1 Introduction

The performance evaluation of state-owned enterprises is a core issue in the modern economic system, and it has received widespread attention from the business and accounting communities (Perrini and Tencati 2006). With the globalization of the world economy, tremendous changes have taken place in the business environment of enterprises, and competition among enterprises has become more intense. The original performance evaluation system of state-owned enterprises has been difficult to adapt to this rapidly changing, uncertain and risky competitive environment (Valiente et al. 2012). For this reason, if an enterprise wants to maintain its competitive advantage in the fierce competition, it needs to establish an evaluation system that can quickly, comprehensively and objectively evaluate the performance of state-owned enterprises (Joshi et al. 2003). In this way, an enterprise can measure the degree of achievement of its established goals and the degree of contribution of various departments, teams, and individuals within the organization, judge the effectiveness of the organization's various activities in achieving organizational goals, and thereby improve the performance of state-owned enterprises (Rami 2016). The government can better formulate and implement relevant industrial policies, monetary policies, tax policies and social security policies. Better implement classified guidance to enterprises, support and grasp a number of superior enterprises and enterprise groups, so as to enhance the country's comprehensive economic strength (Sheu and Lo 2005). The public can then make better specific decisions on the flow of personal consumption and capital investment. Financial institutions can then determine the scale and strategy of credit to enterprises. The owner of the enterprise can determine the specific flow of capital and the reward and punishment measures to be imposed on the enterprise operator.

In countries like China, the current state-owned enterprise performance evaluation system is mainly based on the financial efficiency of the enterprise, i.e., profit maximization as the guide to evaluate the performance of the enterprise (Dixon and Whittaker 1999). This has a large short-term behavior and cannot provide a good guide for the future development of the enterprise. Now, with the gradual maturity of China’s market economy, the level of internationalization of corporate management has been significantly improved, and traditional corporate management concepts are being transformed into strategic management concepts. As one of the modern corporate management methods, the state-owned enterprise performance evaluation system is also in urgent need compatible with corporate strategic management (Guo-Liang et al. 2011). The corporate performance evaluation method that only focuses on short-term benefits has weakened the guidance of the company's long-term sustainable development capabilities and is facing the challenge of rapid economic development and corporate management methods (Liu 1995). Both theory and practice urgently require us to improve and perfect our country's current corporate performance evaluation system in order to improve the management level of enterprises, enhance their market competitiveness, and adapt to the needs of China’s economic and social development and international trends. Therefore, this article starts from the actual situation of Chinese enterprises and draws on the research results at home and abroad and introduces the fuzzy neural network method into the field of performance evaluation of state-owned enterprises for research (Du et al. 2013; Hong and Kim 2017).

The contributions of the proposed research are as follows;

  • To propose construction ideas and construction principles for the state-owned enterprise performance evaluation index system, and finally established the enterprise performance evaluation index system examined in this article.

  • The Takagi-Sugeno fuzzy neural network is used to build a financial indicator model and the BP neural network is used to build a non-financial indicator model as part of the construction of a state-owned enterprise performance evaluation portfolio model based on fuzzy neural networks.

  • To compare between those results and those from the BP neural network performance evaluation model after sample data have been selected, standardization processing has been applied, and a simulation of the network model has been trained.

The paper is structured as: Sect. 2 discusses the related work to the proposed research. The method of the proposed research study is mentioned in Sect. 3. The experiments and discussions are given in Sect. 4. The paper conclusion is given in Sect. 5.

2 Related work

The details of the literature are categorized into the following:

  • Balanced Score Card Method American management scientists Robert. S. K and David. P. Norton introduced the balanced scorecard method (Fullerton and Mcwatters 2002; Kuwaiti and Kay 2000; Shaeffer 1996; Neely et al. 1995) through several companies that were in a leading position in performance evaluation at the time. The best solution advocated that business managers observe their company from the most important indicators such as customers, innovation and learning, internal business processes, and financial indicators. The balanced scorecard method not only considers the company's results, but also considers the company's potential for future development. Then, we evaluate the company's operation level from both the customer and the business, fully integrate the company's long-term goals and short-term actions, and turn the company into an excellent system, so that performance evaluation can fully participate in the final goal decision.

  • Guanjian Performance Index Method The essence of the key performance indicator method (Gargeya 2005) depends on clearly setting out the configuration of corporate performance indicators, which need to be linked to the company’s future decision-making with the main problem to be solved strategically. However, some foreign companies have obtained the evaluation of the individual performance of the employees and then summarized some major indexes that can better reveal the performance of the employees and use this as a method to evaluate the performance. The effect of KPI's implementation in the development strategy of foreign companies has begun to be significant. Using its purpose to establish a certain system, the development strategy will be transformed into the company's internal activities, and it has indeed promoted the company's process of stably maintaining high profits in the market competition.

  • Management by objectives From the goal management method (Dixon et al. 1990), the performance appraisal method has stabilized. This method originated from the goal management method of the management master Peter F. Drucker. In order to ensure the smooth development of an enterprise, human resource management must be done: The order of building goals must be precise and rigorous, so that the method can be successfully implemented and achieved. This method focuses on planning rather than comments; focusing on communication rather than imagination; focusing on tomorrow rather than yesterday. The personnel department must reposition the target performance and pay close attention to what the managers and employees are after. To this regard performance appraisal as a means to ensure the victory of goals and then realize that the essence of goal management is actually a process of mutual discussion among people. The real purpose of management by objectives is to mobilize all forces to mobilize the enthusiasm of employees, improve the company's own labor productivity, and enable entire staff to unite and successfully move toward the organizational goals.

  • Multi-evaluator method There are numerous methods for performance appraisal. Most of these companies use the multi-evaluator method (or 360-degree performance appraisal) to assess the performance of their employees (Bassioni et al. 2004). To bring employees as fair as possible evaluation results, managers maximize the integration of all relevant aspects of the dynamics, including bosses, colleagues, subordinates, customers, etc. Therefore, this method fundamentally describes that managers can effectively extract employee dynamics and improve employee loyalty. Although there are many unrecognized controversies about 360-degree performance appraisal (Luu et al. 2008; Bond 1999; Zhu et al. 2012; Pan and Qian 2010).

3 Method

3.1 Construction of performance evaluation index system

The performance evaluation of state-owned enterprises refers to the evaluation subject's scientific and standardized evaluation methods. We use a specific evaluation index system and unified evaluation standards to evaluate the business benefits of the enterprise in a specific operation period through specific procedures and quantitative and qualitative comparative analysis. Make an objective, fair, accurate, comprehensive evaluation based on the operators' performance. The enterprise performance evaluation index system is a collection of interrelated and interdependent indexes that make a thorough evaluation of the enterprise's overall business performance and operational efficiency. Through this indicator system, investors can predict the future operating performance of different companies to obtain maximum benefits. Creditors can determine the loan and receipt of their funds to ensure the safety of capital and the maximization of interest income. Business operators can obtain the operating performance and development trends of themselves and their competitors and provide a basis for formulating competitive strategies. Therefore, to evaluate the performance of state-owned enterprises, we must first build a scientific and reasonable evaluation index system.

3.1.1 Thoughts on constructing performance evaluation index system

The construction of the evaluation index system in this article is mainly considered from the following aspects. (1) Take input and output as the starting point. For an enterprise to survive and develop, it must achieve the goal of earning more than it costs. Under the conditions of a market economy, the operating activities of an enterprise are the operating process of enterprise capital, and the objectives of enterprise operations are embodied in capital preservation and capital appreciation. The relationship between income and expenses is transformed into the relationship between the realization of capital value and capital advances. The pursuit of maximum value-added capital is the production purpose of the enterprise. Pay attention to quantitative indicators and combine qualitative indicators in the input–output analysis to make up for the shortcomings of quantitative analysis. (2) Attach importance to establishing additional evaluation indicators and handling coordination among various indicators. At present, cash flow status plays an essential role in corporate performance, so evaluation indicators related to cash flow should be appropriately increased. This article considers the addition of cash flow indicators from the aspects of the company's operating ability, debt solvency, and income quality. It uses the sales cash ratio, the main business income-to-revenue ratio, and the cash flow debt ratio to make up for accounting shortcomings. (3) Improve evaluation and analysis methods and standards. The complete corporate performance evaluation system is a multi-level and diversified system. The traditional method builds an evaluation system with multiple weights based on different evaluation subjects. As a result, the setting of indicator weights is more cumbersome, and the weight settings are different for other evaluation subjects. This article uses the artificial intelligence method so that the evaluation subject does not need to care about the weight setting of each evaluation index of different industries and companies when evaluating the performance of the enterprise. These are automatically learned and revised according to the sample data during the network training process.

3.1.2 Determination of performance evaluation index system

According to the construction ideas and principles of corporate performance evaluation indicators and with the help of specific methods, the final indicator system includes financial indicators and non-financial indicators. Financial indicators are mainly constructed from four aspects: corporate profitability, asset operation ability, debt repayment ability, and development ability. Non-financial indicators are mainly considered from the four elements of business operators, employees, customers, and corporate contributions to society. A combination of several factors determines each aspect.

3.2 Performance evaluation combination based on fuzzy neural network

Because the fuzzy rules increase exponentially with the increase of the input dimension of the fuzzy neural network, this inevitably increases the structural complexity and training time of the fuzzy neural network, reduces the performance and generalization ability of the network, and affects work efficiency. Therefore, to simplify the network structure and improve the network training speed and accuracy, this paper proposes a combined model based on a fuzzy neural network. Finally, the financial index model uses the learning results of the above two models to build a company performance evaluation model based on Takagi Sugeno fuzzy neural network. The construction process is described as follows: According to the enterprise performance evaluation indicator system determined in the previous section, combined with previous research experience and expert opinions, select various indicator variable combinations that can fully reflect enterprise performance, including n samples, and each sample contains p indicators—sample preprocessing: including sample data cleaning and sample data standardization. Sample data cleaning is to remove incomplete data and data that does not meet the requirements so that the remaining samples are all samples with attribute values. Standardization of sample data is to transform each indicator value to the same level. Select the appropriate combined model structure and training algorithm and use the combined model to learn and train the preprocessed sample data. Use the BP neural network to learn and train the selected corporate financial indicators, use the fuzzy neural network to learn and train-related corporate non-financial indicators, select the appropriate membership function and adjust the relevant parameters and build the final fuzzy neural network network-based corporate performance evaluation model.

3.2.1 BP neural network to construct financial index model

BP neural network is a multilayer feedforward neural network with hidden layers. It is currently the most representative and most widely used network model. However, because the input quantities processed by BP neural network are all accurate values, the processing errors for fuzzy and inaccurate quantities are relatively large. Here, we only use it to process financial index data and construct a three-layer network structure financial index model. The network structure is shown in Fig. 1.

Fig. 1
figure 1

Financial index BP neural network structure

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Among them, \(n\) and \(m\), respectively, represent the number of input layer neurons and hidden layer neurons. According to Kolmogorov's theorem, "the number of hidden layer neurons is twice the number of input layer neurons plus 1". Combined with the empirical formula and the method of experimentation, the value of \(m\) is finally determined. The learning of the BP neural network is trained by the gradient descent method, which minimizes the error function and attributes the error of the network output to the fault of the network connection weight. The error of the output layer unit is propagated back to the input layer to apportion it to the neurons of each layer, thereby adjusting the corresponding connection weight of each layer until the error value of the network output is the smallest.

3.2.2 Fuzzy neural network model based on Takagi-Sugeno

Takagi-Sugeno type fuzzy neural network is a fuzzy neural network based on Takagi-Sugeno type rules. In this fuzzy neural network, the consequence of the fuzzy rules is a function of the input language variables, usually a linear combination of the input variables. The most widely used fuzzy neural network is the self-organizing fuzzy inference system (ANFIS). It was first proposed by Jang and uses a network to divide the input space. The fuzzy rules are in the form of Takagi-Sugeno. The network is trained using a hybrid learning algorithm that alternates between the BP algorithm and the least square algorithm (LSE). This paper uses this network model to construct a non-financial indicator model and a corporate performance evaluation model. The network model structure based on Takagi-Sugeno is shown in Fig. 2.

Fig. 2
figure 2

Takagi-Sugeno type fuzzy neural network structure

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The model is comprised of two sub-networks, the antecedent network and the subsequent network. The antecedent network is used to match the antecedents of fuzzy rules and produces a corresponding degree of applicability for each fuzzy rule. The subsequent network is used to implement the subsequent fuzzy rules. The final output of the network is equal to the weighted sum of all the subsequent elements of the rules, and the weighting coefficient is the applicability of each fuzzy rule output by the antecedent network. From the overall function, the model is equivalent to a fuzzy system; from the topological structure, the model is a neural network composed of nodes and weights. Therefore, the fuzzy neural network model has the advantages of fuzzy system and neural network at the same time and has strong mapping and adaptive ability.

The antecedent network consists of four layers. The first layer is the input layer and each node of it is directly connected with each component \({x}_{i}\) of the input vector, and it plays the role of directly transmitting the input value \(x={({x}_{1},{x}_{2},\dots ,{x}_{n})}^{T}\) to the next layer. The number of nodes in this layer is \({N}_{1}=n\). The second layer is the blurring layer and each node represents the value of a language variable, and 4 language variables are selected in this article. The function of this layer is to calculate the membership degree of each input component belonging to each fuzzy set of linguistic values:

$$ \mu_{i}^{j} = \mu_{{A_{i}^{j} }} \left( {x_{i} } \right), i = 1,2, \ldots ,n. i = 1,2, \ldots ,m_{i} , m_{i} = 4 $$
(1)

\(n\) is the dimension of the input, and the number of fuzzy divisions of \({x}_{i}\) is 4. In this paper, the membership function adopts Gaussian function:

$$ \mu_{i}^{j} = e^{{ - \left( {x_{i} - c_{ij} } \right)^{2} /\sigma_{ij}^{2} }} $$
(2)

\(c_{ij}\) and \(\sigma_{ij}\) represent the center and width of the membership function, respectively. The total number of nodes in this layer is:

$$ N_{2} = 4n $$
(3)

The third layer is the rule reasoning layer and ach node in this layer represents a fuzzy rule, and its function is to match the antecedents of the fuzzy rule and calculate the applicability of each rule, which is:

$$ \alpha_{j} = \mu_{1}^{{i_{1} }} \mu_{2}^{{i_{2} }} \ldots \mu_{n}^{{i_{n} }} $$
(4)

The total number of nodes in this layer \({N}_{3}=m\). For a given input, only those linguistic variable values near the input point have a greater degree of membership, and the linguistic variable values far away from the input point have either small or zero membership. When the degree of membership is very small (such as less than 0.05), it can be approximately taken as 0. The fourth layer is the normalization layer. The number of nodes in this layer is the same as that of the third layer, namely \({N}_{4}=m\), which implements normalized calculation, which is:

$$ \overline{\alpha }_{j} = {{\alpha_{j} } \mathord{\left/ {\vphantom {{\alpha_{j} } {\sum\limits_{i = 1}^{m} {\alpha_{i} } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{i = 1}^{m} {\alpha_{i} } }} $$
(5)

The first layer of the subsequent network is the input layer, which transmits input variables to the second layer. The input value \({x}_{0}=1\) of the 0th node in the input layer, its function is to provide the constant term in the subsequent part of the fuzzy rule. There are m nodes in the second layer, and each node represents a rule. The function of this layer is to calculate the consequence of each rule, namely:

$$ y_{1j} = p_{j0}^{1} + p_{j1}^{1} x_{1} + p_{j2}^{1} x_{2} + \cdots + p_{jn}^{1} x_{m} $$
(6)

The third layer is the output layer, and the output value is the evaluation result value:

$$ y_{1} = \sum\limits_{j = 1}^{m} {y_{1j} \overline{\alpha }_{j} } $$
(7)

\({y}_{1}\) is the weighted sum of the consequence of each rule, and the weighting coefficient is the normalized applicability of each fuzzy rule, that is, the output of the antecedent network is used as the connection weight of the third layer of the subsequent network. ANFIS based on Takagi-Sugeno provides a learning algorithm for fuzzy modeling that can extract the corresponding fuzzy rules from the data set. It is a hybrid learning algorithm. It is completed in two steps in one iteration. The first step is to fix the antecedents. Parameters, when the input information is passed along the network to the 4th layer, the least square algorithm is used to adjust the subsequent parameters. After the adjustment, the information continues to pass along the network to the 5th layer (output layer). The second step is to fix the subsequent parameters, propagate the obtained error signal back along the network, and use the BP algorithm to adjust the antecedent parameters. In this way, the parameters of the front and rear parts are adjusted iteratively. After several trainings, the network can achieve high accuracy.

Suppose the network has \(L\) layers, and the number of nodes in the kth layer is \(\#(k)\). \((k,i)\) represents the \(i\)-th node of the \(k\)-th layer, and its node function or node output is denoted as \({O}_{i}^{k}\), which is the function of the combined parameter set of the input signal, which can be denoted as:

$$ O_{i}^{k} = O_{i}^{k} \left( {O_{1}^{k - 1} , \ldots ,O_{{\# \left( {k - 1} \right)}}^{k - 1} } \right) $$
(8)

The given training data set has \(N\) groups, and the error measurement function of the \(p\)-th group of sample data is defined as:

$$ E_{p} = \sum\limits_{m = 1}^{\# \left( L \right)} {\left( {T_{m,p} - O_{m,p}^{L} } \right)^{2} } $$
(9)

where \({T}_{m,p}\) represents the target output value of the \(m\)-th element of the \(p\)-th group of samples, and \({O}_{m,p}^{L}\) represents the actual output value of the \(m\)-th element of the \(p\)-th group of samples. The total error of the network is:

$$ E = \sum\limits_{p = 1}^{N} {E_{p} } $$
(10)

Reuse gradient descent method to update network weights.

Adjust the parameters of the subsequent parts. Assuming that the vector of unknown linear parameters is \(\theta\), the matrix equation can be obtained from the known \(P\) group of sample data:

$$ A\theta = y $$
(11)

Then, the optimal solution of \({\Vert A\theta -y\Vert }^{2}\) to minimize \(\theta \) is the least squares estimator \(^{*}\):

$$ \theta^{*} = \left( {A^{T} A} \right)^{ - 1} A^{T} y $$
(12)

where \({A}^{T}\) is the transpose of \(A\), if \({A}^{T}A\) is not singular, then \({({A}^{T}A)}^{-1}{A}^{T}\) is the pseudo-inverse of \(A\). Due to the large amount of calculation for matrix transposition, at this time, recursive LSE can be used to calculate \({\theta }^{*}\).

4 Experiment and discussions

4.1 Application of BP neural network financial index model

The BP neural network financial indicator model constructed in this paper is a three-layer network structure. The number of neurons in the input layer (the number of financial indicators determined in this paper) is 23, and the number of neurons in the output layer is 1. (The output is the financial performance of the enterprise.) The number of neurons in the hidden layer is determined according to the Kolmogorov theorem mentioned in Chapter 3, combined with the empirical formula \(m={(n+1)}^{1/2}+\alpha \) (where \(m\) is the number of neurons in the hidden layer, \(n\) is the number of neurons in the input layer, and 1 is the number of neurons in the output layer. \(\alpha \) is a constant between 1 and 10) and trial and error method; then, the number of hidden layer neurons training selection is shown in Fig. 3.

Fig. 3
figure 3

Hidden layer neuron number selection

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It is not difficult to find that when the number of neurons in the hidden layer is 14, the output error of the network reaches the minimum at this time. Combined with the complexity of the network structure, when the number of neurons in the hidden layer increases, the complexity of the network structure increases, and the evaluation time will increase accordingly. Therefore, the study finally determined that the number of neurons in the hidden layer was 14, and the network structure of the approximate optimal network was determined: 23-14-1. Finally, the trained BP neural network is used to evaluate the test sample data, and the selected test results are shown in Fig. 4. The mean square error of the test data is 0.0414. It can be seen that the training effect of the network is better, and you can enter the next step of practical application.

Fig. 4
figure 4

Partial sample test results

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4.2 Performance evaluation model of state-owned enterprise

The study uses a fuzzy neural network to build a non-financial indicator model. The model structure consists of seven non-financial indicators as input and one output. The fuzzification layer consists of 28 neurons. In the training process of the above-mentioned fuzzy neural network, the research calls the command function anfis provided in the MATLAB fuzzy toolbox, which can adaptively construct fuzzy rules through the given sample data and adjust the parameters in the membership function. The result value obtained by the above model training is used as the input of the enterprise performance evaluation model, and the number of inputs of the evaluation model is 2 (respectively, the financial status of the enterprise and the non-financial status of the enterprise). The number of output is 1, which is the result of enterprise performance evaluation. The fuzzification layer of the network contains 8 neurons, and the number of fuzzy rules is 16. Mean square error curve of the training data is shown in Fig. 5.

Fig. 5
figure 5

MSE curve of training data

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The mean square error curve of the training data of the enterprise performance evaluation model is relatively smooth, with the mean square error reaching 7.8678e-005, and the network training effect is good. Finally, we input 10 sets of test sample data into the trained fuzzy neural network model to evaluate corporate performance. The expected output of the test data and the actual output of the network model are shown in Fig. 6.

Fig. 6
figure 6

Expected output and actual output of network model check data

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The study divides the performance of enterprises into four grades, and the performance grades of enterprises evaluated by the test sample data through the network are shown in Table 1.

Table 1 Partial inspection sample evaluation grade
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In order to test the final evaluation grade results of the network, here, an evaluation performance index is proposed; the accuracy of the network evaluation result \(\sigma \).

$$ \sigma = H/S $$
(13)

where \(H\) represents the number of network output evaluation results that match the actual performance of the enterprise. \(S\) represents the total number of companies participating in the evaluation. According to the network evaluation results of part of the test samples in Table 1, we get the accuracy of the network evaluation results \(\sigma =0.8\). May wish to analyze the reasons for the deviation between the network output evaluation level and the actual performance level of the company. The error between the network model’s output evaluation value for some errors and the level boundary value that we have artificially delineated the company is very small, and it is basically numerical. It can be determined that the evaluation value output by the network is consistent with the actual performance, so that the accuracy of the network evaluation result can reach \(\sigma =0.95\) after revision.

4.3 Model comparison and analysis

To compare and illustrate the evaluation effect of the combined model proposed in this research, a corporate performance evaluation model based on BP neural network is constructed. The number of neurons in the input layer of the model structure is 30, the total number of financial indicators and non-financial indicators determined in this article, and the number of neurons in the output layer is 1, and the method for determining the number of neurons in the hidden layer is similar to the previous description. To repeat, the final determined number of hidden layer neurons is 16, and the mean square error of the network output is 0.0442 at this time. The accuracy of the network evaluation result of the BP neural network model is \(\sigma =0.75\). Excluding the influence of artificial performance grade factors, the accuracy of the network evaluation results of the BP network model is \(\sigma =0.85\). The performance comparison of the above two evaluation models is shown in Table 2.

Table 2 Performance comparison between BP and combination model
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We can easily get that the accuracy of the network evaluation results of the combination model of enterprise performance evaluation based on the fuzzy neural network proposed in this research is significantly higher than that of the BP neural network model. The final network output error of the combined model is smaller than the output error of the BP network, and the output accuracy is significantly improved. Through the above comprehensive analysis, it can be obtained that the state-owned enterprise performance evaluation combination model based on fuzzy neural network is effective and feasible and is better than the enterprise performance evaluation model based on BP neural network. Therefore, the introduction of the fuzzy neural network combination model in this paper provides new research ideas for enterprise performance evaluation and then enriches and perfects the theories and methods of state-owned enterprise performance evaluation.

5 Conclusion

As an important method and means of modern enterprise management, performance evaluation of state-owned enterprises has attracted widespread attention in the business and theoretical circles. For state-owned enterprises to have a strong competitive advantage in the context of economic globalization, it must make a true evaluation of their own performance. This is of great significance to business operators, owners, and other stakeholders of the business. Based on the existing research on performance evaluation of enterprises, this research summarizes the performance evaluation methods of state-owned enterprises and points out some of the defects. The fuzzy neural network method is introduced into the field of performance evaluation of state-owned enterprises, and the self-learning ability of fuzzy theory and neural network is used to minimize the influence of human subjective factors in the acquisition of non-financial data and the setting of indicator weights. Aiming at the problem that the number of fuzzy rules of the fuzzy neural network increases exponentially with the increase in the number of input evaluation indicators. A combination model of performance evaluation of state-owned enterprises based on fuzzy neural network is established, and the BP neural network is used to build a financial indicator model. The network constructs a non-financial index model and finally uses the two results as input to construct the final fuzzy neural network state-owned enterprise performance evaluation model. The empirical results of the study show that the method is effective and feasible.