistic\noptions can be taken into consideration. The benefit of this Dealing with multiple objectives, GP needs an a priori\nweighting of the different goals. In contrast, epsilon constraint\nmethod uses the concept of Pareto efficiency in which the\nsolution is optimal in which an increase of a target would\nresult in a decrease of another target. The advantage of GP is\nthat one optimal solution can be determined whereas in the a\nposteriori method, the decision-maker is encouraged to decide\nfor a compromise. Besides these two methods for multiple\nobjective decision making, there exist further methods, which\ncannot be discussed here. A general overview including the\npros and cons is given by Hwang and Masud (1979). Rectangular LP models for choosing technologies have\nbeen used in input-output economics. Duchin and Levine\n(2011, 2012) introduced a rectangular model to study the Fig. 5 Pareto front (50%\nscenario). h: environmental\nimpacts; z: profit Fig. 5 Pareto front (50%\nscenario). 4 Discussion But instead of sectors as in input-output\nmodels, the columns of the technology matrix in LCA repre-\nsent processes, which are usually modeled in more detail. Furthermore, these processes can be multi-functional (e.g.,\nKätelhön et al. 2016), which is contrary to input-output\nmodeling in which the assumption of homogenous sector out-\nputs is implied by creating the technical coefficient matrix\n(Miller and Blair 2009). In our example, only the biorefinery\nprocesses are multi-functional. Solving the multifunctional\nproblem for these technologies can be interpreted as a system\nexpansion approach. Furthermore, this approach is equivalent\nto substitution method, since the implicit aim of the introduced\nmodels is the substitution of current processes by less pollut-\nant processes. To compare the biorefineries, the systems are\nexpanded by current substitutable processes which produce\nthe same type of products. Due to that a programming prob-\nlem results, in which we seek to find the optimal substitution\nof current processes by a set of biorefinery alternatives. Other\nprocesses in the example are mono-functional. However, if\nother multioutput processes would be considered within the\nupstream processes of the biorefinery or the reference system,\nthe multi-functionality can be solved by the program\ninterpreting it as a kind of surplus method (Heijungs and\nSuh 2002). When minimizing the total environmental impacts\nof the system under study, larger amounts of supplied products\nwould be allowed compared with those in the final demand. It\nis crucial to take this into consideration, since more products\nand hence more functions are possible in the final supply\nvector of the optimized system. On the other hand, if products\nof current technologies are not provided by the new technol-\nogies, the program would ensure that at least the amounts of\nthe current system are generated. In this case a study would\nonly be meaningful, if processes are modeled in a sufficient\ndetailed manner. In other words, it must be decided which Compared to common consequential LCA approaches\nwhich use substitution method to solve the multifunctional\nproblem, the definition of determining products wherein all\nother co-products are summarized into one avoided product\ngroup (e.g., Weidema 2001; Suh et al. 2010) is not necessary. Here, in contrast, all products of the new technologies are\nconsidered as determining products without distinguishing\nbetween the multiple products. For each biorefinery product,\nsubstitutable reference products are determined. 4 Discussion h: environmental\nimpacts; z: profit Int J Life Cycle Assess (2019) 24:2191–2205 2203 Table 5\nResults of the epsilon-constraint method (50% scenario), per annum\nP1\nP2\nP3\nh (kg CO2-eq.)\n7.15709438·1011\n7.15778471·1011\n7.15778469·1011\nz (€)\n2.514068·107\n4.492112·107\n4.509840·107\nBiorefinery 1, Leuna (pc.)\n1\n0\n0\nBiorefinery 2, Leuna (pc.)\n0\n0\n1\nBiorefinery 1, Zeitz (pc.)\n0\n0\n0\nBiorefinery 2, Zeitz (pc.)\n0\n0\n0\nBiorefinery 1, Böhlen (pc.)\n0\n0\n0\nBiorefinery 2, Böhlen (pc.)\n0\n1\n0\nBiorefinery 1, Schwarzheide (pc.)\n0\n0\n0\nBiorefinery 2, Schwarzheide (pc.)\n0\n0\n0 Table 5\nResults of the epsilon-constraint method (50% scenario), per annum process, e.g., a certain ethylene plant, is substitutable by the\nnew process. Even in LCA, which uses more disaggregated\nprocesses than input-output analysis does, this is usually not\nthe case. For instance, in our example only one process for\nethylene generation represents a bundle of ethylene plants. Furthermore, the ethylene production is mono-functional,\nwhich does not represent real-life complexity, since ethylene\nusually is produced with other co-products (e.g., propylene). Including those co-products, however, can lead to binding\nconstraints that are not achievable by the new technologies. In our example this becomes obvious if the bundled process of\nfossil ethylene production would additionally provide propyl-\nene. Since it seems not realistic to achieve a model that repre-\nsents all production processes at a plant level, the only viable\nway is to make a model choice. The question is thus, what\nimplicit assumption is appropriate in terms of the goal of the\nstudy (Zamagni et al. 2012). A general discussion is beyond\nthe scope of this article. However, in terms of the predominant\ngoal of the exemplary case studies, to identity the best alter-\nnative from a set of new biorefinery options, we argue that\nusing aggregated mono-functional processes with average da-\nta seems to be sufficient. To increase the reliability of future\ncase studies, however, the assessments should be enhanced by\nsensitivity analyses using different approaches for allocating\nthe environmental impacts to the substitutable reference prod-\nucts (e.g., partitioning by physical and monetary factors). optimal choice of technologies. Before that, Carter (1970)\napplied a square choice-of-technology model using linear pro-\ngramming in a similar manner. Our model works in principle\nthe same way. References Azapagic A, Clift R (1995) Life cycle assessment and linear program-\nming – optimization of product system. Comput Chem Eng 19:229–\n234 Azapagic A, Clift R (1998) Linear programming as a tool in life cycle\nassessment. 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Delft, CE\nDelft, March 2010 4 Discussion In our opin-\nion, these choices for all products of the biorefineries seem\ncloser to reality than whether clustering several co-products 2204 Int J Life Cycle Assess (2019) 24:2191–2205 Acknowledgements This work has been carried out within the project\nSpitzencluster BioEconomy (031A078A). Acknowledgements This work has been carried out within the project\nSpitzencluster BioEconomy (031A078A). into one avoided product group or declaring the multi-\nfunctional processes to be mono-functional by using\npartitioning method. Funding information The German Federal Ministry of Education and\nResearch financially supported this project. Towards a comprehensive LCSA framework, some\npotential directions for further research shall be\nbroached. The characterization matrix Q can be easily\nmodified by corresponding characterization factors to\ntake into account additional environmental impact cate-\ngories (e.g., midpoint or endpoint categories). Targets for\nvarious impact categories can be considered separately\nwithin a multi-objective framework. On the other hand,\nnormalization and weighting could be introduced within\na single environmental target function. Thus the optimi-\nzation could be carried out in terms of a single environ-\nmental score which takes into account various normal-\nized and weighted impact indicator results. In the same\nmanner various social categories could be dealt with. By\nimposing a single score for each of the three pillars of\nsustainability (LCA, LCC, social LCA), the assessment\nof technological alternatives would be extended toward a\ncomprehensive LCSA within a multi-objective frame-\nwork. 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