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Allocation Games: Addressing the Ill-Posed Nature of Allocation in Life-Cycle Inventories Rebecca J. Hanes,† Nathan B. Cruze,‡ Prem K. Goel,§ and Bhavik R. Bakshi*,† †

William G. Lowrie Department of Chemical and Biomolecular Engineering, and §Department of Statistics, The Ohio State University, Columbus, Ohio 43210, United States ‡ United States Department of Agriculture (USDA) National Agricultural Statistics Service (NASS), Washington, D.C. 20004, United States S Supporting Information *

ABSTRACT: Allocation is required when a life cycle contains multi-functional processes. One approach to allocation is to partition the embodied resources in proportion to a criterion, such as product mass or cost. Many practitioners apply multiple partitioning criteria to avoid choosing one arbitrarily. However, life cycle results from different allocation methods frequently contradict each other, making it difficult or impossible for the practitioner to draw any meaningful conclusions from the study. Using the matrix notation for life-cycle inventory data, we show that an inventory that requires allocation leads to an ill-posed problem: an inventory based on allocation is one of an infinite number of inventories that are highly dependent upon allocation methods. This insight is applied to comparative life-cycle assessment (LCA), in which products with the same function but different life cycles are compared. Recently, there have been several studies that applied multiple allocation methods and found that different products were preferred under different methods. We develop the Comprehensive Allocation Investigation Strategy (CAIS) to examine any given inventory under all possible allocation decisions, enabling us to detect comparisons that are not robust to allocation, even when the comparison appears robust under conventional partitioning methods. While CAIS does not solve the ill-posed problem, it provides a systematic way to parametrize and examine the effects of partitioning allocation. The practical usefulness of this approach is demonstrated with two case studies. The first compares ethanol produced from corn stover hydrolysis, corn stover gasification, and corn grain fermentation. This comparison was not robust to allocation. The second case study compares 1,3-propanediol (PDO) produced from fossil fuels and from biomass, which was found to be a robust comparison.

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INTRODUCTION Life-cycle assessment (LCA) aims to account for all material and energy inputs required to produce a functional unit, from resource extraction through the use phase to end-of-life processing.1,2 Frequently, the processes that make up the life cycle are multi-functional and provide more than one useful product or service. Multi-functional processes provide two types of outputs: primary products that are consumed within the life cycle and byproducts that are consumed outside the life cycle. Because a life cycle inventory should include only the materials and energy embodied in the functional unit, the inputs and environmental impacts embodied in the byproducts must be removed from the inventory. Two popular methods for removing this excess are system expansion and partitioning allocation.3,4 System expansion or displacement entails gathering inventory data on processes external to the original life cycle that produce primary products that are functionally equivalent to the original byproducts.5 The inventory for the external, avoided processes is removed from the original inventory, leaving only the material and energy inputs embodied in the © 2015 American Chemical Society

functional unit. System expansion has several downsides related to choosing external processes. Gathering inventory data for the external processes adds time and effort to life-cycle studies. More problematically, external processes that produce the correct primary products may themselves be multi-functional, requiring more and more external processes to account for the additional byproducts. This is known as the endless regression problem.6,7 In partitioning allocation, multi-functional processes are divided into several mono-functional sub-processes, each of which provide a single product.8 The sub-processes that produce byproducts are removed from the inventory, leaving only the part of the original multi-functional process that provides the primary product(s). Division into sub-processes is commonly done according to the relative amounts of each product measured in units, such as mass, energetic content, or Received: Revised: Accepted: Published: 7996

May 23, 2014 May 20, 2015 June 10, 2015 June 10, 2015 DOI: 10.1021/acs.est.5b01192 Environ. Sci. Technol. 2015, 49, 7996−8003

Article

Environmental Science & Technology monetary value.9 The choice of units is called the partitioning criterion. Inputs and impacts are occasionally allocated separately, according to several different partitioning criteria. This type of allocation is less common in practice than is singlecriterion allocation. Which partitioning criterion to use is a subject of debate among LCA practitioners.10,11 There is no consensus on which criterion is best: different organizations and practitioners prefer or recommend different partitioning criteria.12,13 Moreover, it is difficult to defend any single criterion, because no one criterion provides logical partitioning for all products. Mass allocation works well in manufacturing processes but cannot be applied to energetic outputs, such as electricity; energetic allocation is not intuitive for products not commonly quantified according to energy content; and economic allocation is problematic for products that are typically treated as waste and products with volatile prices. For many inventories, allocation has such a drastic effect that it is possible to “game” life-cycle impacts by assigning allocation weights that correspond with the objectives of a particular study. For instance, cellulosic ethanol produced from corn stover can appear either superior or inferior to corn ethanol, depending upon whether or not stover is treated as a waste. Applying multiple partitioning criteria and performing impact assessment on multiple allocated inventories has recently become popular as a way to avoid making and justifying a particular allocation decision.14,15 However, differences in the allocated inventories can lead to contradictory conclusions, particularly for comparative life-cycle studies that examine several functionally equivalent products with the goal of determining the preferred or lowest impact product. Many case studies have shown that the preferred product is not consistent across different partitioning criteria.16−22 In such cases, the comparisons are not robust to allocation; thus, no definite conclusions can be drawn from the case study. Recent work has used linear model theory to gain insight into the issues surrounding partitioning allocation.23,24 Using the matrix notation for life-cycle inventory data,25 inventory analysis for a multi-functional inventory is seen as an illposed problem with infinite solutions that are highly sensitive to the applied allocation method. The implication for LCA is that any allocated inventory is only one of infinitely many possible inventories, and inventories obtained from different partitioning criteria vary widely. Because allocation assumptions made in the inventory phase are propagated through to the impact assessment and interpretation phases, any decisions made based on allocated inventories are controlled by the allocation method rather than differences in the inventory data. If only partitioning allocation is applied in a comparative study, there is no way to determine conclusively whether or not results of the study are robust to allocation. When multiple partitioning criteria are applied, the end result is a finite subset of infinitely many possible inventories. Conclusions based on partitioning, therefore, ignore a large amount of information and may be incorrect; this is true for not only comparative studies but also LCA studies in general. In particular, conclusions reached via partitioning allocation may be incorrect because of non-robustness not detectable by conventional allocation and sensitivity analyses. We propose a general and systematic calculation procedure, the Comprehensive Allocation Investigation Strategy (CAIS), for parametrizing allocation within a multi-functional inventory and performing inventory analysis without restricting allocation decisions to individual partitioning criteria. A key advantage of CAIS is the ability to

represent all allocation decisions simultaneously, which allows for the effects of allocation to be examined in detail as well as visualized. Unlike partitioning allocation, applying CAIS can identify all non-robustness to allocation, even that which is undetectable by standard sensitivity analyses. CAIS does not solve the ill-posed allocation problem but rather provides a method for extracting useful information from multi-functional inventories that can then be used in subsequent LCA phases and eventually in decision support. By applying CAIS during inventory analysis, the impact assessment and interpretation phases will be based on complete information rather than the partial set of information obtainable via partitioning allocation.

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MOTIVATING EXAMPLE Ethanol is a transportation fuel that can be produced from a variety of feedstocks and conversion pathways. Allocation is required in virtually all ethanol production pathways, and previous studies have shown that ethanol inventories are sensitive to allocation.27,28 Consider three ethanol production pathways: corn grain fermentation, corn stover hydrolysis, and corn stover gasification. It is of interest to decide which of these pathways should be implemented on the basis of which pathway has the lowest life-cycle CO2 emissions. Inventory data for ethanol production is available in the literature; the data used in this study was obtained from refs 29−31, and the inventories are given in the Supporting Information. Allocation is required in all three pathways at the farming process and in the grain fermentation and stover hydrolysis pathways at the ethanol production process. Both mass and economic allocation are applied. Results given in Table 1 indicate that different Table 1. Life-Cycle CO2 Emissions for Three Ethanol Production Pathways under Mass and Economic Allocationsa pathway corn grain fermentation corn stover hydrolysis corn stover gasification a

mass allocation (kg of CO2 emissions/L of ethanol)

economic allocation (kg of CO2 emissions/L of ethanol)

5.46

7.88

1.77

0.16

3.14

0.14

A bold entry indicates the preferred pathway.

pathways are preferred under the two allocation methods. Therefore, the comparison is not robust, and using only partitioning allocation, we can only conclude that grain ethanol should probably not be implemented. Moreover, on the basis of only mass or only economic allocation, it is not apparent that the comparison is not robust. The CAIS provides complete information on how the inventories are affected by allocation decisions. Figure 1 compares results from CAIS (vertical lines) and partitioning allocation (points). Each line represents the range of values that can be assigned to CO2 emissions by changing the allocation method. Note the large overlap between the lines for the three fuel life cycles; within this region of overlap, different partitioning allocation methods will prefer different processes. For example, the CO2 emissions of the two corn stover pathways fall within this region: stover hydrolysis is preferred under mass allocation (dots), while stover gasification is preferred under economic allocation (triangles). From the overlapping CAIS results, we can easily determine that the 7997

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Here, s represents the scale at which each process must operate to deliver the functional unit. The intervention data in B is scaled by s to calculate life-cycle impacts for the functional unit in the form of a r × 1 impact vector g. Assuming that the life cycle contains only mono-functional processes such that p = n and A is a square matrix and is nonsingular,33 the calculation of g is trivial.8 g = BA−1f

(2) −1

When A is rectangular, A does not exist and eq 2 cannot be applied immediately. Partitioning allocation enables the calculation of one particular value of A−1 by expanding A to a square p × p matrix, A*, as follows: A

A

⎡ a11 − a12 − a13 ⎤ ⎢− a ⎥ ⎢ 21 a 22 − a 23 ⎥ partitioning allocation ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎢ 0 a32 0 ⎥ ⎢ ⎥ ⎢⎣ − a41 − a42 a43 ⎥⎦

Figure 1. Life-cycle CO2 emissions for three ethanol production pathways under mass allocation, economic allocation and CAIS. Results from CAIS are shown as bars that indicate the variation in CO2 emissions as a result of allocation decisions in each pathway; partitioning allocation results are shown as points.

* ⎡ a − w a − (1 1 12 ⎢ 11 ⎢− a 21 a 22 ⎢ 0 ⎢ 0 ⎢ ⎣ − a41 − w2a42 − (1

− w1)a12 − a13 ⎤ ⎥ 0 − a 23 ⎥ ⎥ a32 0 ⎥ ⎥ − w2)a42 a43 ⎦

(3)

The expansion separates the multi-functional process into two artificial mono-functional sub-processes.5,9 Inputs and intervention data for the original process are divided among the subprocesses with allocation weights w, and each of the subprocesses is scaled independently. In eq 3, w1 and w2 may be calculated from a common partitioning criterion (w1 = w2) or calculated from different criteria (w1 ≠ w2). B is similarly expanded to a r × p matrix B*, and s expands to a p × 1 vector s* with the addition of scaling factors for each allocated subprocess. The impact vector for the allocated system, g*, is calculated as above.

comparison is not robust to allocation. We still cannot decide which pathway should be implemented, but we may be able to make a decision when we examine results of CAIS in more detail.

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ALLOCATION AS AN ILL-POSED PROBLEM Partitioning Allocation in Matrix Notation. A brief discussion of the matrix notation for life-cycle inventory data is given here,8,25 and an illustrative example is given in the Supporting Information. We assume throughout the following discussion that any rectangular technology matrix is rectangular because of the presence of multi-functional processes; therefore, the inventory contains more products than processes. We further assume, for simplicity, that each multi-functional process provides exactly one primary product and one byproduct and that no product is produced by multiple processes. While CAIS and associated concepts apply in situations where these assumptions do not hold true, modifications are necessary to the calculation procedure to accommodate differences in the way allocation decisions are modeled. Only the situation that corresponds to these simplifying assumptions is discussed in detail. The exchange of p distinct products among the n processes in a life cycle is represented in the p × n technology matrix A. Each column in A represents a single process. Negative entries indicate inputs, and positive entries indicate outputs. The data in A is in mixed physical units normalized to some unit of time chosen such that all processes can be assumed to operate at steady state.32 Environmental intervention data are organized in the r × n intervention matrix B. Each column in B contains intervention data for a single process. Interventions may include emissions produced, natural resources consumed, and inputs that come from outside the system boundary. Data in the B matrix are normalized to the level of output of each process given in A. Finally, the functional unit is represented as a p × 1 vector f, in which each element corresponds to one of the p products. A and f are used to calculate a n × 1 vector of process scaling factors s such that the following equation holds: As = f

A s =f **

(4) −1

g =B A f (5) * * * Other Methods for Multi-functional Inventory Analysis. Emergy is defined as the total solar energy consumed directly or indirectly to provide a product or service.34 Unlike energy and mass, emergy is not necessarily conserved,35 implying that, when emergetic content is used as a partitioning criterion, energy and mass are not necessarily conserved within a life cycle. Emergetic allocation is thus fundamentally different from partitioning allocation, which preserves energy and mass balances. However, emergetic and partitioning allocation are implemented similarly within the matrix notation. A

A

⎡ a11 − a12 − a13 ⎤ ⎢− a a 22 − a 23 ⎥ emergetic allocation ⎥ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎢ 21 ⎢− a31 a32 − a33 ⎥ ⎥ ⎢ ⎣ − a41 − a42 a43 ⎦

* ⎡ a11 − a12 − a12 − a13 ⎤ ⎢ ⎥ 0 − a 23 ⎥ ⎢− a 21 a 22 ⎢−a 0 a32 − a33 ⎥ ⎢ 31 ⎥ ⎢⎣ − a41 − a42 − a42 a43 ⎥⎦ (6)

Moreover, emergetic and partitioning allocation yield identical results for specific allocation weight values. A demonstration is given in the Supporting Information. Emergetic allocation can thus be viewed as a special case of partitioning allocation, despite the methodological differences between the two methods. System expansion is another way to calculate g* for a multifunctional life cycle. In system expansion, the life cycle is expanded to include external processes with primary products

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One non-partitioning attempt at solving the ill-posed allocation problem used ordinary, total, and data least-squares methods36,37 to find minimal corrections to A and f, such that eq 1 has a solution. This allows g to be calculated for multifunctional inventories without performing partitioning allocation. However, corrections made to the inventory data A may not be consistent with the physical system being represented: small flows can reverse direction, zero flows can become nonzero, and vice versa. There is no way to guarantee that the corrections made to A and f are consistent with system-wide mass and energy balances, and thus, no guarantee that g and further results based on g calculated from these techniques are valid.23,24,33 If least-squares methods are to yield consistent, meaningful results, further constraints must be imposed on the methods, including but not limited to mass and energy balances and constraints that prevent process data from becoming unrealistic.

that are functionally equivalent to the byproducts of the original life cycle.3,5 To exclude process inputs and interventions that are embodied in unused byproducts, these external processes are represented in A* as running backward: outputs of the external processes are negative, and inputs are positive. This sign reversal is interpreted as avoided production of the byproduct. When the original multi-functional process and the new external processes are combined by adding the corresponding columns of A*, the result is an artificial monofunctional process with only inputs and impacts associated with the primary product. Cherubini et al.26 showed that, when inputs and impacts are allocated separately, allocation weights can be derived from external process data used in system expansion. In this case, system expansion and partitioning allocation provide the same results. System expansion, like emergetic allocation, can thus be viewed as a special case of partitioning allocation. Further details are given in the Supporting Information. Insights from the Linear Model Theory. Equation 1 is a system of linear equations. When A is square, the system has one unique solution, which is used to calculate g. When p > n, eq 1 becomes an overdetermined system of equations ⎡f⎤ ⎡ a11 −a12 −a13 ⎤ ⎢ 1⎥ s ⎡ 1⎤ ⎢ f ⎥ ⎢− a ⎥ −a 23 ⎢ ⎥ a 2 ⎢ 21 22 ⎥ s 2 = ⎢ ⎥ ⎢ 0 a32 0 ⎥⎢ ⎥ ⎢ f ⎥ ⎢ ⎥⎢⎣ s3 ⎥⎦ ⎢ 3 ⎥ ⎣ −a41 −a42 a43 ⎦ ⎢⎣ f4 ⎥⎦

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METHODOLOGY: COMPREHENSIVE ALLOCATION INVESTIGATION STRATEGY CAIS is applicable to life-cycle inventories that require allocation in an arbitrary number of multi-functional processes. Flows in the inventory that must be allocated are written as functions of allocation weights w. Thus, the inventory itself, specifically the matrices A and B, become functions of w. This allows the impact vector g* to be calculated, also as a function of w. This section provides a brief overview of the calculations involved in CAIS. The first step in CAIS is locating multi-functional processes. For small inventories, this can be done by inspection, and for large inventories, the process is easily automated. The technology matrix A is decomposed into a make matrix V′ and a use matrix U,38 as follows:

(7)

and, in general, has no exact solution. Using partitioning allocation to expand A to A* allows g* to be calculated; however, the system of equations now involves the allocation weights w, which are themselves independent variables. The system is now underdetermined with four equations and either five or six unknowns, as follows: ⎡ a11 −w1a12 −(1 ⎢ ⎢−a 21 a 22 ⎢ 0 ⎢ 0 ⎢ ⎣ −a41 −w2a42 −(1

− w1)a12 −a13 ⎤ ⎥ −a 23 ⎥ 0 ⎥ a32 0 ⎥ ⎥ − w2)a42 a43 ⎦

⎧ aij if aij > 0 vji = ⎨ ⎩ 0 otherwise

(9)

⎧|aij| if aij < 0 uij = ⎨ ⎩ 0 otherwise

(10)

⎪

⎪

⎡f⎤ ⎡ s1 ⎤ ⎢ 1 ⎥ ⎢s ⎥ ⎢ f ⎥ ⎢ 2a ⎥ = ⎢ 2 ⎥ ⎢ s 2b ⎥ ⎢ f ⎥ ⎢ s ⎥ ⎢ 3⎥ ⎣ 3⎦ ⎢ ⎥ ⎣ f4 ⎦

⎪

⎪

V′ contains only process outputs, while U contains only process inputs such that eq 11 holds. A = V′ − U

(8)

(11)

Multi-functional processes correspond to columns in V′ with more than one non-zero element. Next, the multi-functional processes must be divided into mono-functional processes and inputs to the mono-functional processes written as functions of w. V′ is expanded by moving off-diagonal elements to the diagonal and adding columns where necessary, resulting in the diagonal matrix V*′. U is expanded by adding columns where multi-functional processes are being divided; elements in these new columns and in the original columns corresponding to multi-functional processes are written as functions of w, resulting in the square matrix U*(w). B is expanded using the procedure for U. Finally, V*′ and U*(w) are recombined using eq 11 to obtain A*(w). Equation 2 is now applied to obtain g*(w), a vector of functions that give the magnitude of each intervention of interest r. A diagram of the procedure is shown in Figure 2. Further details on the calculations and an illustrative example are provided in the Supporting Information.

The underdetermined nature of the system leads to the insight that allocation is a fundamentally ill-posed problem with the following characteristics: there are infinitely many values of s* and w that satisfy eq 8, and s* and g* are highly sensitive to allocation choices represented by the value of w.33 An ill-posed problem can be turned into a well-posed problem by incorporating data external to the original problem. A well-posed problem, if it has a solution, has one unique solution. In the current context, equations that impose further restrictions on s* and w must be added to eq 8. Partitioning allocation appears to solve the ill-posed problem using product mass, energetic content, or other information to fix the value of w. This does not provide an adequate solution, because the information used to fix w is not external to the life-cycle inventory: all commonly used partitioning criteria are based on process output data, which is already in the inventory, plus some conversion factor to change the units. 7999

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Figure 2. Diagram of the CAIS procedure.

Figure 3. N2O preference boundaries and regions for the ethanol production pathways.

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REPRESENTING, INTERPRETING, AND APPLYING RESULTS After obtaining the impact function vector g*(w) for each product, the information in g*(w) must be interpreted correctly and compared between products. The focus of this section is to determine if a given product comparison is robust to allocation and, if not, to determine the extent of non-robustness. The most direct way to determine robustness is to compare the rth impact function for each product and determine if equality sets, sets of points in the allocation space (0 ≤ w ≤ 1) for which two or more impact functions are equal, exist for any of the impacts of interest. A equality set for products A and B and impact r is defined as the set {we} for which

preference boundaries that exist between the three ethanol production pathways for N2O emissions. Equation 12 was applied to locate the equality sets. As seen in Figure 3, the product preference on either side of the two equality sets is different; thus, both equality sets are also preference boundaries. Further details on locating equality sets and identifying preference boundaries using eqs 12 and 13 are given in the Supporting Information. For product comparisons involving two or fewer allocation weights, such as the comparison between ethanol pathways discussed in the motivating example, preference regions and boundaries can be represented graphically. Figure 4 shows two such preference plots for CO2 emissions (Figure 4a) and CO emissions (Figure 4b). Different preference regions are indicated by color, and preference boundaries divide one region from another. In Figure 4, wF and wP indicate the allocation weight for the corn and stover farming process and the ethanol production process, respectively. wF is the fraction of farming inputs and impacts allocated to the primary product, which is corn grain for the grain fermentation pathway and corn stover for the other two pathways. wP is the fraction of ethanol production inputs and impacts allocated to ethanol; thus, 1 − wP is the fraction assigned to the co-product, which is dried distiller’s grains and solubles for grain fermentation and lignin electricity for stover hydrolysis. Because stover gasification does not involve allocation at the ethanol production process, wP does not exist for that pathway. The impact functions for stover gasification, therefore, do not depend upon wP. Figure 4 also shows the impact magnitude of the preferred product at each point, demonstrating that both the preferred product and the impacts of the preferred product are sensitive to allocation. For CO2 emissions shown in Figure 4a, stover hydrolysis is the preferred product over the majority of the allocation space. Stover gasification is preferred only when wF < 0.1 and wP > 0.8, and grain fermentation is preferred only when wP < 0.1. In contrast, grain fermentation and stover hydrolysis are preferred almost equally for CO emissions shown in Figure 4b, with stover gasification being preferred only when wF < 0.4 and wP > 0.8. Using CAIS Results for Decision Support. In practice, impact assessment would be performed on the impact functions prior to using them for decision support. Here, we use the impact functions as-is for illustrative purposes.

g Ar (we) = g Br (we), ∀ we ∈ {we} (12) * * The existence of an equality set for two or more products implies that the impact functions for those products overlap, as seen in Figure 1 for the three ethanol pathways. If no equality sets exist, then the product comparison is robust. If equality sets exist among the products and impacts of interest, then the comparison is potentially non-robust. The equality set must be tested to determine if it is also a preference boundary or the equality set that also divides a region of the allocation space in which one product is preferred (“preference region”) from an adjacent region in which another product is preferred. The preference boundary for products A and B and impact r is defined as the equality set {wp} for which the following condition holds: ∀ wp ∈ {wp},

∃ Δwp

such that

min{g Ar (wp + Δwp), g Br (wp + Δwp), ...} = g Ar (wp + Δwp) * * * and min{g Ar (wp − Δwp), g Br (wp − Δwp), ...} = g Br (wp − Δwp) * * * (13)

Equation 13 imposes the requirement that the product preference, the product with minimum impact r, among all products being compared, changes across the preference boundary. This requirement is necessary because, when more than two products are being compared, it is possible to have an intersection of two impact functions where neither product is preferred on either side of the intersection. Figure 3 shows two 8000

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Figure 4. Preference plots show the different preference regions of the allocation space. For comparisons involving up to two allocation weights, preference plots can be used to locate preference boundaries instead of or in addition to eq 13.

Table 2. Preference Percentages by Emission for the Ethanol Production Pathways pathway grain fermentation stover hydrolysis stover gasification pathway grain fermentation stover hydrolysis stover gasification

kg of CO2 9.1 89.2 1.7 kg of SO2 17.3 81.8 0.8

kg of CO

kg of CH4

50.4 44.6 5.0 kg of coal 83.5 13.2 3.3

kg of NOx

kg of N2O

9.1 15.7 75.2 kg of crude oil

50.4 48.8 0.8 kg of natural gas

9.1 47.9 43.0 kg of water

9.1 66.1 24.8

91.7 5.0 3.3

91.7 2.5 5.8

Table 3. Preference Percentages by Emission for Fossil- and Biomass-Based PDO process

g of CO

g of CO2

fossil-based biomass-based process

0 100 g of N2O

0 100 CH4 (g of CO2 equiv)

fossil-based biomass-based process

20.0 80.0

0 100

fossil-based biomass-based

MJ of non-renewable energy

g of NOx

0 100

5.5 94.5 g of SOx

g of PM10 0 100

CFC (g of CO2 equiv)

g of lead

0 100 g of VOC

0 100

0 100

0 100

of the preference region of product j expressed as a percentage of the total allocation space. Preference percentages can also be calculated by defining a grid of points in the allocation space and finding the preferred product at each point and for each impact r. Then, the number of points for which product j is preferred, PPj, divided by the total number of points, N, is approximately the preference percentage for that product and impact.

In Figure 4a, stover hydrolysis is preferred over the majority of the allocation space. It is, therefore, not unreasonable to conclude that, when considering CO2 emissions, stover hydrolysis is the preferred pathway. When there is no one pathway preferred over the majority of the allocation space, as for the CO emissions shown in Figure 4b, the preference is less obvious. More quantitative information is required to determine a preference, and even then, it may not be possible to reach a definite conclusion. The size of each preference region expressed as a fraction of the allocation space can be used to determine if one product is preferred over the majority of the space or if products are preferred more equally. Comparing an arbitrary number of products yields a set of M preference boundaries {wp}1, ..., {wp}M and some number of preference regions for an impact r. The rth preference percentage for product j is equal to the size

1 PPj ≈ N

N

∑ Hj , j=1

⎧ ⎪ 1 product j is preferred Hj = ⎨ ⎪ otherwise ⎩0

(14)

Preference percentages estimated from eq 14 are given for the ethanol pathway comparison in Table 2. Stover gasification is preferred only in small fractions of the space, except for CH4 and N2O emissions; grain fermentation performs slightly 8001

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external to the original life cycle. If the endless regression problem can be resolved, this may be the most straightforward way to find a valid solution to the allocation problem.

better; and stover hydrolysis appears to have the overall best performance. Case Study: Production of 1,3-Propanediol (PDO). Urban and Bakshi performed a cradle-to-gate comparative LCA of PDO produced from fossil- and biomass-based feedstocks.39 The fossil-based pathway involves the reaction of syngas and ethylene oxide over a silver catalyst to produce PDO; the fossilbased PDO inventory does not require allocation. The biomassbased pathway uses corn as a feedstock, which is milled to produce starch and starch byproducts. The starch, which contains glucose, is fermented to produce PDO. The corn wet milling process is multi-functional, providing both starch, the primary product, and valuable starch byproducts. After applying the CAIS procedure, it was found that, regardless of allocation, biomass-based PDO was preferred over fossil-based PDO for 9 of 11 impacts considered. The comparison was not robust for N2O and NOx emissions, most likely because of the farming process in the biomass-based PDO life cycle. For these two emissions, fossil-based PDO was preferred only for relatively high values of w: w > 0.8, and w > 0.945, respectively. Table 3 contains the preference percentages for the 11 emissions considered. Because biomass-based PDO is preferred exclusively for the majority of impacts considered and because it is preferred in a small fraction of the allocation space even for N2O and NOx, it is reasonable to conclude that it is the preferred pathway. Inventory data and further results are given in the Supporting Information.

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ASSOCIATED CONTENT

* Supporting Information S

Further information on the CAIS calculation procedure, lifecycle inventory data for the ethanol production pathway comparison and for the PDO case study, and further CAIS results for both comparisons (PDF). The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.est.5b01192.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: 1-614-292-4904. Fax: 1-614-292-3769. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Partial funding for this work was provided by the National Science Foundation (CBET-0829026) and the United States Department of Agriculture (2012-38202-19288).

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EXTENSIONS AND FUTURE WORK While the preference plots of Figures 3 and 4 provided convenient visualizations for the ethanol production pathway comparison, such visualizations cannot easily be created for inventories containing more than two multi-functional processes because of the higher dimensional plots required. However, the CAIS procedure and analysis techniques given by eqs 13 and 14 can be applied to inventories with an arbitrary number of multi-functional processes; it is not necessary to visualize the results to obtain useful insights from CAIS. Another issue to be addressed is how to compare life cycles that have few or no multi-functional processes in common. This situation can be addressed by treating impact functions as constants over any allocation weights that do not appear in the function, as was done for corn stover gasification and the ethanol process allocation weight. Results of CAIS can thereby be analyzed and interpreted just as they would be for life cycles that share a common allocation space. CAIS systematically parametrizes an allocated inventory and provides extensive information on the effects of allocation on the inventory, but it does not solve the ill-posed allocation problem. A complete, mathematically valid solution to allocation requires the addition of data external to the original inventories. The external data is used to impose a new relationship or relationships between the variables s and w, creating additional equations in eq 1. Ideally, after incorporating external data, the problem will no longer be underdetermined and will have one unique solution. Partitioning allocation relies on data from the original inventories and, therefore, does not adequately solve the problem. Work is ongoing on what type of external data are best and how to guarantee that the problem will have a solution. One possibility is to use the method by Cherubini et al.26 and derive partitioning weights from system expansion data, which is 8002

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DOI: 10.1021/acs.est.5b01192 Environ. Sci. Technol. 2015, 49, 7996−8003