*2.3. Fractional Factorial Analysis*

The USWS involves a number of economic sectors. These sectors' solid waste production may be interrelated to each other, increasing the complexity of the decision-making process. Fractional factorial analysis (FFA) can be employed to recognize the main factors and detect their interactions on the response variables of the USWS. Sectoral solid waste production (*e<sup>i</sup>* in **E**) and sectoral direct consumption coefficient (*aij* = *zij*/*x<sup>j</sup>* ) can be chosen as factors, which are divided into multiple levels. Solid waste production intensity (abbreviated as SPI) can be selected as the response when SPI = direct solid waste production (Mg)/gross domestic product (10<sup>4</sup> RMB¥ = 1542 USD). Using a fractional factorial analysis can screen main *e<sup>i</sup>* and *aij* as well as quantify their interactions with reduced experimental cost. Researchers select an appropriate experimental matrix based on the number of *e<sup>i</sup>* and *aij* [35]. A set of SPI values are gained by running the EIOM based on the matrix. Fractional factorial analysis quantifies the sensitivity of SPI to important factors and their combinations through addressing the curve traits of SPI when factors change at various levels. The quadratic sum for single factor and two-factor combinations are presented as follows [36,37]:

$$\text{SS}\_A = \sum\_{i=1}^{I} \left( \sum\_{j=1}^{J} \sum\_{k=1}^{K} \mathbf{Y}\_{ijk} \right)^2 / J\mathbf{K} - \left( \sum\_{i=1}^{I} \sum\_{j=1}^{J} \sum\_{k=1}^{K} \mathbf{Y}\_{ijk} \right)^2 / I\mathbf{J}\mathbf{K} \tag{18}$$

$$SS\_B = \sum\_{j=1}^{I} \left( \sum\_{i=1}^{I} \sum\_{k=1}^{K} Y\_{ijk} \right)^2 / IK - \left( \sum\_{i=1}^{I} \sum\_{j=1}^{I} \sum\_{k=1}^{K} Y\_{ijk} \right)^2 / IIK \tag{19}$$

$$\text{SS}\_{A \times B} = \sum\_{i=1}^{I} \sum\_{j=1}^{J} \left( \sum\_{k=1}^{K} \text{Y}\_{ijk} \right)^2 / K - \left( \sum\_{i=1}^{I} \sum\_{j=1}^{J} \sum\_{k=1}^{K} \text{Y}\_{ijk} \right)^2 / IIK - \text{SS}\_A - \text{SS}\_B \tag{20}$$

$$\text{SST}\_{T} = \sum\_{i=1}^{I} \sum\_{j=1}^{J} \sum\_{k=1}^{K} \mathbf{Y}^{2}\_{ijk} - \left( \sum\_{i=1}^{I} \sum\_{j=1}^{J} \sum\_{k=1}^{K} \mathbf{Y}\_{ijk} \right)^{2} / I \text{JK} \tag{21}$$

where *I* and *J* are the designed levels of factors *A* and *B*, respectively; *yijk* is the observed value in the *Kth* replication when *A* and *B* are at level *Ith* and *Jth*; *SSA*, *SSB,* and *SSA*×*<sup>B</sup>* denote the square sum of *A*, *B*, and their combinations and *SS<sup>T</sup>* is the total of squares. The contribution of each factor is calculated as the sum of its squares to the sum of the total squares.
