Using the Reliability Theory for Assessing the Decision Confidence

Article pubs.acs.org/est

Using the Reliability Theory for Assessing the Decision Confidence Probability for Comparative Life Cycle Assessments Wei Wei,† Pyrène Larrey-Lassalle,‡ Thierry Faure,† Nicolas Dumoulin,† Philippe Roux,‡ and Jean-Denis Mathias*,† †

Irstea, UR LISC, 9 Avenue Blaise Pascal, F-63178 Aubière, France Irstea, UMR-Itap, 361 rue Jean-Francois Breton, F-34196 Montpellier, France

‡

S Supporting Information *

ABSTRACT: Comparative decision making process is widely used to identify which option (system, product, service, etc.) has smaller environmental footprints and for providing recommendations that help stakeholders take future decisions. However, the uncertainty problem complicates the comparison and the decision making. Probability-based decision support in LCA is a way to help stakeholders in their decision-making process. It calculates the decision confidence probability which expresses the probability of a option to have a smaller environmental impact than the one of another option. Here we apply the reliability theory to approximate the decision confidence probability. We compare the traditional Monte Carlo method with a reliability method called FORM method. The Monte Carlo method needs high computational time to calculate the decision confidence probability. The FORM method enables us to approximate the decision confidence probability with fewer simulations than the Monte Carlo method by approximating the response surface. Moreover, the FORM method calculates the associated importance factors that correspond to a sensitivity analysis in relation to the probability. The importance factors allow stakeholders to determine which factors influence their decision. Our results clearly show that the reliability method provides additional useful information to stakeholders as well as it reduces the computational time.

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• LCA modeling is based on assumptions (for example about the system definition: system boundaries, consequential vs attributional LCA, time horizon, etc.) but also on the modeling choice (what systems are modeled). Besides, errors arise with the choice of the characterization method. • Uncertainties due to parameters propagation through LCA calculation yielding uncertainties in results. • Life cycle impact assessment (LCIA) model uncertainty: at midpoint level, about 10 to 18 impacts models can be used and then combined at end point level thus adding an additional level of uncertainty. Each of these models may be subjected to an assessment of the uncertainty such as done, for instance, in the IPCC report,4 which assesses the uncertainty of Radiative Forcing models commonly used in LCIA, or in Rosenbaum et al.5 for the uncertainty of toxicity models. However, to the knowledge of the authors, there is no study about an operational synthesis of all uncertainties associated with all environmental models used in LCIA.

INTRODUCTION Life cycle assessment (LCA) has been widely applied as a decision support tool to identify the important environmental factors in product systems. The purpose is to evaluate and compare the environmental performance of various products and systems (for example biofuels and biomaterials compared to petroleum chemistry products). So LCA consists in modeling complex systems that usually include a large number of emissions to biosphere compartments and several natural resources consumptions during all life cycle stages within the technosphere. This implies a lot of assumptions and a large amount of associated uncertainties. These uncertainties have been described extensively, e.g. by Reap et al.1 and Williams et al.2 Generally two types of uncertainties are considered: (i) epistemic uncertainty relates to an incomplete state of knowledge;3 (ii) stochastic uncertainty originates from the inherent variability of the natural system. Uncertainties arise in LCA in many different ways: • Life cycle inventory (LCI) input data used in LCA contain uncertainties due to inaccurate measurements, but also due to the variability of the considered processes (for example, for the industrial process of steal production, there are spatial and temporal variability). LCI uncertainties can also be associated with emissions model choices. © XXXX American Chemical Society

Received: July 30, 2015 Revised: January 19, 2016 Accepted: January 27, 2016

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DOI: 10.1021/acs.est.5b03683 Environ. Sci. Technol. XXXX, XXX, XXX−XXX

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Environmental Science & Technology • Stakeholders interpret LCIA with uncertainties in different ways depending on their knowledge and on how the results are shown (i.e. diagrams with error bars are difficult to interpret, etc.). In the example application of this article, we do not consider the uncertainties associated with LCA modeling and characterization method choice as well as LCIA model uncertainty. But the methodology we developed is general enough to be applied. Generally, the uncertainty assessment of the foreground system is based on a statistical analysis of the first hand available data (from measurements or models of the studied activity). On the contrary, the uncertainty coming from the huge number of background activities used from databases cannot be assessed conventionally and various methods were developed to quantify these uncertainties. For example, for the mainly used ecoinvent database, the pedigree method is used to handle uncertainties.6 This method transforms qualitative data (expert knowledge, data quality indicators, etc.) to quantitative uncertainties. Then these uncertainties are handled with statistical methods (Monte Carlo, adapted sampling methods (Latin hypercube sampling, quasi Monte Carlo sampling), interval calculation, fuzzy logic, Gaussian error propagation, Taylor expansion, etc.). Readers can find more details in Groen et al.,7 and Heijungs et al.8 for explanation and comparison of various methods. Monte Carlo (MC)-based approaches are classical methods currently implemented in LCA software (Simapro, OpenLCA, etc.). One of the main drawback of MC-based methods is the number of sampling needed to explore the parameters space. Typically, in LCA analysis of a product system, the number of uncertain parameters is between hundreds and thousands: as a consequence, the MC analysis will needs a huge number of runs to estimate impacts with an acceptable level of error. Comparative decision making process identifies which product has smaller environmental footprints and provides recommendations that help stakeholders to take future decisions. However, the comparison is difficult in the presence of uncertainty. To take into account the impact of uncertainties in decision-making process, several decision support methods have been proposed to evaluate the differences of options increasingly over the past decade. Discernibility analysis aims to test if option A is statistically discernible from option B.9 PradoLopez et al.10 proposed a trade-off identification approach that measures the overlap area between probability distribution and provides a way to evaluate overall trade-offs significance when faced with more than two options. Henriksson et al. used the statistical testing with dependent sampling done in comparative carbon footprints for comparative decision.11 Further, sensitivity analysis (SA) is used to study the robustness of results and their sensitivity to uncertainty factors, and has been intensively studied in LCA during recent years.12 It provides the indicators which allow the LCA practitioners and stakeholders to highlight the most important set of model parameters and thus to determine whether data quality needs to be improved, and to enhance interpretation of results. Up to now, almost all of the available SA were done on the results of impact assessment.9,13,14 Such indicators based on impact assessment are very useful during an ecodesign process but they are not directly in relation with decision making. This raises the issue of decision and of the associated indicators needed by stakeholders. However, SA on the decision has not been applied in LCA yet.

Reliability theory aims to describe the satisfactory probability of a system and has been successfully applied in engineering science for three decades.15 It focuses on assessing the probability of a system to resist to constraints replacing the conventional determinist approach by a reliability perspective (ex. probability of a structure to resist to an earthquake: that is, probability that the uncertain loads are smaller than the uncertain resistance). In our LCA context, reliability methods would approximate the decision confidence probability which is the probability of a product to have a smaller environmental footprint than the one of another product.16 Therefore, the probability support method can be based on reliability methods. Moreover, reliability methods provide the associated importance factors which are the results of a sensitivity analysis performed in relation to the decision. The goal of the present study is to apply the reliability framework for LCA comparative decision and to provide importance factors about decision. First, we describe the different levels of decision support methods and associated sensitivity analysis. Conventional deterministic LCA, stochastic LCA and probability support method are presented. Second, we introduce the reliability theory in LCA. Finally, we compare the Monte Carlo method with the first-order reliability method (FORM). Both methods give us an approximation of the decision confidence probability. Moreover, the FORM method also gives this probability with fewer simulations than the Monte Carlo method. Indeed, assessing uncertainties consists in scanning the uncertainty space by calculating the LCIA results for different sets of randomized LCI parameters. The Monte Carlo method scans the entire uncertainty space for all stochastic variables of the LCI (by randomizing all parameters on all their respective range of uncertainty). Doing such a computation with an acceptable level of confidence needs thousands of runs (commonly up to tens of thousands for LCAs). The reliability approaches (such as FORM) only scan the uncertainty space in the neighboring of the design point and the response surface. That is the reason why the number of runs is really lower (for the same level of confidence) compared to Monte Carlo method and this is the case especially for highdimensional problems such as LCA (hundreds/thousands of LCI parameters). Moreover, the FORM method calculates the associated importance factors (according to the decision confidence probability). As an illustration, reliability analysis is carried out by comparing two insulation systems (rock wool and glass wool). We consider that all the random variables are independent of each other in this paper for the sake of simplicity. Note that the case of dependent variables can be also addressed with the current approach (by considering the correlation matrix as an input for instance). We emphasize that the proposed methods are generic and can be applied to all LCA systems.

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PROBLEM STATEMENT We give at first a table of the notations and their description that we will use in the following (see Table 1). The Context of Multicriteria Decision Making in LCA. This study is of interest within the context of comparative LCAs (i.e., comparing two options providing the same function quantified by the functional unit, FU) and not for LCAs conducted to identify hotspots of a system for eco-design purposes. For comparative LCAs, indicators rarely result in a single best option and LCA main output is identifying trade-off between impacts categories and/or life cycle stages to identify B


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Let’s consider A and B, two options compared with LCA. hA,i(xA) and hB,i(xB) are, respectively for A and B options, the impact assessment results for the impact category i, calculated from the deterministic input parameters xA and xB. The comparison problem between these two options may be described as the following function:

Table 1. Summary of the Notations symbol

description

A,B xA, xB XA, XB h i j hA,i, hB,i Gi(X) P(Gi(X) < 0)

options evaluated (system, product, service, etc.) deterministic parameters respectively for A and B options stochastic parameters respectively for A and B options impact impact category index parameter impact (i) of option A or B limit decision state function for impact category i decision confidence probability for impact category i

Gi(x) = hA , i(xA) − hB , i(xB)

(1)

where x = {xA, xB} is represented as a vector which corresponds to the set of all input parameters in the two systems. The decision can be made thanks to Gi(x). The option A is better than the option B for the category i if Gi(x) < 0, and vice versa. Deterministic LCA concerns most of basic practices that does not take into account the uncertainty problem. However, lack of knowledge or of data leads to significant uncertainties in LCA. Decisions made without regarding the uncertainties may thus be flawed. From the end of the 1990s, in order to overcome the limitations of deterministic LCA, an advanced practice, called stochastic LCA, was developed in a statistical framework:18,19 uncertain parameters are considered as random variables, and defined by different probability distributions. Stochastic LCA consists in randomly sampling values in the probability distribution of uncertain parameters, to obtain the frequency distribution of hi.20,21 hi is commonly presented by its average, noted μ and its standard deviation (SD), noted σ. In this context, we compare the average value, but we cannot directly compare with SD, especially when the distributions are overlapping or crossing. Figure 1b shows a typical example of impacts comparison of two options in stochastic LCA. Scenario A is better than scenario B by only comparing average. But, it becomes difficult to compare the options in the presence of SD. In this latter case, the comparison stochastic LCA problem becomes:

the environmental implications of a decision.10 The proposed method (FORM) does not claim to solve the trade-off issue in a multicriteria context but it gives a consolidated probabilistic approach in order to compare options and to identify decision importance factors for each impact category. The question of decision-making in a multicriteria context is not addressed by the proposed approach and remains due to the numbers of LCA indicators (depending on LCIA methods, the number of midpoint indicators is in the range of 10−18 while it is 3−4 at end point level). From Deterministic LCA to Stochastic LCA. LCA has been broadly used for providing a set of indicators on different environmental impact categories (e.g., climate change, freshwater ecotoxicology, eutrophication, resource depletion, etc.) to stakeholders in a context of decision-making. We note hi the impact assessment result for the category i. Several LCIA methods were developed for calculating hi. The main method is the matrix-based approach17 which has been widely used in software tools to support the calculation of LCA. The LCIA (impact assessment) calculations produce a vector of indicators as a result (generally, 10−18 categories for midpoint LCIA methods and 3−4 categories for end point LCIA methods). Then, hi is often represented as a vector, where i is the category of impact. LCA decision is based on the comparative process to find which option has the smaller hi. Therefore, the problem in LCA is the comparison of two options. Currently, the primary study in LCA is called deterministic LCA that does not take into account the uncertainty factors. In this context, hi values are deterministic. Figure 1a shows a example of comparing the impacts of two options in deterministic LCA. The associated decision is that option A is better than option B. However, the associated decision lacks any information related to uncertainty.

Gi(X ) = hA , i(XA) − hB , i(XB)

(2)

where X = {XA, XB} is a random vector which is the set of all random variables in the two systems. Indeed, the vector x in the eq 1 is a realization of this random vector X. Figure 2 shows the graphical representation for the calculation of limit decision state surface Gi(X) = 0 with two uncertain parameters X = {X1, X2} using Monte Carlo method. The green points represent the realizations of X if Gi(X) > 0. The red points represent the realizations of X with Gi(X) < 0. Then, the boundary between the two domains is the limit decision state surface where Gi(X) = 0. The limit decision state function is non linear, because the

Figure 1. Comparing impacts of two options in LCA. (a) Comparing deterministic LCA. (b) Comparing stochastic LCA. C


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Figure 2. Calculation of the limit state surface with two random variables. Each axis (x1, x2) represents the range of uncertain values that each variable of the inventory (LCI) can take.

the results. Conversely, a low SI means that the uncertainty factors have little influence on the results. We note that several type of SI can meet this objective. Each SI has its own properties and needs different computational methods.12 For example, perturbation analysis uses relative sensitivity coefficients9 which estimate by how many percents the environmental impact will change if an input parameter changes by 1%; global screening analysis uses screening sensitivity indicators28 to provide qualitative measures of sensitivity; global sensitivity analysis uses Sobol indices which determine uncertainty sensitivity over the entire domain of variation, together with interactions between input parameters.12 However, these sensitivity analyses are performed on hi. Reliability methods provide the importance factors which correspond to the sensitivity indexes in relation to the probability P(Gi(X) < 0). Therefore, the importance factors represent the sensitivity indexes in relation to decision. Sensitivity analysis on the decision provides a further useful source of information for decision makers. We note f(Xj) the importance factor of the variable Xj according to the probability of Gi(X) < 0, defined as follows:

uncertain parameters are realized with different probability distributions that are often non linear. We note that, in practice, the number of random variables used is usually several hundreds. Therefore, the limit decision state surface is an hyper-surface in LCA. Stochastic LCA gives the information in relation to uncertainty problem. Therefore, the associated decision has no conclusion due to uncertainty. To overcome this problem, several decision support methods were proposed for the decision making during last ten years. Analytical propagation of uncertainty using matrix formulation22 or Taylor series expansion;23 stochastic multiattribute analysis for comparative LCA;24 the statistical testing done in comparative carbon footprints11 are some examples of decision support methods. Furthermore, probability-based decision support is widely used to evaluate the differences of options in impact categories given probabilistic results, especially as this method is implemented in the SimaPro software.25 Probability-Based Decision Support: Decision Confidence Probability. A probability is a measure of the likelihood that an event will occur. The probability support method aims to calculate the probability of having ”Gi(X) < 0″ and therefore ”Gi(X) ≥ 0″. The probability of Gi(X) < 0 is called decision conf idence probability, defined as follows: P(Gi(X ) < 0) = P(hA , i − hB , i < 0)

f (X j ) =

∂P(Gi(X ) < 0) ∂Xj

(4)

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(3)

MATERIALS AND METHODS Reliability Theory. Reliability theory was initially developed in the field of mechanical engineering29−32 and was broadly diffused in other domains, especially in environmental management.33,34 In these applications, the system, subject to uncertainties, belongs either to a failure domain or to a safety domain. The purposes of the reliability methods are (1) to approximate the limit decision state surface (G(X) = 0) that separates the failure domain (G(X) < 0) and the reliability domain (G(X) > 0); (2) to deduce (from the approximation of the limit decision state surface) the probability that the system belongs to the failure domain. For this purpose, a performance function G(X) is used to determine if the system belongs to the failure domain. If G(X) < 0, the system fails. If G(X) ≥ 0, the system does not fail. The limit state surface is therefore defined by G(X) = 0. In the current LCA problem defined beforehand, the problem remains the same: the failure probability for the category i corresponds to the decision confidence probability P(Gi(X) < 0). In previous stochastic LCA problems, Monte Carlo-based methods were used to estimate P(Gi(X) < 0) in the case of a low number of random parameters. As explained above,

The value of the decision confidence probability P(Gi(X) < 0) is between 0 and 1. A high value of decision confidence probability (close to 1) means that the option A is better than the option B. Conversely, a low value of decision confidence probability (close to 0) means that the option B is better. Monte Carlo method is a way to estimate this probability: from a large number of realizations of X with associated probability distributions known, we calculate the number of cases for which the hypotheses are verified. However, a large number of simulations is needed to ensure the accuracy of probability results with the Monte Carlo method, especially when the number of uncertain parameters is significant. Fortunately, the problem stated eq 3 is a reliability problem already addressed in the literature from which several reliability methods are available to approximate P(Gi(X) < 0) (see section ”Reliability Theory”). Sensitivity Analysis on Decision: Importance Factors. Sensitivity index (SI) is a measure used to describe the influence of uncertain parameters on the results. They are useful indicators provided by sensitivity analysis, recommended to limit the number of uncertainty factors.26,27 A high value of SI means that the uncertainty factors have a strong influence on D


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Figure 3. Standardized and centered transformation in 2 dimensions. Step 1 is the reliability problem in real space. Step 2 is the reliability problem in standardized and centered space. Step 3 corresponds to the approximation of P(Gi(X) < 0) using FORM method. β is the reliability index. U* is the design point.

imation of P(Gi(X) < 0) using FORM method. The minimal distance between the origin and the limit decision state D(U) = 0 represents the reliability index β. The closest point on the limit state represents the design point U*. P(Gi(X) < 0) is then approximated from the index β:

reliability-based methods are expected to address problems exhibiting a larger number of random variables. Several reliability methods are available in the literature. For example, first-order reliability method (FORM) estimates the failure probability using linear approximation;35 second-order reliability method (SORM) uses quadratic approximation instead of linear approximation and SMART (Support vector Margin Algorithm for Reliability esTimation) is based on Support Vector Machine.36 Among the reliability-based methods, FORM is one of the most popular used by engineers because this method requires a low number of simulations, the latter being a critical issue in engineering due to computation times. This method is used in the next section for approximating the limit decision state surface. FORM Method. In reliability context, most of the information is contained in a small area of space, called ”critical zone” where the set of x for which the probability of failure density is significant. The point where the probability of failure density is maximum is called ”design point”, noted x*. The FORM method is an approximation method for finding the design point and determining the failure probability associated with this design point and the limit state. It consists in replacing the hyper-surface by a hyperplane tangent at the design point.37 Whereas Monte Carlo approach approximates the failure probability by counting the number of simulations within the entire failure domain (G(X) < 0), the FORM method assesses the failure probability by approximating the limit decision state surface (G(X) = 0) with the hyperplane tangent. A standardized and centered space transformation has to be done because random variables may be defined by different probability distributions.20 The standardized and centered space transformation converts the random vector X in the physical space into a random vector U in the standard Gaussian space. The limit state in the standard Gaussian space is denoted as D(U) = 0 (see, S1, Supporting Information for more details on standardized and centered space transformation). Figure 3 shows the graphical representation of standardized and centered space transformation and the approximation of P(Gi(X) < 0) using FORM method. The step 1 shows the reliability problem in real space and the step 2 shows the reliability problem in standardized and centered space. Di(U) = 0 is the limit decision state surface in this space (the Di(U) = 0 is different from the Gi(X) = 0 because of the standardized and centered transformation). The step 3 represents the approx-

P(Gi(X ) < 0) ≈ Φ( −β)

(5)

where Φ(.) is the standard normal distribution. We can also determine the importance factor f(Xj) of parameter Xj: it corresponds to a sensitivity index of the probability P(Gi(X) < 0) to the parameter Xj. The importance factor of the variable Xj is the square of the cofactors of the design point in standardized and centered space: 2

f (Xj) = (αj) =

(u*j )2 (∥u∥)2

(6)

For example, in the step 3 of the Figure 3 the variables X2 has more influence than X1, because α22 > α21. Case Study. We compare two insulation systems: glass wool and rock wool materials. Both studied systems come from the LCA database Ecoinvent (data v2.1, July 2009).38 The complete description of the studied system is provided in the Life Cycle Inventories of Building Products report39 (see Figure S1, Supporting Information for schematic diagram). Glass Wool Mat System. Data for the glass wool mat system in Ecoinvent are provided by a Swiss company that has worked to a high technical standard, but the data refer to the situation before 1995. Energy required for the fusion process is primarily from electricity and natural gas. The amount of glass waste used as raw material is about 65%. The system comprises the following processes: melting, fiber forming and collecting, hardening and curing, and internal processes (workshop, etc.). Additionally transportation of raw materials and energy supply for the furnace, packing and infrastructure are included. For the heat needed, energy modules are used, and the electricity needed comes from the Swiss grid. Rock Wool System. The company works on a technically very high level. The fuel for the melting process is coke. The furnace is a cupola melting furnace. The emissions are abated by postcombustion with energy recuperation. E


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RESULTS In this section, we illustrate the proposed methodology for comparing the two product systems. We use the data and their uncertainties as given by the ecoinvent database, and the characterization method IMPACT 2002+ (version 2.0), end point.40 For reliability calculations, we need to couple OpenLCA software with an external reliability module. Among the available reliability modules, we choose the scientific library ”OpenTURNS v1.1”.41 More details about FORM method implementation are available in http://www. openturns.org/. In the following, we denote hg,i the impact of the glass wool system on the category i, and hr,i the impact of the rock wool system on the category i. Thus, the limit state function G(X) is defined as follows: Gi(X ) = hg , i(Xg ) − hr , i(X r )

CO2ToAir_R for rock wool or AmmoniaToAir_G for glass wool) of the systems. Decision Confidence Probability. Figure 4a shows the computed decision confidence probabilities using the FORM method. Considering this example, we are confident that rock wool is better than glass wool for the category Climate Change, because its decision confidence probability is significantly very low. The decision confidence probalities for other categories are also close to 50% and therefore a decision regarding these categories is not consistent. We can see that this probability gives a clear indicator about the confidence of an impact category when comparing these two systems. The Monte Carlo method is a way to find these probabilities, but it requires a large number of simulations to ensure the good estimation of LCA safety probability. For example, Figure 4b shows the error of the Monte Carlo method with different amounts of simulations. We have taken the same uncertain parameters as defined in the Table 2 and achieved 10 replicates for each amount of simulations. As we can see, the FORM method has required 131 simulations for computing the target probability, while the Monte Carlo method needed 1 000 000 simulations to reach the same precision. Indeed, the variability of the computed probability with Monte Carlo method is a clear proof that an insufficient amount of simulations (such as 100 replicates in our case) can lead to a random decision, as some replicates give a probability near to 60% or near 25% leading to opposite decisions. Importance Factors. Other important indicators provided by FORM are the importance factors, which play a significant role in the reliability analysis. Figure 5 shows the percentage of the parameter influence on the results for different systems. Figure 5a shows the importance factors of the random parameters of the Table 2 for the category of Human Health. AmmoniaToAir_G is the most important parameter because it is involved in about 39% of variance of results (in term of decision confidence probability). It means that the ammonia emission to air in the glass wool system has the highest importance on the decision, regarding the category of Human Health. Importance factors correspond to sensitivity indexes according to the decision confidence probability, which are different from the sensitivity indexes related to the impact. Indeed, sensitivity indexes on the impact can be useful when studying an isolated system, but when one want to study the comparison between two uncertain systems we should consider indicators on the sensitivity of the comparison. Figures 5b and c give sensitivity indexes using relative sensitivity coefficients

(7)

Therefore, the decision confidence probability P(Gi(X) < 0) corresponds to the probability that the impact of the glass wool system is smaller than the impact of the rock wool system. For clarity, we have restricted the comparison by selecting (with a local sensitivity analysis) to ten uncertain parameters, namely five for each system as listed in Table 2. These parameters can Table 2. Uncertain Parameters and Its Probability Distributions Selected from the LCA Database Ecoinvent (data v2.1, July 2009)38a

a The first column gives the name of the parameter, the second one the description of parameters, and the third one the probability density function of uncertain factors. lnN(μ, σ) is lognormal distribution with mean μ and standard deviation σ.

either be technological flux (e.g., diesel_R for rock wool or Phenol_G for glass wool) or environmental flux (e.g.,

Figure 4. Decision confidence probability for that the glass wool is better than rock wool (cf.eq 7). (a) LCA safety probability using FORM. (b) Computational efficiency of FORM versus Monte Carlo: Error associated with the number of simulations per replicate for Human Health category. F


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Figure 5. Percentage of the parameter influence on the results (Human Health category) for different systems. (a) Sensitivity analysis on decision (P(Gi(X) < 0)): importance factors of the uncertain parameters in the Table 2. (b): Sensitivity analysis using relative sensitivity coefficients for rock wook system. (c): Sensitivity analysis using relative sensitivity coefficients for glass wool system.

importance factors of AmmoniaToAir_G for all categories of the method IMPACT2002+, end point. AmmoniaToAir_G is involved in about 39% and 44% of variance of impacts for categories Human Health and Climate Change (in term of decision confidence probability). It is involved in about 10% on Resources and has hardly an influence on Ecosystem quality.

which are widely used in LCA for performing the sensitivity analysis on the impact.9,13,42 Relative sensitivity coefficients use the OFAT method43 and give the sensitivity indicators on the impact for each system (see S2, Supporting Information for more details). We can see that the importance factors of the Figure 5a reveal some sensitivity from the parameters like ammonia from glass wool and lubricant from rock wool that were not spotted by the relative sensitivity coefficients of the Figures 5b and c. As LCA is a multicriteria problem: the sensitivity analysis has to be conducted for all categories. Figure 6 shows the

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DISCUSSION In this paper, we applied the FORM method to estimate the decision confidence probability, because the comparative LCAs problem corresponds to the reliability problem in the literature. We compared the Monte Carlo method and the FORM method. Both methods give the same results in terms of decision confidence probability. However, the FORM method contributes to reduce the computation time compared to the Monte Carlo method. Moreover, the FORM method provides the importance factors which correspond to the sensitivity indexes on the decision. These importance factors give very useful information and they have never yet been used in LCA to our knowledge. It may help stakeholders by providing decision in terms of probability and sensitivity index toward the decision. Table 3 lists all the methods used in the paper. Impact assessments are raw environmental results. They give the impacts of each option for different impact categories. The objective of LCA is to find the scenario which has the smaller value of impact. Relative sensitivity coefficient is a sensitivity

Figure 6. Importance factors of AmmoniaToAir_G for all categories of the method IMPACT2002+, end point.

Table 3. Overview of the Definitions in LCA indicators in LCA

description

calculation method

formula

reference

impact assessment conventional relative sensitivity coefficient probability indicator

environmental results at midpoint or end-point level sensitivity index on each studied option

conventional LCAs sensitivity analysis

hi

decision support confidence using probability

P(Gi(X) < 0)

25,49

importance factor

sensitivity index on the decision confidence probability (i.e., the difference between the two studied options)

monte carlo results or proposed reliability approach proposed reliability approach

∂P(Gi(X ) < 0) ∂Xi

proposed by the authors

G

∂hi(X ) ∂Xi

44−48 9,13,42


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indicator in relation to the impact as well as importance factor is a sensitivity indicator in relation to the decision. It aims to study the influence of input parameters on the decision. Probability indicator is given by the probability support method which calculates the decision confidence probability and help stakeholders to make their decision. We emphasize that the methods we propose are generic and can be applied to all LCA systems. Note also that, if there are more than two options, pair comparison should be conducted. The next step will be to test other available methods in the literature (e.g., the SORM method uses quadratic approximation instead of linear approximation). Then, a module of reliability will be implemented in the LCA software OpenLCA. Further, reliability analysis needs the probability distribution on the uncertainty factors. However, this information can be obtained by data quality indicators (DQI) which are the reference in LCA.6 Therefore, we need more and more accurate uncertainty data in the future.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.est.5b03683.

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Additional methods and calculations (PDF)

AUTHOR INFORMATION

Corresponding Author

*Phone: +33 473440680; fax: +33 473440696; e-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the French National Research Agency (ANR-10-ECOT-004 grant, project DEMETHER), Céréales Vallée and ViaMéca for their financial support. We also thank all the members of ELSA research group (www.elsa-lca.org) for their precious advice.

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REFERENCES

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Using the Reliability Theory for Assessing the Decision Confidence

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