*2.1. Physical Input-Output Model*

The PIOM originates from the monetary IOM proposed by Leontief, and can reflect urban solid waste flows among sectors and investigate the multiple sectoral linkages [5,23]. The basic form of IOM can be presented as [27]:

$$\mathbf{x}\_{i} = \sum\_{j=1}^{n} z\_{ij} + f d\_{i} \text{ for } i = 1 \text{ to } n \tag{1}$$

where *x<sup>i</sup>* is the total output of sector *i*, *zij* is the amount of goods *i* that sector *j* consumes, and *fd<sup>i</sup>* is the final demand of sector *i*. Solid waste intensity is then introduced to transform the monetary IOM into PIOM as follows [28,29]:

$$
\mathbf{E} + \varepsilon \mathbf{Z} = \varepsilon \mathbf{X} \tag{2}
$$

$$\boldsymbol{\varepsilon} = \mathbf{E}(\mathbf{X} - \mathbf{Z})^{-1} \tag{3}$$

$$\mathbf{F} = \operatorname{diag}(\varepsilon) \ast \mathbf{Z} \tag{4}$$

where **E** = [*e<sup>i</sup>* ]1×*<sup>n</sup>* is the amount of sectoral solid waste; ε = [*ε<sup>i</sup>* ]1×*<sup>n</sup>* is the solid waste intensity vector, *ε<sup>i</sup>* is the embodied solid waste per unit of monetary value of sector *i*; **Z** = [zij]*n*×*n*, *zij* is the amount of goods *i* that sector *j* consumes; **X** = [*x<sup>j</sup>* ]1×*<sup>n</sup>* is the total economic output and **F** = [*fij*]*n*×*<sup>n</sup>* is the solid waste flows among various sectors. By physical units, it is referred to mass units for presenting waste flows (e.g., Mg). Direct solid waste production equals the initial input of the monetary-physical input-output table, and the indirect solid waste production of each sector equals the sum of its column elements in the physical input-output table. For instance, sector *i* produces 1 Mg solid waste per unit product production, meaning 1 Mg is the amount of direct solid waste in sector *i*. Sector *i* sells product to sector *j,* implying that the 1 Mg solid waste is indirectly transferred to sector *j* (i.e., the amount of indirect solid waste in sector *j* is 1 Mg).

Then, the amount of sectoral indirect solid waste and sectoral total flows can be calculated based on Equations (5) and (6) [30]:

$$IF\_{\bar{j}} = \sum\_{i=1}^{n} f\_{\bar{i}\bar{j}} \tag{5}$$

$$T\_i = \sum\_{j=1}^{n} f\_{i\bar{j}} + e\_{\bar{i}} \tag{6}$$

where *fij* is the direct solid waste flowing from sector *i* to sector *j*; *e<sup>i</sup>* is the amount of direct solid waste; *IF<sup>j</sup>* is the amount of indirect sectoral solid waste and *T<sup>i</sup>* is the total amount of waste. Taking all pathway flows with different lengths between two sectors into account, the dimensionless integral solid waste flow intensity matrix (**N**) can be obtained through:

$$g\_{i\dot{j}} = f\_{i\dot{j}} / T\_{\dot{i}} \tag{7}$$

$$\mathbf{N} = (\mathbf{G})^0 + (\mathbf{G})^1 + (\mathbf{G})^2 + \dots \\ (\mathbf{G})^\infty = (\mathbf{I} - \mathbf{G})^{-1} \tag{8}$$

where *gij* is the dimensionless input-oriented intercomponent flow from sector *i* to sector *j*; **G<sup>n</sup>** is the dimensionless integral flow intensity matrix with *n* path length and **I**(*n*×*n*) is the identity matrix.
