Waste input-output model

The Waste Input-Output (WIO) model is an innovative extension of the environmentally extended input-output (EEIO) model. It enhances the traditional Input-Output (IO) model by incorporating physical waste flows generated and treated alongside monetary flows of products and services . In a WIO model, each waste flow is traced from its generation to its treatment, facilitated by an allocation matrix . Additionally, the model accounts for the transformation of waste during treatment into secondary waste and residues, as well as recycling and final disposal processes. By including the end-of-life (EoL) stage of products, the WIO model enables a comprehensive consideration of the entire product life cycle, encompassing production, use, and disposal stages within the IO analysis framework. As such, it serves as a valuable tool for life cycle assessment (LCA).

Background

With growing concerns about environmental issues, the EEIO model evolved from the conventional IO model appended by integrating environmental factors such as resources, emissions, and waste. The standard EEIO model, which includes the economic input-output life-cycle assessment (EIO-LCA) model, can be formally expressed as follows

$E=F(I-A)^{-1}y$

Here $A$ represents the square matrix of input coefficients, $F$ denotes releases (such as emissions or waste) per unit of output or the intervention matrix, $y$ stands for the vector of final demand (or functional unit), $I$ is the identity matrix, and $E$ represents the resulting releases (For further details, refer to the input-output model). A model in which $F$ represents the generation of waste per unit of output is known as a Waste Extended IO (WEIO) model. In this model, waste generation is included as a satellite account.

However, this formulation, while well-suited for handling emissions or resource use, encounters challenges when dealing with waste. It overlooks the crucial point that waste typically undergoes treatment before recycling or final disposal, leading to a form less harmful to the environment. Additionally, the treatment of emissions results in residues that require proper handling for recycling or final disposal (for instance, the polluiton abatement process of sulfur dioxide involves its conversion into gypsum or sulfuric acid). Leontief's pioneering pollution abatement IO model did not address this aspect, whereas Duchin later incorporated it in a simplified illustrative case of wastewater treatment.

In waste management, it is common for various treatment methods to be applicabel to a single type of waste. For instance, organic waste might undergo landfilling, incineration, gassification, or composting. Conversely, a single treatment proecess may be suitable for various types of waste; for example, solid waste of any type can typically be dispposed of in a landfill. Formally, this implies that there is no one-to-one correspondence between teratment methods and types of waste.

A theoreteical drawback of the Leontief-Duchin EEIO model is that it consders only cases where this one-to-one correspondence between teratment methods and types of waste applies, which makes the model difficul to apply to real waste management issues. The WIO model addresses this weakness by introducing a general mapping between treatment methods and types of waste, establishing a highly adaptable link between waste and treatment. This results in a model that is applicable to a wide range of real waste management issues.

The Methodology

We describe below the major features of the WIO model in its relationship to the Leontief-Duchin EEIO model, starting with notations.

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. Typical examples include animal waste from livestock, slag from steel mills, sludge from paper mills and the chemical industry, and meal scrap from manufacturing processes. The recyling of animal waste in fertilizer production can be accounted for as a negative input of the former by the latter.
• $X_{T}$: 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 (secondary) waste in waste treatment sectors. Typical examples include ashes generated from incineration processes, sludge produced during wastewater treatment, and residues derived from automobile shredding facilities.
• $y_{P}$: 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 balance of goods and waste

Using the notations introduced above, we can represent the supply and demand balance between products and waste for treatment by the following system of equations:

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

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. The first line petains to the standard balance of goods and services with the left-hand side referring to the demand and the right-hand-side supply. Similarly, the second line refers to the balance of waste, where the left-hand side signifies the generation of waste for treatment, and the right-hand side denotes the waste designated for treatment. It is important to note that increased recycling reduces the amount of waste for treatment $w$.

The IO model with waste and waste treatment

We now 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 table below gives an example of $S$ for seven waste types and three treatment processes. Note that $S$ represents the allocation of waste for treatment, that is, the portion of waste that is not recycled.

Allocation of various types of waste to treatment processes

	Garbage	Waste Paper	Waste Plastics	Metal scrap	Green waste	Ash	Bulky waste
Incineration	0.90	0.93	0.59	0.01	0.99	0	0
Landfill	0.10	0.07	0.41	0.99	0.01	1	0
Shredding	0	0	0	0	0	0	1

The application of the allocation matrix $S$ transforms equation (2) into the following fom:

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

Note that, different from (2), the vaiable $x_{T}$ occurs on the bith side of the equation. This sytem of equatins is thus solvable (provided it exists), with the solution given by:

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

Equation (5) gives the Waste Input-Output (WIO) quantity model, which extends the Leontief-Duchin EEIO model to accomodate waste management challenges characterized by the lack of a one-to-one correspondence between waste types and treatment methods. This distinctive characteristic sets it apart and warrants its designation as the Waste Input-Output (WIO) model.

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

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

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). The MOE-WIO table is abvaialble here.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
• Waste treatment technologies

External links

• WIO table compiled by the Japanese Ministry of the Environment.