## 4 Equality-Enhanced Parallel Resource Allocation Model  
The Gini coefficient is an important measure of inequality, which has been used to capture the degree of equality in income distribution. Additionally, it has been widely applied to guide resource allocation [8; 9; 40]. To compute the Gini coefficient, one needs to first identify a service unit, which is the service (or goods) that need to be allocated  
Figure 1: Parallel structure system. according to an equality indicator [41; 42]. For example, in this paper, we argue that a carbon emission reduction quota should be allocated according to the GDP proportion of a province in the national GDP. In this case, the carbon reduction quota serves as a service unit and the GDP as an equality unit. Let us denote \(S_{j}(j=1,\ldots,n)\) the service units and \(L_{j}(j=1,\ldots,n)\) the equality units. According to [PERSON] [43], the Gini coefficient can be written as the following analytical formula:  
\[G=\frac{\sum\limits_{t}\sum\limits_{k>t}|l_{k}S_{t}-l_{t}S_{k}|}{\sum\limits_{ t}S_{t}},\]  
where \(l_{t}=\frac{L_{t}}{\sum\limits_{j}L_{j}}\). From the above formula, it can be seen that \(G\in[0,\;1]\). Below we introduce the Gini coefficient of carbon emissions of the three major industries as an indicator to measure the equality of carbon quota allocation to capture the equality of allocation among the three industries [44]. Specifically, the initial carbon emission of an industry in a province is taken as a service unit and the GDP of the industry in the province is taken as the equality unit before carbon quota allocation. The Gini coefficient of the initial industrial carbon emission indicates the equality of carbon emission allocation in terms of its GDP. Its analytical formula is given below. \[G^{p}=\frac{\sum\limits_{t}\sum\limits_{k>t}\left|l_{k}^{p}R_{t}^{p}-l_{t}^{p }R_{k}^{p}\right|}{\sum\limits_{t}R_{t}^{p}},\]  
where \(l_{t}^{p}=\frac{y_{t}^{p}}{\sum\limits_{j}y_{j}^{p}}\) denotes the proportion of provincial-industry level GDP in the national GDP of industry \(p\). \(R_{t}^{p}\), \(R_{k}^{p}\) indicate the carbon emissions of SDMU\({}_{t}^{p}\) and SDMU\({}_{k}^{p}\) before the allocation, respectively. Similarly, after the allocation of carbon emission, using the expected carbon emission level as the service unit, the industry carbon emission Gini coefficient is defined as follows:  
\[G^{tp}=\frac{\sum\limits_{t}\sum\limits_{k>t}\left|l_{k}^{p}\left(R_{t}^{p}-b_ {t}^{p}\right)-l_{t}^{p}\left(R_{k}^{p}-b_{k}^{p}\right)\right|}{\sum\limits_{ t}\left(R_{t}^{p}-b_{t}^{p}\right)}.\]  
To ensure that the equality of the allocation scheme after the carbon quota allocation is not inferior to the initial carbon quota scheme, it is reasonable to require that the after-allocation carbon emission Gini coefficient of the industry is less than or equal to the initial carbon emission Gini coefficient, that is, \(G^{tp}\leq G^{p}\). Based on the model (1) and the Gini coefficient of industrial carbon emissions, this paper proposes the equality-enhanced parallel resource allocation model (2). The 5 th constraint ensures that the equality of the allocation does not deteriorate. In this way, model (2) is able to combine the efficiency and equality criteria into a model so that the trade-offs between the two criteria can be handled effectively. \[