## 3 Parallel Structure DEA Allocation Model  
Suppose there are \(n\) DMU\({}_{j}(j=1,\ldots,n)\), where each DMU\({}_{j}\) contains \(q\) parallel SDMU\({}_{j}^{p}(p=1,\ldots,q)\), and each DMU and SDMU contains \(m\) inputs, \(s\) desirable outputs, and \(l\) undesirable outputs. As shown in Figure 1, the input, desirable outputs, and undesirable output vectors of the \(p\)-th SDMU of DMU\({}_{j}\) are \(X_{ij}^{p}(i=1,\ldots,m)\), \(Y_{rj}^{p}(r=1,\ldots,s)\), \(R_{dj}^{p}(d=1,\ldots,l)\), then \(X_{ij}=\sum\limits_{p=1}^{q}x_{ij}^{p}\), \(Y_{rj}=\sum\limits_{p=1}^{q}y_{rj}^{p}\), and \(R_{dj}=\sum\limits_{p=1}^{q}r_{dj}^{p}\), respectively. Let \(B_{d}(d=1,\ldots,l)\) be the total undesirable output to be allocated, the undesirable output allocated to DMU\({}_{j}\) is \(b_{dj}\), and the undesirable output allocated to DMU\({}_{j}^{p}\) within DMU\({}_{j}\) is \(b_{dj}^{p}\), then \(B_{d}=\sum\limits_{j=1}^{n}\sum\limits_{p=1}^{q}b_{dj}^{p}\). Note that we only study the one desirable output and one undesirable output production system in this paper that perfectly serves our research purpose. We use the gross domestic product (GDP) as the desirable output and carbon emissions as the undesirable output, which are typical indicators documented in many studies [11; 38]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Allocation**} & \multicolumn{2}{c}{**Resource Allocation**} \\ \cline{2-5}
**Reference** & \multicolumn{1}{c}{**Efficiency Equality**} & \multicolumn{1}{c}{**Carbon Quota**} & \multicolumn{1}{c}{**Others**} & \multicolumn{1}{c}{**Sectors**} \\ \hline \multirow{2}{*}{[PERSON] [29] (2009)} & \(\surd\) & & & \\  & [PERSON] [30] (2020) & \(\surd\) & & & Urban water industry \\  & [PERSON] [17] (2014) & \(\surd\) & & & \\  & [PERSON] et al. [31] (2015) & \(\surd\) & & & \\  & Esmaeilzadeh and Kazemi [18] (2019) & \(\surd\) & & & \\  & [PERSON] and [PERSON] [32] (2021) & \(\surd\) & & & \\  & [PERSON] et al. [33] (2022) & \(\surd\) & \(\surd\) & & \\  & [PERSON] et al. [34] (2018) & \(\surd\) & & & \(\surd\) \\  & [PERSON] et al. [35] (2020) & \(\surd\) & & & \(\surd\) \\  & [PERSON] et al. [36] (2020) & \(\surd\) & & & \(\surd\) \\  & [PERSON] et al. [37] (2022) & \(\surd\) & & & \(\surd\) \\  & This paper & \(\surd\) & \(\surd\) & \(\surd\) & Three major industries \\ \hline \hline \end{tabular} Note: Where \(\surd\) indicates that the item is included in the literature.  
\end{table} 
Table 2: This paper vs. the literature of parallel DEA models. The DEA models designed in this paper all satisfy the following assumptions:  
1. Variable returns-to-scale (VRS). 2. Weak disposability. Reducing the undesirable outputs under the current technology will inevitably reduce the desirable outputs. 3. Null joint-ness. The only way to eliminate the undesirable output is to shut down production. Combing the VRS assumption and the weak disposability and null joint-ness assumptions pose a challenge to modeling in the literature. [PERSON] et al. [39] proposed a new set of environmental production possibilities to achieve this purpose. In this paper, the parallel allocation model (1) is provided based on the allocation model under the VRS assumption proposed by [PERSON] et al. [38]. \[ \begin{array}{l}\max\quad\sum\limits_{p=1}^{q}\sum\limits_{h=1}^{n}\sum \limits_{j=1}^{n}\lambda_{hj}^{p}y_{h}^{p}\\ s.t.\quad\sum\limits_{j=1}^{n}\lambda_{hj}^{p}x_{ij}^{p}\leq\sigma_{h}^{p}x_{ ih}^{p}\qquad i=1,\ldots,m;\ p=1,\ldots,q;\ h=1,\ldots,n\\ \quad\sum\limits_{j=1}^{n}\lambda_{hj}^{p}R_{j}^{p}=R_{h}^{p}-b_{h}^{p}\quad p =1,\ldots,q;\ h=1,\ldots,n\\ \quad\sum\limits_{h=1}^{n}\sum\limits_{p=1}^{q}b_{h}^{p}=B\\ \quad\sum\limits_{j=1}^{n}\lambda_{hj}^{p}=\sigma_{h}^{p}\qquad\qquad p=1, \ldots,q;\ h=1,\ldots,n\\ \lambda_{hj}^{p}\geq 0\qquad\qquad\qquad\quad p=1,\ldots,q;\ h=1,\ldots,n;\ j=1, \ldots,n\\ \varepsilon\leq\sigma_{h}^{p}\leq 1\qquad\qquad\quad p=1,\ldots,q;\ h=1,\ldots,n\\ R_{h}^{p}-b_{h}^{p}\geq 0\qquad\qquad\quad p=1,\ldots,q;\ h=1,\ldots,n\\ \end{array}