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
Here represents the square matrix of input coefficients, denotes releases (such as emissions or waste) per unit of output or the intervention matrix, stands for the vector of final demand (or functional unit), is the identity matrix, and represents the resulting releases (For further details, refer to the input-output model). A model in which 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 pollution 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 applicable to a single type of waste. For instance, organic waste might undergo landfilling, incineration, gasification, or composting. Conversely, a single treatment process may be suitable for various types of waste; for example, solid waste of any type can typically be disposed of in a landfill. Formally, this implies that there is no one-to-one correspondence between treatment methods and types of waste.
A theoretical drawback of the Leontief-Duchin EEIO model is that it considers only cases where this one-to-one correspondence between treatment methods and types of waste applies, which makes the model difficult 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 producing sectors (each producing a single primary product),Byproducts without primary producers are categorized as waste, whereas those with primary producers can be registered as negative inputs waste treatment sectors, and waste categories. Now, let's define the matrices and variables: : an matrix representing the flow of products among producing sectros. : an matrix representing the generation of wastes 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. : an matrix representing the use (recycling) of wastes by producing sectors. Typical examples include the use of animal waste in fertilizer production and iron scrap in steel production based on an electric arc furnace. : an matrix representing the net flow of wastes: . : an matrix representing the flow of products in waste treatment sectors. : an matrix representing the net generation of (secondary) waste in waste treatment sectors: ( and are defined similar to and ). Typical examples of include ashes generated from incineration processes, sludge produced during wastewater treatment, and residues derived from automobile shredding facilities. : an vector representing the final demand for products. : an vector representing the generation of waste from final demand sectors, such as the generation of kitchen waste and end-of-life consumer appliances. : an vector representing the quantity of products produced. : an vector representing the quantity of waste for treatment.
Variables with or pertain to conventional components found in an IO table and are measured in monetary units. Conversely, variables with or 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:
Here, dednotes a vector of ones () used for summing the rows of , and similar definitions apply to other terms. The first line pertains 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. Increased recycling reduces the amount of waste for treatment .
The IO model with waste and waste treatment We now define the input coefficient matrices and waste generation coefficients as follows
Here, refers to a diagonal matrix where the element is the -th element of a vector .
Using and as derived above, the balance () can be represented as: This equation () 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. This system of equations is generally unsolvable due to the presence of on the left-hand side and 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:
This assumption () 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 Nakamura and Kondo addressed the above problem by introducing the allocation matrix of order that assigns waste to treatment processes:
Here, the element of of represents the proportion of waste treated by treatment . Since waste must be treated in some manner (even if illegally dumped, which can be considered a form of treatment), we have:
Here, stands for the transpose operator. Note that the allocation matrix is essential for deriving from . The simplifying condition () corresponds to the special case where and is a unit matrix.
The table below gives an example of for seven waste types and three treatment processes. Note that 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 processesGarbageWaste PaperWaste PlasticsMetal scrapGreen wasteAshBulky wasteIncineration0.900.930.590.010.9900Landfill0.100.070.410.990.0110Shredding0000001
The application of the allocation matrix transforms equation () into the following form:
Note that, different from (), the variable occurs on both sides of the equation. This system of equations is thus solvable (provided it exists), with the solution given by:
The WIO counterpart of the standard EEIO model of emissions, represented by equation (), can be formulated as follows:
Here, represents emissions per output from production sectors, and denotes emissions from waste treatment sectors. Upon comparison of equation () with equation (), it becomes clear that the former expands upon the latter by incorporating factors related to waste and waste treatment.
Finally, the amount of waste for treatment induced by the final demand sector can be given by:
The Supply and Use Extension (WIO-SUT) In the WIO model (), 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, given below, results in a symmetric WIO model that does not require the conversion of waste flows into treatment flows.
</p>It is worth noting that despite the seemingly different forms of the two models, the Leontief inverse matrices of WIO and WIO-SUT are equivalent.
The WIO Cost and Price Model Let's denote by , , , and the vector of product prices, waste treatment prices, value-added ratios of products, and value-added ratios of waste treatments, respectively.
The case without waste recycling In the absence of recycling, the cost counterpart of equation () becomes:
which can be solved for and as:
The case with waste recycling When there is a recycling of waste, the simple representation given by equation () must be extended to include the rate of recycling and the price of waste :
Here, is the vector of waste prices, is the diagonal matrix of the vector of the average waste recycling rates, , and ( and are defined in a similar fashion).
Rebitzer and NakamuraRebitzer, Gerald, and Shinichiro Nakamura. "Environmental life cycle costing." Environmental life cycle costing. CRC Press, 2008. 35-58. used () to assess the life-cycle cost of washing machines under alternative End-of-Life scenarios. More recently, Liao et al. applied () to assess the economic effects of recycling copper waste domestically in Taiwan, amid the country's consideration of establishing a copper refinery to meet increasing demand.
A caution about possible changes in the input-output coeffcieints of treatment processes when the composition of waste changes The input-output relationships of waste treatment processes are often closely linked to the chemical properties of the treated waste, particularly in incineration processes. The amount of recoverable heat, and thus the potential heat supply for external uses, including power generation, depends on the heat value of the waste. This heat value is strongly influenced by the waste's composition. Therefore, any change in the composition of waste can significantly impact and .
To address this aspect of waste treatment, especially in incineration, Nakamura and Kondo recommended using engineering information about the relevant treatment processes.