### 2.1 Total industrial carbon emissions and carbon emissions intensity

Industrial carbon emissions mainly come from the consumption of fossil energy [5]. Commonly used carbon emission accounting methods include actual measurement method [6], material balance algorithm and coefficient method [7]. The actual measurement method is to calculate the total carbon emissions by measuring the flow rate [8], velocity and concentration of the exhaust gas through the instruments and facilities recognized by the relevant departments. The material balance algorithm is based on the law of conservation of material [9], which quantitatively analyzes the amount of material used in the energy consumption process or the production process to calculate carbon emissions [10]. It is divided into a material balance algorithm based on the production department and a material balance algorithm based on carbon source energy [11]. The coefficient method is divided into the coefficient method based on the production field and the coefficient method based on fossil energy [12]. The coefficient method based on the production field refers to the calculation of the total carbon emissions by calculating the average value of the carbon emissions produced by the production unit product or the unit energy consumption under the general technical and economic management conditions [13]. The coefficient method based on fossil energy obtains the carbon emission coefficient of various energy sources according to different countries or regions [14], different energy structures and technological levels, and the sum of the products of standard coal consumption and oxidation rate converted to various energy sources is the carbon emission Total amount [15]. Due to specific issues, it is difficult to obtain the energy consumption data of the counties and districts at the municipal level, and it is impossible to directly calculate the industrial carbon emissions of the counties and districts based on the coefficient method based on fossil energy [16]. Studies have shown that industrial added value is positively correlated with industrial carbon emissions [17]. Therefore, in this paper, according to the proportion of the industrial added value of each county in the city's industrial added value, the total industrial carbon emissions are proportionally allocated to obtain the industrial carbon emissions of each county [18].

### 2.2 Low-carbon economic theory

Since the 1990s, scholars have gradually paid attention to the issue of carbon dioxide emissions [19], trying to explore the relationship between economic development and carbon dioxide emissions by consulting relevant literature and empirical research [20]. With the deepening of research, scholars at home and abroad generally believe that economic growth plays a positive role in driving carbon dioxide emissions [21], and rapid economic growth has brought about continuous increases in carbon dioxide emissions [22]. Therefore, researchers must pay attention to the issue of carbon emission reduction while developing the economy and the theory of low-carbon economy came into being [23].

The concept of "low-carbon economy" first appeared in the energy white paper "The Future of Our Energy: Creating a Low-Carbon Economy" published in the United Kingdom [24]. The book states that economic development should focus on ecological and environmental protection, instead of using a large amount of energy consumption and carbon dioxide emissions. In exchange for economic growth [25], it is necessary to change the economic growth mode based on fossil fuels such as coal and use more clean energy such as solar energy, wind energy, and tidal energy. This economic growth model of "low energy consumption, low pollution, low emissions" and "high efficiency, high efficiency, and high efficiency" three lows and three highs has attracted global attention and attention.

Low-carbon economy is based on a variety of means such as technological progress, industrial transformation, institutional innovation, and new energy development to form a series of economic forms such as low-carbon industry, low-carbon life, low-carbon energy, low-carbon technology, and low-carbon development. Change the mode of economic development to obtain more economic output with less energy use and lower carbon dioxide emissions. The development of a low-carbon economy does not mean not developing or developing at the expense of the economy, but to achieve a harmonious unity of economy, society, and ecology through a change in the concept of development [26].

From a policy perspective, although my country’s current carbon intensity policy helps curb the growth of carbon emissions; it is far from enough to achieve the peak of carbon dioxide emissions. He Jiankun, deputy director of the National Committee of Experts on Climate Change, believes that in 2030, the target carbon intensity of a relative decline in GDP can be measured by the decline in China’s carbon dioxide emissions. This is also determined by the characteristics of my country’s current industrialization and urbanization, and now we are in during the period of sustained rapid economic growth, energy demand and carbon dioxide emissions will continue to grow for a long period of time. A relative decline in GDP is a necessary condition to ensure sustained and stable economic development, but carbon emission reduction policies need to be further strengthened. So as to vigorously promote the extensive promotion of energy conservation and emission reduction [27].

### 2.3 Overview of interval multi-objective optimization problems

The constrained optimization problem is expressed as Eq. (1):

$$\begin{aligned} & Q:\mathop {\min }\limits_{x} F(x) = (f_{m} (x)),\quad m = 1,2, \ldots ,z \\ & s.t.g_{j} (x) \ge a_{j} ,\quad j = 1,2, \ldots ,n \\ & h_{k} (x) = b_{k} ,\quad k = 1,2, \ldots ,n^{*} \\ & x = (x_{1} , \ldots ,x_{q} ) \in X \subset R \\ \end{aligned}$$

(1)

Among them, all the solutions that meet the constraints in the target space are called feasible solutions.

Formula (1) describes the general form of multi-objective optimization problems, and in engineering problems, optimization problems are all with ambiguity and uncertainty. In order to solve these problems, we have proposed some research methods. This method is called non-Deterministic mathematical methods.

The constrained interval multi-objective optimization problem is shown in the following Eqs. (2), (3), (4), (5):

$$Q:\mathop {\min }\limits_{x} F(x,u) = (f_{m} (x,u)),\quad m = 1,2, \ldots ,z$$

(2)

*X* is the *q*-dimensional decision space:

$$s.t.g_{j} (x,u) \ge a_{j} = [a_{j} ,\overline{{a_{j} }} ],\quad j = 1,2, \ldots ,n$$

(3)

*x* is the *q*-dimensional decision vector:

$$h_{k} (x,u) \ge b_{k} = [b_{k} ,\overline{{b_{k} }} ],\quad k = 1,2, \ldots ,n^{*}$$

(4)

Define *x*:

$$x = (x_{1} , \ldots ,x_{q} ) \in X \subset R^{q} ,x_{i} = [x_{i} ,\overline{{x_{i} }} ],\quad i = 1,2, \ldots ,q$$

(5)

Define *u*:

$$u = (u_{1} , \ldots ,u_{p} ) \subset R^{p} ,u_{1} = [u_{1} ,\overline{{u_{1} }} ],\quad l = 1,2, \ldots ,p$$

(6)

Among them, several important data can be functionalized to obtain formulas (7), (8), (9):

$$f_{z} (x) = \mathop {\min }\limits_{u} f_{z} (x,u),\overline{{f_{z} (x)}} = \mathop {\max }\limits_{u} f_{z} (x,u)$$

(7)

$$g_{j} (x) = \mathop {\min }\limits_{u} g_{j} (x,u),\overline{{g_{j} (x)}} = \mathop {\max }\limits_{u} g_{j} (x,u)$$

(8)

$$h_{k} (x) = \mathop {\min }\limits_{u} h_{k} (x,u),\overline{{h_{k} (x)}} = \mathop {\max }\limits_{u} h_{k} (x,u)$$

(9)

On this basis, Taylor expands to *u*:

$$\mathop {\min }\limits_{x} F(x,u) = ((f_{1} (x),\overline{{f_{1} (x)}} ), \ldots ,(f_{2} (x),\overline{{f_{2} (x)}} ))$$

(10)

Processing in the formula:

$$\begin{aligned} f_{i} (x) & = f_{i} (x,u^{c} ) - \sum\limits_{i = 1}^{p} {|\frac{{\partial f_{i} (x,u^{c} )}}{{\partial u_{l} }}|} u_{l}^{r} \\ \overline{{f_{i} (x)}} & = f_{i} (x,u^{c} ) + \sum\limits_{i = 1}^{p} {|\frac{{\partial f_{i} (x,u^{c} )}}{{\partial u_{l} }}|} u_{l}^{r} \\ \end{aligned}$$

(11)

We use the same approach to the constraint function:

$$\begin{aligned} g_{j} (x,u) & = [g_{j} (x),\overline{{g_{j} (x)}} ] \\ h_{k} (x,u) & = [h_{k} (x),\overline{{h_{k} (x)}} ] \\ \end{aligned}$$

(12)

where:

$$\begin{aligned} p_{j} (x) & = p_{j} (x,u^{c} ) - \sum\limits_{i = 1}^{p} {|\frac{{\partial p_{j} (x,u^{c} )}}{{\partial u_{l} }}|} u_{l}^{r} \\ \overline{{p_{j} (x)}} & = p_{j} (x,u^{c} ) + \sum\limits_{i = 1}^{p} {|\frac{{\partial p_{j} (x,u^{c} )}}{{\partial u_{l} }}|} u_{l}^{r} \\ j_{k} (x) & = j_{k} (x,u^{c} ) - \sum\limits_{i = 1}^{p} {|\frac{{\partial j_{k} (x,u^{c} )}}{{\partial u_{l} }}|} u_{l}^{r} \\ \overline{{j_{k} (x)}} & = j_{k} (x,u^{c} ) + \sum\limits_{i = 1}^{p} {|\frac{{\partial j_{k} (x,u^{c} )}}{{\partial u_{l} }}|} u_{l}^{r} \\ \end{aligned}$$

(13)

The Taylor series expansion is used to effectively reduce the amount of calculation. In summary, the interval multi-objective optimization problem formula (1) is organized into the standard form of the constrained multi-objective interval optimization problem. The algorithm simplification is completed, and the experiment is ready to begin.