1 Introduction
2 Background, aim, and scope
Symbol | Name | Dimension (rows × columns) | Defined in |
---|---|---|---|
f
| Final demand vector | Economic flows × 1 | Goal and scope definition |
A
| Technology matrix | Economic flows × processes | Inventory analysis |
B
| Intervention matrix | Environmental flows × processes | Inventory analysis |
Q
| Characterisation matrix | Categories × environmental flows | Impact assessment |
\( {\mathbf{\dot{g}}} \)
| Intervention totalsa
| Environmental flows × 1 | Impact assessment |
\( {\mathbf{\dot{h}}} \)
| Category totalsa
| Categories × 1 | Impact assessment |
w
| Weighting factors | 1 × categories | Impact assessment |
Symbol | Name | Dimension (rows × columns) | Equation |
---|---|---|---|
s
| Scaling factors | processes × 1 |
\( {\mathbf{s}} = {{\mathbf{A}}^{ - 1}}{\mathbf{f}} \)
|
g
| Inventory results | Environmental flows × 1 |
\( {\mathbf{g}} = {\mathbf{Bs}} \)
|
h
| Characterization results | Categories × 1 |
\( {\mathbf{h}} = {\mathbf{Qg}} \)
|
Λ
| Intensity matrix | Environmental flows × economic flows |
\( {\mathbf{\Lambda }} = {\mathbf{B}}{{\mathbf{A}}^{ - 1}} \)
|
\( {\mathbf{\tilde{h}}} \)
| Normalization results | Categories × 1 |
\( \forall k:{\tilde{h}_k} = \frac{{{h_k}}}{{{{\dot{h}}_k}}} \)
|
\( {\mathbf{\dot{h}}} \)
| Category totalsa
| Categories × 1 |
\( {\mathbf{\dot{h}}} = {\mathbf{Q\dot{g}}} \)
|
W
| Weighted index | 1 × 1 |
\( W = {\mathbf{w\tilde{h}}} \)
|
-
the treatment of allocation and cutoff (Heijungs and Frischknecht 1998);
-
how to connect a process-based LCI to an input–output table (Suh and Huppes 2005);
-
how to efficiently compute an answer to the inventory problem (Peters 2007);
-
how to analyze the feedback structure of the system (Suh and Heijungs 2007);
-
how to calculate sensitivity coefficients (Heijungs 1994).
3 Materials and methods
3.1 Basic equations for LCA
-
A vector of intervention totals, \( {\mathbf{\dot{g}}} \), can be defined for the reference situation. For instance, one can collect data on the emissions of CO2, SO2, etc., which then represent \( {\dot{g}_1} \), \( {\dot{g}_2} \), etc. These then can be processed by the same characterization model to yield the vector of category totals, \( {\mathbf{\dot{h}}} \). This then forms the basis of the normalization. Changing \( {\mathbf{\dot{g}}} \) will induce a change in \( {\mathbf{\dot{h}}} \), but \( {\mathbf{\dot{h}}} \) itself will not be changed directly by the LCA practitioner. We will refer to this as normalization case 1.
-
Alternatively, the vector of category totals \( {\mathbf{\dot{h}}} \) can be known without a detailed specification of the underlying interventions. In that case, \( {\mathbf{\dot{h}}} \) is not an output result (in the sense of belonging to Table 2), but input data (in the sense of belonging to Table 1). Thus, \( {\mathbf{\dot{h}}} \) can be changed directly, and it will affect the normalization results and the weighted index. We will refer to this as normalization case 2.
3.2 General theory of sensitivity coefficients
4 Results
4.1 Sensitivity coefficients for LCA
\( {{\partial {s_k}} \mathord{\left/{\vphantom {{\partial {s_k}} \cdots }} \right.} \cdots } \)
|
\( {{\partial {g_k}} \mathord{\left/{\vphantom {{\partial {g_k}} \cdots }} \right.} \cdots } \)
|
\( {{\partial {h_k}} \mathord{\left/{\vphantom {{\partial {h_k}} \cdots }} \right.} \cdots } \)
|
\( {{\partial {{\tilde{h}}_k}} \mathord{\left/{\vphantom {{\partial {{\tilde{h}}_k}} \cdots }} \right.} \cdots } \)
|
\( {{\partial W} \mathord{\left/{\vphantom {{\partial W} \cdots }} \right.} \cdots } \)
| |
---|---|---|---|---|---|
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {a_{ij}}}}} \right.} {\partial {a_{ij}}}} \)
|
\( - {\left( {{{\mathbf{A}}^{ - 1}}} \right)_{ki}}{s_j} \)
| −λ
ki
s
j
|
\( - {s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} \)
|
\( - \frac{{{s_j}}}{{{{\dot{h}}_k}}}\sum\limits_l {{q_{kl}}{\lambda_{li}}} \)
|
\( - {s_j}\sum\limits_k {\frac{{{w_k}}}{{{{\dot{h}}_k}}}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \)
|
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {b_{ij}}}}} \right.} {\partial {b_{ij}}}} \)
| 0 |
s
j
δ
ik
|
q
ki
s
j
|
\( \frac{{{q_{ki}}{s_j}}}{{{{\dot{h}}_k}}} \)
|
\( {s_j}\sum\limits_k {\frac{{{w_k}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}{q_{ki}}} \)
|
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {q_{ij}}}}} \right.} {\partial {q_{ij}}}} \)
| 0 | 0 |
g
j
δ
ik
|
\( \left( {\frac{{{g_j}}}{{{{\dot{h}}_k}}} - \frac{{{h_k}{{\dot{g}}_j}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}} \right){\delta_{ik}} \)
|
\( {w_i}\left( {\frac{{{g_j}}}{{{{\dot{h}}_i}}} - \frac{{{h_i}{{\dot{g}}_j}}}{{{{\left( {{{\dot{h}}_i}} \right)}^2}}}} \right) \)
|
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {{\dot{g}}_i}}}} \right.} {\partial {{\dot{g}}_i}}} \) (normalization case 1) | 0 | 0 | 0 |
\( - \frac{{{h_k}{q_{ki}}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}} \)
|
\( - \sum\limits_k {\frac{{{w_k}{h_k}{q_{ki}}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}} \)
|
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {{\dot{h}}_i}}}} \right.} {\partial {{\dot{h}}_i}}} \) (normalization case 2) | 0 | 0 | 0 |
\( - \frac{{{h_k}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}{\delta_{ik}} \)
|
\( - \frac{{{w_i}{h_i}}}{{{{\left( {{{\dot{h}}_i}} \right)}^2}}} \)
|
\( { \cdots \mathord{\left/{\vphantom { \cdots {\partial {w_i}}}} \right.} {\partial {w_i}}} \)
| 0 | 0 | 0 | 0 |
\( {\tilde{h}_i} \)
|
4.2 Perturbation analysis
Multiplier | Definition | Formula |
---|---|---|
σ
k
(a
ij
) |
\( \frac{{\partial {s_k}/{s_k}}}{{\partial {a_{ij}}/{a_{ij}}}} \)
|
\( - \frac{{{a_{ij}}}}{{{s_k}}}{\left( {{{\mathbf{A}}^{ - 1}}} \right)_{ki}}{s_j} \)
|
γ
k
(a
ij
) |
\( \frac{{\partial {g_k}/{g_k}}}{{\partial {a_{ij}}/{a_{ij}}}} \)
|
\( - \frac{{{a_{ij}}}}{{{g_k}}}{\lambda_{ki}}{s_j} \)
|
γ
k
(b
ij
) |
\( \frac{{\partial {g_k}/{g_k}}}{{\partial {b_{ij}}/{b_{ij}}}} \)
|
\( \frac{{{b_{ij}}}}{{{g_k}}}{s_j}{\delta_{ik}} \)
|
η
k
(a
ij
) |
\( \frac{{\partial {h_k}/{h_k}}}{{\partial {a_{ij}}/{a_{ij}}}} \)
|
\( - \frac{{{a_{ij}}}}{{{h_k}}}{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} \)
|
η
k
(b
ij
) |
\( \frac{{\partial {h_k}/{h_k}}}{{\partial {b_{ij}}/{b_{ij}}}} \)
|
\( \frac{{{b_{ij}}}}{{{h_k}}}{q_{ki}}{s_j} \)
|
η
k
(q
ij
) |
\( \frac{{\partial {h_k}/{h_k}}}{{\partial {q_{ij}}/{q_{ij}}}} \)
|
\( \frac{{{q_{ij}}}}{{{h_k}}}{g_j}{\delta_{ik}} \)
|
\( {\tilde{\eta }_k}\left( {{a_{ij}}} \right) \)
|
\( \frac{{\partial {{\tilde{h}}_k}/{{\tilde{h}}_k}}}{{\partial {a_{ij}}/{a_{ij}}}} \)
|
\( - \frac{{{a_{ij}}}}{{{{\tilde{h}}_k}}}\frac{{{s_j}}}{{{{\dot{h}}_k}}}\sum\limits_l {{q_{kl}}{\lambda_{li}}} \)
|
\( {\tilde{\eta }_k}\left( {{b_{ij}}} \right) \)
|
\( \frac{{\partial {{\tilde{h}}_k}/{{\tilde{h}}_k}}}{{\partial {b_{ij}}/{b_{ij}}}} \)
|
\( \frac{{{b_{ij}}}}{{{{\tilde{h}}_k}}}\frac{{{q_{ki}}{s_j}}}{{{{\dot{h}}_k}}} \)
|
\( {\tilde{\eta }_k}\left( {{q_{ij}}} \right) \)
|
\( \frac{{\partial \tilde{h}/{{\tilde{h}}_k}}}{{\partial {q_{ij}}/{q_{ij}}}} \)
|
\( \frac{{{q_{ij}}}}{{{{\tilde{h}}_k}}}\left( {\frac{{{g_j}}}{{{{\dot{h}}_k}}} - \frac{{{h_k}{{\dot{g}}_j}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}} \right){\delta_{ik}} \)
|
\( {\tilde{\eta }_k}\left( {{{\dot{g}}_i}} \right) \) (normalization case 1) |
\( \frac{{\partial {{\tilde{h}}_k}/{{\tilde{h}}_k}}}{{\partial {{\dot{g}}_i}/{{\dot{g}}_i}}} \)
|
\( - \frac{{{{\dot{g}}_i}}}{{{{\tilde{h}}_k}}}\frac{{{h_k}{q_{ki}}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}} \)
|
\( {\tilde{\eta }_k}\left( {{{\dot{h}}_i}} \right) \) (normalization case 2) |
\( \frac{{\partial {{\tilde{h}}_k}/{{\tilde{h}}_k}}}{{\partial {{\dot{h}}_i}/{{\dot{h}}_i}}} \)
|
\( - \frac{{{{\dot{h}}_i}}}{{{{\tilde{h}}_k}}}\frac{{{h_k}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}{\delta_{ik}} \)
|
ω(a
ij
) |
\( \frac{{\partial W/W}}{{\partial {a_{ij}}/{a_{ij}}}} \)
|
\( - \frac{{{a_{ij}}}}{W}{s_j}\sum\limits_k {\frac{{{w_k}}}{{{{\dot{h}}_k}}}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \)
|
ω(b
ij
) |
\( \frac{{\partial W/W}}{{\partial {b_{ij}}/{b_{ij}}}} \)
|
\( \frac{{{b_{ij}}}}{W}{s_j}\sum\limits_k {\frac{{{w_k}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}{q_{ki}}} \)
|
ω(q
ij
) |
\( \frac{{\partial W/W}}{{\partial {q_{ij}}/{q_{ij}}}} \)
|
\( \frac{{{q_{ij}}}}{W}{w_i}\left( {\frac{{{g_j}}}{{{{\dot{h}}_i}}} - \frac{{{h_i}{{\dot{g}}_j}}}{{{{\left( {{{\dot{h}}_i}} \right)}^2}}}} \right) \)
|
\( \omega \left( {{{\dot{g}}_i}} \right) \) (normalization case 1) |
\( \frac{{\partial W/W}}{{\partial {{\dot{g}}_i}/{{\dot{g}}_i}}} \)
|
\( - \frac{{{{\dot{g}}_i}}}{W}\sum\limits_k {\frac{{{w_k}{h_k}{q_{ki}}}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}} \)
|
\( \omega \left( {{{\dot{h}}_i}} \right) \) (normalization case 2) |
\( \frac{{\partial W/W}}{{\partial {{\dot{h}}_i}/{{\dot{h}}_i}}} \)
|
\( - \frac{{{{\dot{h}}_i}}}{W}\frac{{{w_i}{h_i}}}{{{{\left( {{{\dot{h}}_i}} \right)}^2}}} \)
|
ω(w
i
) |
\( \frac{{\partial W/W}}{{\partial {w_i}/{w_i}}} \)
|
\( \frac{{{w_i}}}{W}{\tilde{h}_i} \)
|
4.3 Uncertainty analysis
Uncertainty | Equation |
---|---|
var(s
k
) |
\( \sum\limits_{i,j} {{{\left( {{s_j}} \right)}^2}{{\left( {{{\left( {{{\mathbf{A}}^{ - 1}}} \right)}_{ki}}} \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) \)
|
var(g
k
) |
\( \sum\limits_{i,j} {{{\left( {{s_j}{\lambda_{ki}}} \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) + \sum\limits_j {{{\left( {{s_j}} \right)}^2}} {\rm var} \left( {{b_{kj}}} \right) \)
|
var(h
k
) |
\( \sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\rm var} \left( {{b_{ij}}} \right) + \sum\limits_j {{{\left( {{g_j}} \right)}^2}} {\rm var} \left( {{q_{kj}}} \right) \)
|
\( {\rm var} \left( {{{\tilde{h}}_k}} \right) \)
|
\( \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}\left[ {\sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\rm var} \left( {{b_{ij}}} \right) + \sum\limits_j {{{\left( {{g_j} - \frac{{{h_k}{{\dot{g}}_j}}}{{{{\dot{h}}_k}}}} \right)}^2}} {\rm var} \left( {{q_{kj}}} \right) + \sum\limits_i {{{\left( {\frac{{{h_k}{q_{ki}}}}{{{{\dot{h}}_k}}}} \right)}^2}} {\rm var} \left( {{{\dot{g}}_i}} \right)} \right]} & {{\hbox{(normalization case 1)}}} \\{\frac{1}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}\left[ {\sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \right)}^2}} {\rm var} \left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\rm var} \left( {{b_{ij}}} \right) + {{\left( {\frac{{{h_k}}}{{{{\dot{h}}_k}}}} \right)}^2}{\rm var} \left( {{{\dot{h}}_k}} \right)} \right]} & {{\hbox{(normalization case 2)}}} \\\end{array} } \right. \)
|
var(W) |
\( \left\{ {\begin{array}{*{20}{c}} {\frac{1}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}\left[ {\sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda _{li}}} } \right)}^2}} {\text{var}}\left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\text{var}}\left( {{b_{ij}}} \right) + \sum\limits_j {{{\left( {{g_j} - \frac{{{h_k}{{\dot{g}}_j}}}{{{{\dot{h}}_k}}}} \right)}^2}} {\text{var}}\left( {{q_{kj}}} \right) + \sum\limits_i {{{\left( {\frac{{{h_k}{q_{ki}}}}{{{{\dot{h}}_k}}}} \right)}^2}} {\text{var}}\left( {{{\dot{g}}_i}} \right)} \right]} & {\left( {{\text{normalization case 1}}} \right)} \\ {\frac{1}{{{{\left( {{{\dot{h}}_k}} \right)}^2}}}\left[ {\sum\limits_{i,j} {{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda _{li}}} } \right)}^2}} {\text{var}}\left( {{a_{ij}}} \right) + \sum\limits_{i,j} {{{\left( {{s_j}{q_{ki}}} \right)}^2}} {\text{var}}\left( {{b_{ij}}} \right) + {{\left( {\frac{{{h_k}}}{{{{\dot{h}}_k}}}} \right)}^2}{\text{var}}\left( {{{\dot{h}}_k}} \right)} \right]} & {\left( {{\text{normalization case 2}}} \right)} \\ \end{array} } \right. \)
|
4.4 Key issue analysis
Contribution | Formula |
---|---|
ζ(s
k
,a
ij
) |
\( \frac{{{{\left( {{{\left( {{{\mathbf{A}}^{ - 1}}} \right)}_{ki}}{s_j}} \right)}^2}{\rm var} \left( {{a_{ij}}} \right)}}{{{\rm var} \left( {{s_k}} \right)}} \)
|
ζ(g
k
,a
ij
) |
\( \frac{{{{\left( {{s_j}{\lambda_{ki}}} \right)}^2}{\rm var} \left( {{a_{ij}}} \right)}}{{{\rm var} \left( {{g_k}} \right)}} \)
|
ζ(g
k
,b
ij
) |
\( \frac{{{{\left( {{s_j}{\delta_{ik}}} \right)}^2}{\rm var} \left( {{b_{ij}}} \right)}}{{{\rm var} \left( {{g_k}} \right)}} \)
|
ζ(h
k
,a
ij
) |
\( \frac{{{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda_{li}}} } \right)}^2}{\rm var} \left( {{a_{ij}}} \right)}}{{{\rm var} \left( {{h_k}} \right)}} \)
|
ζ(h
k
,b
ij
) |
\( \frac{{{{\left( {{s_j}{q_{ki}}} \right)}^2}{\rm var} \left( {{b_{ij}}} \right)}}{{{\rm var} \left( {{h_k}} \right)}} \)
|
ζ(h
k
,q
ij
) |
\( \frac{{{{\left( {{g_j}} \right)}^2}{\rm var} \left( {{q_{kj}}} \right)}}{{{\rm var} \left( {{h_k}} \right)}} \)
|
\( \zeta \left( {{{\tilde{h}}_k},{a_{ij}}} \right) \)
|
\( \frac{{{{\left( {{s_j}\sum\limits_l {{q_{kl}}{\lambda _{li}}} } \right)}^2}{\text{var}}\left( {{a_{ij}}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\text{var}}\left( {{{\widetilde{h}}_k}} \right)}} \)
|
\( \zeta \left( {{{\tilde{h}}_k},{b_{ij}}} \right) \)
|
\( \frac{{{{\left( {{s_j}{q_{ki}}} \right)}^2}{\rm var} \left( {{b_{ij}}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\rm var} \left( {{{\tilde{h}}_k}} \right)}} \)
|
\( \zeta \left( {{{\tilde{h}}_k},{q_{ij}}} \right) \)
|
\( \frac{{{{\left( {{g_j} - \frac{{{h_k}{{\dot{g}}_j}}}{{{{\dot{h}}_k}}}} \right)}^2}{\rm var} \left( {{q_{kj}}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\rm var} \left( {{{\tilde{h}}_k}} \right)}} \)
|
\( \zeta \left( {{{\tilde{h}}_k},{{\dot{g}}_i}} \right) \) (normalization case 1) |
\( \frac{{{{\left( {\frac{{{h_k}{q_{ki}}}}{{{{\dot{h}}_k}}}} \right)}^2}{\rm var} \left( {{{\dot{g}}_i}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\rm var} \left( {{{\tilde{h}}_k}} \right)}} \)
|
\( \zeta \left( {{{\tilde{h}}_k},{{\dot{h}}_i}} \right) \) (normalization case 2) |
\( \frac{{{{\left( {\frac{{{h_k}}}{{{{\dot{h}}_k}}}} \right)}^2}{\rm var} \left( {{{\dot{h}}_k}} \right)}}{{{{\left( {{{\dot{h}}_k}} \right)}^2}{\rm var} \left( {{{\tilde{h}}_k}} \right)}} \)
|
ζ(W,a
ij
) |
\( \frac{{{{\left( {{s_j}} \right)}^2}{{\left( {\sum\limits_l {\frac{{{w_l}}}{{{{\dot{h}}_l}}}\sum\limits_k {{q_{lk}}{\lambda_{ki}}} } } \right)}^2}{\rm var} \left( {{a_{ij}}} \right)}}{{{\rm var} (W)}} \)
|
ζ(W,b
ij
) |
\( \frac{{{{\left( {{s_j}} \right)}^2}{{\left( {\sum\limits_l {\frac{{{w_l}}}{{{{\dot{h}}_l}}}{q_{li}}} } \right)}^2}{\rm var} \left( {{b_{ij}}} \right)}}{{{\rm var} (W)}} \)
|
ζ(W,q
ij
) |
\( \frac{{{{\left( {\frac{{{w_i}}}{{{{\dot{h}}_i}}}} \right)}^2}{{\left( {{g_j} - \frac{{{h_i}{{\dot{g}}_j}}}{{{{\dot{h}}_i}}}} \right)}^2}{\rm var} \left( {{q_{ij}}} \right)}}{{{\rm var} (W)}} \)
|
\( \zeta \left( {W,{{\dot{g}}_i}} \right) \) (normalization case 1) |
\( \frac{{{{\left( {\frac{{{w_i}{h_i}}}{{{{\dot{h}}_i}{{\dot{h}}_i}}}} \right)}^2}{\rm var} \left( {{{\dot{h}}_i}} \right)}}{{{\rm var} (W)}} \)
|
\( \zeta \left( {W,{{\dot{h}}_i}} \right) \) (normalization case 2) |
\( \frac{{{{\left( {\frac{{{w_i}{h_i}}}{{{{\dot{h}}_i}{{\dot{h}}_i}}}} \right)}^2}{\rm var} \left( {{{\dot{h}}_i}} \right)}}{{{\rm var} (W)}} \)
|
ζ(W,w
i
) |
\( \frac{{{{\left( {{{\tilde{h}}_i}} \right)}^2}{\rm var} \left( {{w_i}} \right)}}{{{\rm var} (W)}} \)
|