1.1. Related Works

The TOPSIS method can combine fuzzy logic with multi-criteria decision-making, and many experts have studied it. Scholars such as Celikbilek have explored the widely used TOPSIS method in multi-criteria decision-making, utilizing simulation techniques to design multiple scenario simulations, revealing the shortcomings of traditional TOPSIS methods under specific conditions, and quantitatively analyzing the results. Finally, the possible ways was proposed to improve the traditional TOPSIS method ^[4]. To expand the application scope of Multi-Criteria Group Decision-Making (MCGDM) technology, Akram et al. proposed the Complex Spherical Fuzzy TOPSIS for MCGDM in the CSFS. This method used CSFWA operators to integrate independent decisions from multiple experts and rank and evaluate alternative solutions ^[5]. Nazim M et al. reviewed and reviewed existing selection methods and related literature to compare the performance differences between fuzzy Analytic Hierarchy Process (AHP) and Fuzzy TOPSIS used in software requirement selection problems ^[6]. Scholars such as Le Minh Thanh have designed a single neuron PID controller based on a cyclic fuzzy NN identifier for the control problem of trajectory tracking of a 3-degree of freedom Delta robot ^[7]. In response to the issue of improving the performance of the speed curve tracking controller for subway trains, Pu and his team members considered the time-varying parameters of train motion and constructed a train model with dynamic parameters to more accurately describe the train's motion behavior. In practical circuit applications, the proposed controller was superior to the currently used controller ^[8].

The rise of the global DigE has prompted researchers to focus on how DigT shapes and changes economic structure and market patterns. Litvinenko analyzed the probability of utilizing direct DigT in the exploration, design, and utilization of mineral resources to study the impact of the global DigE on the technological development of the world's mineral industry ^[9]. Oguz et al. developed 15 six rotor deformation state maps in the Solidworks program to address the limitations of longitudinal flight control for six rotor unmanned aerial vehicles. They obtained the PID of longitudinal flight for these drawings from the Matlab/Simulink program, created a training set, and then passed it on to DNN to estimate the inertia moment and PID ^[10]. Viriyasitavat et al. analyzed the key indicators involved in Business Process Management (BPM) in the DigE of the Internet of Things era. They are particularly concerned about the conflicting indicators of openness, security, flexibility, scalability, and cost and flexibility ^[11]. Luo et al. explored the role of the DigE in enhancing the green innovation level in the context of sustainable development. They used principal component analysis to evaluate the development urban DigE level, and used some models to explore the direct effects, indirect effects, spatial effects, nonlinear relationships, and the impact of policy DigE on green innovation ^[12]. He et al. analyzed the environmental governance achievements of 9 provinces and 2 cities in the Yellow River Economic Belt (YREB) from 2009 to 2018, and used the Slack-Based Measurement (SBM) and Malmquist index to calculate the green total factor productivity ^[13].

In summary, many experts have conducted research on TOPSIS, NN, and green economy, but there is still room for improving practical applications. This paper is based on TOPSIS and NN, with the Yellow River (YR) Basin as the research objective, aiming to achieve coordinated economy and ecological environment.

2.1. Establishment of an Indicator System Based on CRITIC-TOPSIS

\textls[-10]{This manuscript is designed upon the DigE in the YREB. To comprehensively reflect the research content, a large number of tertiary indicators connected with the DigE will be collected. Before conducting empirical analysis, it is necessary to conduct independence, redundancy, and universality tests to ensure the rationality of the initially constructed indicator system and accurately reflect the DigE situation of the YREB ^[15]. In the construction of an indicator system, if multiple indicators are involved, a correlation coefficient matrix can be used to analyze the degree of correlation between these indicators. Assuming the correlation coefficient matrix is $R^{{p}} $, $p=1$, $2$, $\ldots$, $n$, as shown in Eq. (1).}

The average correlation coefficient RD of the redundancy indicator system is Eq. (2).

In Eq. (2), the value of $RD$ is between 0 and 1. The smaller the value of $RD$, the lesser the redundant signal between the indicators in the indicator system, indicating a lower correlation between the indicators. When $RD\le 0.5$, it can be considered that the correlation between indicators is relatively low. When $RD\ge 0.5$ is present, it means that there is a high correlation between various indicators and there may be excess information ^[16]. The sensitivity test of the indicator system is shown in Eq. (3).

Following each alteration to a parameter, the evaluation outcomes of the indicator system must be recalculated. The final result is Eq. (4).

According to Eq. (4), when the absolute value of $SD$ is less than 5, the structure of the indicator system can be considered reasonable. The calculated value of $SD=1.4$, with an absolute value less than 5, indicates that relatively stable and reliable evaluation results can still be provided in different situations. Considering that $RD$ is 0.29, it indicates that the correlation between indicators in the indicator system is low and there is no excessive information redundancy. On the basis of the economic development data of 9 provinces from 2017 to 2021, data preprocessing work is carried out, as shown in Eq. (5).

The CRITIC is employed to determine the initial weights of each indicator and to establish the original indicator data matrix, as illustrated in Eq. (6).

In Eq. (6), $n$ is the evaluation object amounts. $m$ is the evaluation indicator numbers. $x_{kij} $ shows the indicator values. $k$, $i$, and $j$ respectively represent the year, evaluation object, and evaluation index, with values ranging from ($k=1$, $2$, $\ldots$, $y$), ($i=1$, $2$, $\ldots$, $n$), ($j=1$, $2$, $\ldots$, $m$). The evaluation indicators are shown in Eq. (7).

When the positive correlation between various indicators is stronger, it indicates that they may have strong common influence or duplicate information in the evaluation object, so the conflict between them is lower. The expressions for each indicator are shown in Eq. (8).

In Eq. (8), $S_{j} $, $R_{j} $, $C_{j} $, and $W_{j} $ respectively represent the strength of comparison, the conflict between indicators, the amount of information contained in the evaluation indicators, and the objective weight of the indicators. The positive ideal solution is Eq. (9).

In Eq. (9), $J_{1} $ represents a positive indicator and $J_{2} $ is a negative one. After the raw data is standardized, the range of values for all indicators is scaled to between 0 and 1. The ideal solution refers to the situation where the maximum value of 1 is achieved on all indicators, that is, all indicators have achieved the best performance. Anti-ideal solution refers to the situation where the minimum value of 0 is achieved on all indicators, that is, all indicators have achieved the worst performance ^[17].

In Eq. (10), $D_{i}^{+} $ and $D_{i}^{-} $ represent the distance from the evaluation object to the active and inactive ideal solution, $x^{+} $ and $x^{-} $. $W_{j} $ means the weight of the $j$ indicator. The relative closeness of each evaluation object is Eq. (11).

In Eq. (11), $C_{i}^{*} \in [0$, $1]$, the closer the calculated closeness value is to 1, the closer the calculated closeness value is to 1. Fig. 1 shows the structure diagram.

Fig. 1. BPNN structure.

In Back Propagation Neural Networks (BPNN), the node numbers in the input layer are usually the same as that of indicators in the DigE system. The determination of the nodes in the hidden is an important issue, and there is no fixed standard or formula to decide the nodes in the hidden. It is needed to try different node numbers, then to compare the performance of the model to select the appropriate quantity of nodes ^[18]. The formula is shown in Eq. (12).

In Eq. (13), $k$, $m$, and $n$ respectively represent the number of nodes in the hidden, input and output layers. Through trial and error, the final number of nodes in the hidden layer was ensured to be 13. Besides of determining the node amounts, the weight of evaluation indicators should also be calculated, as shown in Eq. (13).

In Eq. (14), $x=w_{jk} $, and the correlation index is shown in Eq. (14).

In Eq. (15), $y=r_{ij} $, the absolute influence coefficient is Eq. (15).

In Eq. (15), $S$ represents the weight. $i$, $j$, $k$ represent the input unit, output unit, and hidden unit of the NN, respectively, with a value range of ($i=1$, $2$, $\ldots$, $m$), ($j=1$, $2$, $\ldots$, $n$), ($k=1$, $2$, $\ldots$, $P$).

2.2. Construction of a GDES for Economic Level in the YR Built in NNs and TOPSIS

The construction of the economic level and green development evaluation system (GDES) in the YR area is grounded on the research of NN and TOPSIS methods. In Subsection 2.1, the study collects data related to economic level and green development (GD), including economic indicators and environmental factors. Then, the collected data are preprocessed, including data cleaning, missing value processing, and standardization, to make sure the consistency of the data. Next, the TOPSIS is adopted to evaluate the economic level and GD, and an NN is utilized to construct a model for the evaluation system. The evaluation system flowchart is Fig. 2.

Fig. 2. GDES of economic level along the YR.

Fig. 2 shows the flowchart of the economic level and GDES for the YR Basin, covering data collection, preprocessing, NN training, model validation, output generation, integration with TOPSIS methods, and interpretation and analysis of results. Data collection is the foundation for building an evaluation system, followed by data cleaning, missing value processing, and standardization to ensure quality. After preprocessing, the dataset is divided into a training set and a validation set for training and evaluating the NN. During the training process, the training set data are input to learn data patterns, and the model performance is evaluated through validation sets. The output of the NN reflects the evaluation score of economic level and GD, which is then integrated with the TOPSIS method for comprehensive evaluation. The Gini coefficient expression is Eq. (16).

In Eq. (16), $n$ is the quantity of regions participating in the evaluation. $y_{i} $ means the level of DigE and GD in the YR Basin, ranked from low to high, at the level of the $i$-th province and city. $\mu $ is the mean DigE level and GD in the YR Basin, used to measure the average level of the entire basin ^[19,20]. The kernel function generates a smooth curve at each sample point, and all these smooth curves add up to form an estimated probability density function. The expression is Eq. (17).

In Eq. (17), $N$ is the amount of research subjects. $x_{i} $ refers to independent and identically distributed observations. $x$ is the average DigE of 9 observation provinces, used to measure the average level of the entire YREB. $H$ represents the window width, which is a parameter in kernel density estimation that controls the width of the kernel function. The smoothness level $k(*)$ of the impact estimation represents the kernel function. This article adopts Gaussian kernel function, which is often used for smoothing in kernel density estimation ^[21,22]. The standard elliptical deviation is Eq. (18).

The standard deviation ellipse (SDE) has two half axes, i.e., the major-half and the minor-half. The major-half axis means the major distribution direction of geographical features, and its length reflects the degree of concentration of spatial movement of geographical features ^[23]. The minor-half axis refers to the spatial distribution range of geographical features, that is, the distance from the main distribution movement of geographical features. The expressions for the $X$-axis and $Y$-axis standard deviations of the SDE are shown in Eq. (19).

In Eq. (19), $(x_{i} ,y_{i} )$ represents each decision-making unit in the spatial area of the research object. $w_{i} $ and $i$ are the weight and the index or number of each decision unit, respectively. The horizontal and vertical distances of the decision units are $x$ and $y$ from the center point of the ellipse. $\theta $ is the angle formed by the clockwise rotation of the major axis of the SDE with due north. $\sigma _{x} $ and $\sigma _{y} $ are the SDEs in the $x$-axis and $y$-axis directions.

3.1. TOPSIS-based Model and Weight Allocation Analysis in the YR area

The research focuses on the DigE development of the YREB in China and its role in promoting green economic transformation, covering data from 2017 to 2021. The research covers nine provinces: Shandong, Henan, Sichuan, Shaanxi, Gansu, Inner Mongolia, Qinghai, Ningxia, and Shanxi. To ensure data consistency and accuracy, preprocessing steps including data cleaning, missing value processing, and standardization are carried out. The DigE indicator system is a set of key indicators taken to assess the DigE. These indicators usually cover aspects such as digitalization level, information technology application, e-commerce, digital industry, etc., to comprehensively reflect the development of the DigE. Table 1 shows the weights of digital environment in the YR area.

As shown in Table 1, the digital environment is of great significance in the YREB, especially with the proportion of mobile phone base stations and domain names reaching 7.07% and 6.59%, which is the most prominent in digital development. It is worth noting that the weight of the computers used per hundred people and the length of fiber optic cables is relatively low, reflecting that computers have become a necessity in people's daily life and office. Table 2 shows the weights of digital industrialization in the YR region.

As shown in Table 2, the added value of Information Transmission, Software, and Information Technology Services (IT-S-ITS) accounts for the highest proportion of industrial added value, reaching 3.61%. This indicates that accelerating the development of IT-S-ITS industries will effectively promote the growth of the digital industry. In terms of the weight distribution of the three-level indicators for digital industrialization, each indicator is relatively balanced, with a weight of around 2%. Table 3 shows the digital weights of products in the YR area

As shown in Table 3, there are varying degrees of differences in weight allocation in industrial digitization. The highest weight is 6.71%, corresponding to fixed asset investment in IT-S-ITS. In contrast, the lowest weight is only 0.04%, which corresponds to the number of websites owned by every hundred enterprises. The weights of other three level indicators range from 1% to 5%, indicating that industrial digital transformation requires balanced development in multiple aspects. Table 4 shows the weights of social digitization along the YR.

As shown in Table 4, the degree of government digitization is measured by an e-government index weight of 1.79%. In terms of digital finance, it includes the digitalization degree of inclusive finance, the breadth of digital finance coverage, and the depth of digital finance use. Among them, inclusive finance has the highest proportion of digitalization, reaching 3.81%, demonstrating its importance over other aspects. The three indicators of digital education have similar weights, all ranging from 1% to 2%. The data verify that the digital environment is the largest proportion of primary indicators, reaching 42.34%. Secondly, digital industrialization and industrial digitization rank second and third respectively, with weights of 28.87% and 16.87%, respectively. Therefore, in addition to continuously improving digital infrastructure, it is also required to focus on strengthening the application of digital industries. Although social digitization accounts for a relatively small proportion of 11.92% in the primary indicators, its importance cannot be ignored. Compared to the degree of industry digitization, individual digital applications are relatively limited, therefore the weight is lower.

3.2. Analysis of Spatio-temporal Differences Between the Level of DigE and GD in the YR region Based on NN and TOPSIS

To discuss the disparities in DigE and GD among different regions of the YREB, the TOPSIS is taken to calculate the evaluation indicators of DigE and GD in each region, and the corresponding comprehensive evaluation values are obtained. By comparing and analyzing the temporal data of evaluation indicators, the distinctions in the level of DigE and GD in different regions at different time points can be revealed. The trend of the DigE development is Fig. 3.

Fig. 3. Trends in the status of development of the DigE.

Fig. 4. Average level of GD.

There is a significant gap in the development condition of DigE in the YREB, with the highest value being 0.6818 and the lowest value being 0.1425. The comprehensive scores of the five provinces are between 0.15 and 0.35, indicating a relatively low overall DigE and slow growth. Most of the other provinces are above 0.3, with rapid growth. The highest level of DigE development in Shandong Province is 0.5712, followed closely by Sichuan and Shaanxi provinces, while the lowest level in Gansu Province is 0.2105. From a time perspective, the DigE in various regions has gradually grown since 2015, with a general upward trend except for a decline in 2018. Inner Mongolia has the slowest growth rate, only 37.57%, while Henan has the fastest growth rate, exceeding 1. The average level of GD is Fig. 4.

According to Fig. 4, there has been a significant gap in the GD level of the YREB over the past seven years, with the highest value being 0.7312 and the lowest value being 0.2441. However, the overall trend is on the growth. Shandong Province has the highest level of GD and DigE development, both at 0.5149, while Inner Mongolia, Gansu, and Qinghai have relatively high levels of GD. Overall, there is not much distinction in the level of GD among provinces. Table 5 shows the Gini coefficient of the DigE.

According to Table 5, the overall DigE gap in the YR region shows a slow downward trend, but the fluctuations between the gaps are also not low. The Gini coefficient reached its highest value in 2018, while it reached its lowest value in 2020. Compared to the benchmark in 2015, the Gini coefficient has decreased by 0.046% annually. In terms of month on month growth rate, 2018 showed the fastest growth, while 2020 saw the fastest decline rate, reaching 12.19%. The DigE's core density is Fig. 5.

Fig. 5. Core density of DigE.

From the observation of the nuclear density distribution curve, the gross level of DigE development in the YREB has been improving year by year, showing a trend of shifting to the right. The kurtosis analysis showed that from 2015 to 2021, the nuclear density curve changed from a ``tall and thin'' type to a ``short and chubby'' type, with a peak to the right and gradually dispersed intensity, indicating a continuous increase in the level of the DigE. In terms of shape, the development level of DigE in the YREB was basically in a bimodal state, with an overall trend of polarization. Specifically, for each province, Shandong, Henan, Sichuan, and Shaanxi were at a mid to high level, leading to a widening gap in the DigE level among each province year by year. The SDE and Center of Gravity (GC) shift of the DigE are shown in Fig. 6.

Fig. 6. DigE’s SDE and GC migration.

The overall level of DigE in the YREB has been increasing year by year from 2015 to 2021. From the perspective of GC transfer, the overall spatial distribution shifts towards the southwest direction, with a displacement distance of 87.808 kilometers. The fluctuation range in the north-south direction is relatively large, indicating an imbalance in the development of the DigE, while the south is relatively developed. Shandong, Shanxi, Shaanxi, and Henan provinces have relatively high levels of DigE and a large coverage area. However, Inner Mongolia, Qinghai, Ningxia, and Gansu provinces have lower levels of DigE and smaller coverage areas. The tendency of the standard deviation of the long and short axes shows an expansion trend in the north-south and a contraction trend in the east-west. Table 6 shows the Gini coefficient for GD.

Table 6 shows that the GD level gap in the YREB showed an obvious downward trend from 2015 to 2021. Among them, the Gini coefficient reached its highest value of 0.0782 in 2017, but also reached its lowest value of 0.0498 in the same year. Based on 2015, the Gini coefficient has decreased by 3.427% annually. The fastest month on month growth rate occurred in 2017, while in 2018, it showed the fastest decline rate, reaching 33.95%. Overall, the GD gap in the YREB fluctuated significantly from 2015 to 2021, showing an alternating trend of increase and decrease. The core density of GD is Fig. 7.

Fig. 7. DigE SDE and GC migration.

The changing trend of GD level in the YREB is observed in Fig. 7. The nuclear density distribution curve for the first seven years has moved to the right as a whole, indicating an overall increase in GD level year by year. The kurtosis analysis showed that from 2015 to 2021, the kurtosis of the nuclear density curve shifted to the right, and the intensity gradually dispersed. The area from the left to the middle decreased, while the trailing part on the right increased, indicating a continuous improvement in the level of the DigE. About the shape, the GD level in 2015, 2017, and 2020 showed a bimodal state, while in other years, it was basically a single peak and there was no obvious multi-peak trend.

Fig. 8 shows the changes and migration trajectories of GD levels in different periods. Between 2015 and 2021, the turning angle expanded from $71.688^\circ$ to $74.215^\circ$, but the overall change was not significant. The area ratio and average shape index of the SDE fluctuated generally, with a relatively small degree of variation. The azimuth deviation of the SDE of GD level was not significant, only rotating $2.527^\circ$ from northeast to southwest, with a smaller angle. The movement trajectory of the distribution GC indicated that the GD level of the YREB exhibited a predominant east-west directionality, with the distance between the north-south movement of the center being smaller than the east-west movement. The overall trend exhibited a westward to southward movement. Finally, the algorithm performance was tested to evaluate the performance of different algorithms in hypothetical MCDM problems, as listed in Table 7.

Fig. 8. GD's SDE and GC migration.

As shown in Table 7, the TOPSIS accuracy is 74.5%, the running time is 130 seconds, and the robustness score is 7.2. NN has the accuracy of 79.1%, running time of 270 seconds, and the robustness score of 6.5. TOPSIS-NN has the accuracy rate of 84.8%, running time of 390 seconds, and the robustness score of 8.1. The AHP's accuracy is 73.2%, the running time is 105 seconds, and the robustness score is 8.9. The consistency of rankings in TOPSIS is 82.3%, with NN for 84.7%, TOPSIS-NN for 90.4%, and AHP for 79.6%. The overall satisfaction rate in TOPSIS is 85.6%, while NN is 88.3%, TOPSIS-NN is 92.1%, and AHP is 82.8%. The combination method performs the best in accuracy, ranking consistency, and overall satisfaction, although it takes a long time to run. AHP has the shortest running time, but performs relatively poorly in other performance indicators.