Waste Input-Output Model

Waste Input-Output (WIO) model 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)[1].

## Introduction[edit]

The Waste Input-Output (WIO) model [2] 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[3]. 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[4] and Faye Duchin[5] .

## The Model[edit]

### The balance of goods and waste[edit]

Let there be  n P {\displaystyle n_{P}} producing sectors (each producing a single primary product[6]),  n T {\displaystyle n_{T}} waste treatment sectors, and  n w {\displaystyle n_{w}} waste categories.

Now, let's define the matrices and variables:

* X P {\displaystyle X_{P}} : an  n P × n P {\displaystyle n_{P}\times n_{P}} matrix representing the flow of products among producing sectros.
* W P {\displaystyle W_{P}} : an  n W × n P {\displaystyle n_{W}\times n_{P}} matrix representing the net flow of wastes (generation minus use (recycle)) from producing sectors.
* X T {\displaystyle X_{T}} : an  n W × n T {\displaystyle n_{W}\times n_{T}} matrix representing the flow of products in waste treatment sectors.
* W T {\displaystyle W_{T}} : an  n W × n T {\displaystyle n_{W}\times n_{T}} matrix representing the net generation of waste in waste treatment sectors.
* y P {\displaystyle y_{P}} : an  n P × 1 {\displaystyle n_{P}\times 1} vector representing the final demand for products.
* w Y {\displaystyle w_{Y}} : an  n W × 1 {\displaystyle 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 {\displaystyle x_{P}} : an  n P × 1 {\displaystyle n_{P}\times 1} vector representing the quantity of  n P {\displaystyle n_{P}} products produced.
* w {\displaystyle w} : an  n W × 1 {\displaystyle n_{W}\times 1} vector representing the quantity of  n w {\displaystyle n_{w}} waste for treatment.

It is important to note that variables with  X {\displaystyle X} or  x {\displaystyle x} pertain to conventional components found in an IO table and are measured in monetary units. Conversely, variables with  W {\displaystyle W} or  w {\displaystyle 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:

( X P X T W P W T ) ( ι P ι T ) + ( y P w y ) = ( x P w ) . {\displaystyle {\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}}}

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(1)

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

### The IO model with waste and waste treatment[edit]

We define the input coefficient matrices  A {\displaystyle A} and waste generation coefficients  G {\displaystyle G} as follows

A P = X P x ^ P − 1 , A T = X T x ^ T − 1 , G P = W P x ^ P − 1 , G T = W T x ^ T − 1 . {\displaystyle {\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,  v ^ {\displaystyle {\hat {v}}} refers to a diagonal matrix where the  ( i , i ) {\displaystyle (i,i)} element is the  i {\displaystyle i} -th element of a vector  v {\displaystyle v} .

Using  A {\displaystyle A} and  G {\displaystyle G} as derived above, the balance (1) can be represented as:

( A P A T G P G T ) ( x P x T ) + ( y P w y ) = ( x P w ) . {\displaystyle {\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}}}

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(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 {\displaystyle x_{T}} on the left-hand side and  w {\displaystyle w} on the right-hand side, resulting in asymmetry[7]. 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 {\displaystyle x_{T}=w}

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(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[edit]

Shinichiro Nakamura and Yasushi Kondo[8] addressed the above problem by introducing the allocation matrix  S {\displaystyle S} of order  n T × n w {\displaystyle n_{T}\times n_{w}} that assigns waste to treatment processes:

x T = S w . {\displaystyle {\begin{aligned}x_{T}=Sw.\end{aligned}}}

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(4)

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

ι T ′ S = ι w ′ . {\displaystyle {\iota _{T}}^{'}S={\iota _{w}}^{'}.}

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

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

( A P A T S G P S G T ) ( x P x T ) + ( y P S w y ) = ( x P x T ) {\displaystyle {\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}}}

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(5)

the solution of which is given by:

( x P x T ) = ( I − A P − A T − S G P I − S G T ) − 1 ( y P S w y ) . {\displaystyle {\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:

w = ( G P G T ) ( I − A P − A T − S G P I − S G T ) − 1 ( y P S w y ) + w y . {\displaystyle {\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[edit]

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[9]. 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[edit]

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[10].

## References[edit]

1. ^ Suh, Sangwon; Nakamura, Shinichiro (2007). "Five years in the area of input-output and hybrid LCA". The International Journal of Life Cycle Assessment. 12 (6): 351–352. doi:10.1065/lca2007.08.358. ISSN 1614-7502.
2. ^ Nakamura, Shinichiro; Kondo, Yasushi (2002). "Input‐Output Analysis of Waste Management". Journal of Industrial Ecology. 6 (1): 39–63. doi:10.1162/108819802320971632. ISSN 1088-1980.
3. ^ Towa, Edgar; Zeller, Vanessa; Achten, Wouter M.J. (2020). "Input-output models and waste management analysis: A critical review". Journal of Cleaner Production. 249: 119359. doi:10.1016/j.jclepro.2019.119359. ISSN 0959-6526.
4. ^ Wassily Leontief (1970). "Environmental repercussions and the economic structure: an input-output approach". The Review of Economics and Statistics. 52: 262-271.
5. ^ Faye Duchin (1990). "The conversion of biological materials and wastes to useful products". Structural Change and Economic Dynamics. 1: 243-261.
6. ^ Byproducts without primary producers are categorized as waste, whereas those with primary producers can be registered as negative inputs.
7. ^ Nakamura, Shinichiro; Kondo, Yasushi (2018). "Toward an integrated model of the circular economy: Dynamic waste input–output". Resources, Conservation and Recycling. 139: 326–332. doi:10.1016/j.resconrec.2018.07.016. ISSN 0921-3449.
8. ^ Nakamura, Shinichiro; Kondo, Yasushi (2002). "Input‐Output Analysis of Waste Management". Journal of Industrial Ecology. 6 (1): 39–63. doi:10.1162/108819802320971632. ISSN 1088-1980.
9. ^ Lenzen, Manfred; Reynolds, Christian John (2014). "A Supply‐Use Approach to Waste Input‐Output Analysis". Journal of Industrial Ecology. 18 (2): 212–226. doi:10.1111/jiec.12105. ISSN 1088-1980.
10. ^ Nakamura, Shinichiro (2023). "A Practical Guide to Industrial Ecology by Input-Output Analysis". SpringerLink. doi:10.1007/978-3-031-43684-0., Chapter 5