The impact of carbon footprinting aggregation on realizing emission reduction targets

Abstract

A variety of activity-based methods exist for estimating the carbon footprint in transportation. For instance, the greenhouse gas protocol suggests a more aggregate estimation method than the Network for Transport and Environment (NTM) method. In this study, we implement a detailed estimation method based on NTM and different aggregate approaches for transportation carbon emissions in the dynamic lot sizing model. Analytical results show the limitations of aggregate models for both accurate estimation of real emissions and risks of compliance with carbon constraints (e.g., carbon caps). Extensive numerical experimentation shows that the magnitude of errors can be substantial. We provide insights under which limited conditions aggregate estimations can be used safely and when more detailed estimates are appropriate.

Appendix

Property 1

Let $Y^{c*} = \left[ {y_{1}^{c*} y_{1}^{c*} \cdots y_{T}^{c*} } \right]$ and $Q^{c*} = \left[ {q_{1}^{c*} q_{1}^{c*} \cdots q_{T}^{c*} } \right]$ be the optimal inventory policy obtained by model C. Suppose $q_{{1,T^{\prime } }} > q_{{T^{\prime } ,T}}$ and $W_{k} < W_{l}$ such that $W_{k} = W_{1} = \cdots = W_{{T^{'} }}$ and $W_{l} = W_{{T^{\prime } + 1}} = \cdots = W_{T}$ , if $\sum\nolimits_{t = 1}^{T} {y_{t}^{c*} } = T$ and $T^{\prime } \le \frac{T}{2}$ , then $Y^{c*}$ exceeds the carbon cap.

Proof

Since $C\left( {Y^{c*} } \right) \le C$, $C\left( {Y^{c*} } \right) + \gamma = C$ for some γ ≥ 0. If we assume that $Y^{c*}$ is not exceeding the carbon cap, the following relation should always hold: $CFP\left( {Y^{c*} ,Q^{c*} } \right) \le C \to CFP\left( {Y^{c*} ,Q^{c*} } \right) \le C\left( {Y^{c*} } \right) + \gamma$, which implies $\gamma \ge \sum\nolimits_{t = 1}^{T} {\widehat{g}_{t} }q_{t}^{c*} - \overline{D} \sum\nolimits_{t = 1}^{T}{\widehat{g}_{t} } y_{t}^{c*}$ such that $\overline{D} = \frac{{\mathop \sum \nolimits_{j = 1}^{T} d_{j} }}{T}$. Thus, $\sum\nolimits_{t = 1}^{T} {\widehat{g}_{t} } q_{t}^{c*} \ge\overline{D} \sum\nolimits_{t = 1}^{T} {\widehat{g}_{t} }y_{t}^{c*}$. Using the nomenclature q _i,j the condition is reduced to $\widehat{g}_{i} \left( {\frac{{q_{{1,T^{\prime } }} }}{{\overline{D} }} - \sum\nolimits_{t = 1}^{{T^{\prime } }} {y_{t}^{c*} } } \right) + \widehat{g}_{j} \left( {\frac{{q_{{T^{\prime } + 1,T}} }}{{\overline{D} }} - \sum\nolimits_{{t = T^{\prime } + 1}}^{T} {y_{t}^{c*} } } \right) \ge 0$ such that $\widehat{g}_{i}$ and $\widehat{g}_{j}$ are the emission factors related to $W_{{1,T^{\prime } }}$ and $W_{{T^{\prime } + 1,T}}$ respectively. Clearly $q_{{1,T^{\prime } }} + q_{{T^{\prime } + 1,T}} = \sum\nolimits_{t = 1}^{T} {d_{t} }$, and therefore it is possible to define a factor m such that $q_{{1,T^{\prime } }} = m\sum\nolimits_{t = 1}^{T} {d_{t} }$ and $q_{{T^{'} + 1,T}} = \left( {1 - m} \right)\mathop \sum \limits_{t = 1}^{T} d_{t}$. Therefore, $\widehat{g}_{i} \left( {mT - T^{\prime } } \right) + \widehat{g}_{j} \left( {T^{\prime } - mT} \right) \ge 0$. Since condition $q_{{1,T^{'} }} > q_{{T^{'} + 1,T}}$ implies that $m > \frac{1}{2}$, thus $\widehat{g}_{i} \ge \widehat{g}_{j}$. □

Property 2

Let $Y^{d*} = \left[ {y_{1}^{d*} y_{2}^{d*} \cdots y_{T}^{d*} } \right]$ and $Q^{d*} = \left[ {q_{1}^{d*} q_{2}^{d*} \cdots q_{T}^{d*} } \right]$ be the optimal inventory policy obtained by model D. Suppose $W_{1} = \cdots = W_{T}$ , if $C - D\left( {Y^{d*} } \right) < i\frac{{\widehat{g}}}{T}\mathop {\sum\nolimits_{t} {d_{t} } }\limits_{t}$ such that i is the number of $y_{j}^{d*} = 0$ for any $1 \le j \le T$ , then $Y^{d*}$ exceeds the carbon cap.

Proof

Since $D\left( {Y^{d*} } \right) \le C$, and we can define $D\left( {Y^{d*} } \right) + \delta = C$ for some $\delta \ge 0$. If we assume that $Y^{d*}$ is not exceeding the carbon cap, the following relation should always hold: $CFP\left( {Y^{d*} ,Q^{d*} } \right) \le C \to CFP\left( {Y^{d*} ,Q^{d*} } \right) \le D\left( {Y^{d*} } \right) + \delta$, which implies the following condition $\widehat{g}\sum\nolimits_{j = 1}^{T} {d_{j} } \le \widehat{g}\sum\nolimits_{t} {y_{t}^{d*} } \left( {\frac{{\mathop \sum \nolimits_{j = 1}^{T} d_{j} }}{T}} \right) + \delta$. Thus, $\delta \ge \left( {T - \sum\nolimits_{t = 1}^{T} {y_{t}^{d*} } } \right)\widehat{g}\frac{{\mathop \sum \nolimits_{j = 1}^{T} d_{j} }}{T}$. □

Corollary 2.1

Let $Y^{d*} = \left[ {y_{1}^{d*} y_{2}^{d*} \cdots y_{T}^{d*} } \right]$ and $Q^{d*} = \left[ {q_{1}^{d*} q_{2}^{d*} \cdots q_{T}^{d*} } \right]$ be the optimal inventory policy obtained by model D. Suppose $W_{1} = \cdots = W_{T}$ , if $\sum\nolimits_{t = 1}^{T} {y_{t}^{d*} < T}$ and $C = D\left( {Y^{d*} } \right)$ , then $Y^{d*}$ exceeds the carbon cap.

Proof

If $\delta = 0$ thus $\widehat{g}\sum\nolimits_{j = 1}^{T} {d_{j} \le \widehat{g}} \sum\nolimits_{t} {y_{t}^{d*} \frac{{\mathop \sum \nolimits_{j = 1}^{T} d_{j} }}{T}}$. However, since $\sum\nolimits_{t} {y_{t}^{d*} \le T}$ it is clear that the previous condition is just valid for $\sum\nolimits_{t} {y_{t}^{d*} = T}$, but we know that $\sum\nolimits_{t} {y_{t}^{d*} < T}$. □

Property 3

Let $Y^{f*} = \left[ {y_{1}^{f*} y_{2}^{f*} \cdots y_{T}^{f*} } \right]$ $Q^{f*} = \left[ {q_{1}^{f*} q_{1}^{f*} \cdots q_{T}^{f*} } \right]$ be the optimal inventory policy obtained by model F. Suppose $W_{1} = \cdots = W_{T}$ , if $WT - \sum\nolimits_{j = 1}^{T} {d_{j} \in [0,iW)}$ such that i is the number of $y_{j}^{*} = 0$ for any $1 \le j \le T$ , then $Y^{f*} Q^{f*}$ is not the true optimal.

Proof

Since $CFP\left( {Y^{*} ,Q^{*} } \right) \le C$, we can define $CFP\left( {Y^{*} ,Q^{*} } \right) + \beta = C$ for some $\beta \ge 0$. Besides, we notice that $WT \ge \sum\nolimits_{j = 1}^{T} {d_{j} }$ and therefore we define $WT = \varepsilon + \sum\nolimits_{j = 1}^{T} {d_{j} }$for some $\varepsilon \ge 0$.If we assume that the optimal solution of model F is indeed the true optimal solution, the following relation should always hold: $Y^{*} = Y^{f*}$, thus $F\left( {Y^{*} } \right) \le C \to \beta \ge F\left( {Y^{*} } \right) - CFP\left( {Y^{*} ,Q^{*} } \right)$, which implies $\beta \ge \widehat{g}\left( {W\sum\nolimits_{t} {y_{t}^{*} } - \sum\nolimits_{j = 1}^{T} {d_{j} } } \right)$. Thus $W\sum\nolimits_{t} {y_{t}^{*} } - \sum\nolimits_{j = 1}^{T} {d_{j} \ge 0} \to W\sum\nolimits_{t} {y_{t}^{*} } - WT + \varepsilon \ge 0$ and therefore the following condition should hold $\varepsilon \ge W\left( {T - \sum\nolimits_{t} {y_{t}^{*} } } \right)$. □

Corollary 3.1

Let $Y^{f*} = \left[ {y_{1}^{f*} y_{2}^{f*} \cdots y_{T}^{f*} } \right]$ $Q^{f*} = \left[ {q_{1}^{f*} q_{2}^{f*} \cdots q_{T}^{f*} } \right]$ be the optimal inventory policy obtained by model F. Suppose $W_{1} = \cdots = W_{T}$ , if $\sum\nolimits_{t = 1}^{T} {y_{t}^{*} < T}$ and $WT - \sum\nolimits_{j = 1}^{T} {d_{j} = 0}$ then the optimal solution of model F is not the true optimal solution.

Proof

If $\varepsilon = 0$ thus $\sum\nolimits_{t} {y_{t}^{*} - T \ge 0}$. However, since $\sum\nolimits_{t} {y_{t}^{*} \le T}$ it is clear that the previous condition is just valid for $\sum\nolimits_{t} {y_{t}^{*} = T}$, but we know that $\sum\nolimits_{t} {y_{t}^{*} < T}$. □

Property 4

Let $Y^{e*} = \left[ {y_{1}^{e*} y_{2}^{e*} \cdots y_{T}^{e*} } \right]$ and $Q^{e*} = \left[ {q_{1}^{e*} q_{2}^{e*} \cdots q_{T}^{e*} } \right]$ be the optimal inventory policy obtained by model E. If $C - CFP\left( {Y^{*} ,Q^{*} } \right) < \sum\nolimits_{t = 1}^{T} {} \widehat{g}_{t} \left( {W_{t} y_{t}^{*} - q_{t}^{*} } \right)$ then $Y^{e*}$ $Q^{e*}$ is not the true optimal.

Proof

Since $CFP\left( {Y^{*} ,Q^{*} } \right) \le C$ we define $CFP\left( {Y^{*} ,Q^{*} } \right) + \beta = C$ for some $\beta \ge 0$. Furthermore, we notice that $min\left\{ {W_{t} } \right\} \ge \sum\nolimits_{j = 1}^{T} {d_{j} }$. If we assume that model E is a true optimal solution the following relation should always hold: $Y^{*} = Y^{e*}$, thus $E\left( {Y^{*} } \right) \le C \to \beta \ge E\left( {Y^{*} } \right) - CFP\left( {Y^{*} ,Q^{*} } \right)$, which implies $\beta \ge \widehat{g}_{1} \left( {W_{1} y_{1}^{*} - q_{1}^{*} } \right) + \cdots + \widehat{g}_{T} \left( {W_{T} y_{T}^{*} - q_{T}^{*} } \right)$. □

Corollary 4.1

Let $Y^{f*} = \left[ {y_{1}^{f*} y_{2}^{f*} \cdots y_{T}^{f*} } \right]$ and $Q^{f*} = \left[ {q_{1}^{f*} q_{2}^{f*} \cdots q_{T}^{f*} } \right]$ be the optimal inventory policy obtained by model F. If $C - CFP\left( {Y^{*} ,Q^{*} } \right) < \widehat{g}\left( {W\sum\nolimits_{t} {y_{t}^{*} } - \sum\nolimits_{j = 1}^{T} {d_{j} } } \right)$ then $Y^{f*}$ $Q^{f*}$ is not the true optimal.

Proof

Since $W = W_{1} = W_{2} = \cdots = W_{T}$, property 2 is reduced to $\beta < \widehat{g}\left( {W\sum\nolimits_{t} {y_{t}^{*} } - \sum\nolimits_{j = 1}^{T} {d_{j} } } \right)$. □