Waste input-output model

Waste input-output (WIO) model (analysis) is a method based on input-output model, allowing for thorough tracking of product life cycles. This renders it a valuable tool for conducting life cycle assessment (LCA).

Introduction

The waste input-output (WIO) model (analysis) is a hybrid methodology of the environmentally extended input-output (EEIO) model designed to manage waste flow, including recycling, and its treatment processes. Unlike traditional input-output models, the WIO model explicitly incorporates the end-of-life phase, allowing for a comprehensive assessment of environmental impacts across a product's entire life cycle within the input-output (IO) analysis framework. By addressing the complexities of waste management, where waste and its treatment are not directly linked on a one-to-one basis, the WIO model extends the pollution abatement (environmental) IO model pioneered by Wassily Leontief and Faye Duchin .

The Methodology

The balance of goods and waste

Let there be $n_{P}$ producing sectors (each producing a single primary product), $n_{T}$ waste treatment sectors, and $n_{w}$ waste categories.

Now, let's define the matrices and variables:

• $X_{P}$: an $n_{P}\times n_{P}$ matrix representing the flow of products among producing sectros.
• $W_{P}$: an $n_{W}\times n_{P}$ matrix representing the net flow of wastes (generation minus use (recycle)) from producing sectors.
• $X_{T}$: an $n_{W}\times n_{T}$ matrix representing the flow of products in waste treatment sectors.
• $W_{T}$: an $n_{W}\times n_{T}$ matrix representing the net generation of waste in waste treatment sectors.
• $y_{P}$: an $n_{P}\times 1$ vector representing the final demand for products.
• $w_{Y}$: an $n_{W}\times 1$ vector representing the generation of waste from final demand sectors, such as the generation of kitchen waste and end-of-life consumer appliances.
• $x_{P}$: an $n_{P}\times 1$ vector representing the quantity of $n_{P}$ products produced.
• $w$: an $n_{W}\times 1$ vector representing the quantity of $n_{w}$ waste for treatment.

It is important to note that variables with $X$ or $x$ pertain to conventional components found in an IO table and are measured in monetary units. Conversely, variables with $W$ or $w$ typically do not appear explicitly in an IO table and are measured in physical units.

The equation below illustrates the balance between products and waste:

${\begin{aligned}{\begin{pmatrix}X_{P}&X_{T}\\W_{P}&W_{T}\end{pmatrix}}{\begin{pmatrix}\iota _{P}\\\iota _{T}\end{pmatrix}}+{\begin{pmatrix}y_{P}\\w_{y}\end{pmatrix}}={\begin{pmatrix}x_{P}\\w\end{pmatrix}}.\end{aligned}}$

1

Here, $\iota _{P}$ dednotes a vector of ones ($n_{P}\times 1$) used for summing the rows of $X_{P}$, and similar definitions apply to other $\iota$ terms. It is important to note that increased recycling reduces waste for treatment $w$.

The IO model with waste and waste treatment

We define the input coefficient matrices $A$ and waste generation coefficients $G$ as follows

${\begin{aligned}A_{P}=X_{P}{\hat {x}}_{P}^{-1},A_{T}=X_{T}{\hat {x}}_{T}^{-1},G_{P}=W_{P}{\hat {x}}_{P}^{-1},G_{T}=W_{T}{\hat {x}}_{T}^{-1}.\end{aligned}}$

Here, ${\hat {v}}$ refers to a diagonal matrix where the $(i,i)$ element is the $i$-th element of a vector $v$.

Using $A$ and $G$ as derived above, the balance (1) can be represented as:

${\begin{aligned}{\begin{pmatrix}A_{P}&A_{T}\\G_{P}&G_{T}\end{pmatrix}}{\begin{pmatrix}x_{P}\\x_{T}\end{pmatrix}}+{\begin{pmatrix}y_{P}\\w_{y}\end{pmatrix}}={\begin{pmatrix}x_{P}\\w\end{pmatrix}}.\end{aligned}}$ 2

This equation (2) represents the Duchin-Leontief environmental IO model, an extension of the original Leontief model of pollution abatement to account for the generation of secondary waste. It is important to note that this system of equations is generally unsolvable due to the presence of $x_{T}$ on the left-hand side and $w$ on the right-hand side, resulting in asymmetry. This asymmetry poses a challenge for solving the equation. However, the Duchin-Leontief environmental IO model addresses this issue by introducing a simplifying assumption:

$x_{T}=w$

3

This assumption (3) implies that a single treatment sector exclusively treats each waste. For instance, waste plastics are either landfilled or incinerated but not both simultaneously. While this assumption simplifies the model and enhances computational feasibility, it may not fully capture the complexities of real-world waste management scenarios. In reality, various treatment methods can be applied to a given waste; for example, organic waste might be landfilled, incinerated, or composted. Therefore, while the assumption facilitates computational tractability, it might oversimplify the actual waste management processes.

The WIO model

Shinichiro Nakamura and Yasushi Kondo addressed the above problem by introducing the allocation matrix $S$ of order $n_{T}\times n_{w}$ that assigns waste to treatment processes:

${\begin{aligned}x_{T}=Sw.\end{aligned}}$

4

Here, the element of $S_{kl}$ of $S$ represents the proportion of waste $l$ treated by treatment $k$. Since waste must be treated in some manner (even if illegally dumped, which can be considered a form of treatment), we have:

${\iota _{T}}^{'}S={\iota _{w}}^{'}.$

Here, $'$ stands for the transpose operator. Note that the allocation matrix $S$ is essential for deriving $x_{T}$ from $w$. The simplifying condition (3) corresponds to the special case where $n_{T}=n_{w}$ and $S$ is a unit matrix.

The application of the allocation matrix $S$ transforms equation (2) into a solvable form, the Waste Input-Output (WIO) quantity model:

${\begin{aligned}{\begin{pmatrix}A_{P}&A_{T}\\SG_{P}&SG_{T}\end{pmatrix}}{\begin{pmatrix}x_{P}\\x_{T}\end{pmatrix}}+{\begin{pmatrix}y_{P}\\Sw_{y}\end{pmatrix}}={\begin{pmatrix}x_{P}\\x_{T}\end{pmatrix}}\end{aligned}}$ 5

the solution of which is given by:

${\begin{aligned}{\begin{pmatrix}x_{P}\\x_{T}\end{pmatrix}}={\begin{pmatrix}I-A_{P}&-A_{T}\\-SG_{P}&I-SG_{T}\end{pmatrix}}^{-1}{\begin{pmatrix}y_{P}\\Sw_{y}\end{pmatrix}}.\end{aligned}}$

The amount of waste for treatment can then be given by:

${\begin{aligned}w={\begin{pmatrix}G_{P}&G_{T}\end{pmatrix}}{\begin{pmatrix}I-A_{P}&-A_{T}\\-SG_{P}&I-SG_{T}\end{pmatrix}}^{-1}{\begin{pmatrix}y_{P}\\Sw_{y}\end{pmatrix}}+w_{y}.\end{aligned}}$

Equation (5) presents an extension of the Leontief-Duchin pollution abatement IO, tailored to address waste management challenges characterized by the lack of a direct correspondence between waste types and treatment methods. This unique attribute distinguishes the approach, hence termed Waste Input-Output (WIO).

The Supply and Use Extension

In the WIO model (5), waste flows are categorized based solely on treatment method, without considering the waste type. Manfred Lenzen addressed this limitation by allowing both waste by type and waste by treatment method to be presented together in a single representation within a supply-and-use framework. This extension of the WIO framework results in a symmetric WIO model that does not require the conversion of waste flows into treatment flows.

WIO Tables and Applications

The WIO table compiled by the Japanese Ministry of the Environment (MOE) for the year 2011 stands as the only publicly accessible WIO table developed by a governmental body thus far. This MOE-WIO table distinguishes 80 production sectors, 10 waste treatment sectors, 99 waste categories, and encompasses 7 greenhouse gases (GHGs). It is accessible via the provided link. Additonally, many researchers have independently created their own WIO datasets and utilized them for various applications, encompassing different geographical scales and process complexities.

See also

• Anthropogenic metabolism
• Energy accounting
• Environmentally extended input-output analysis
• Industrial ecology
• Industrial metabolism
• Life cycle assessment
• Material criticality
• Material flow analysis
• Material flow accounting
• Social metabolism
• Sustainability
• Urban metabolism