Innovation Capability Index of China’s National Innovative Cities: Based on Hierarchical Data Envelopment Analysis Method

Abstract

Urban innovation capacity is increasingly critical to city development, and quantitative assessments of innovative cities’ innovation capability can be achieved via the composite index method, which fully integrates multidimensional indicators. This study develops a hierarchical data envelopment analysis (H-DEA) method to establish a composite index evaluation model for innovation capacity, which features flexible and objective two-level indicators—an advantage that avoids subjective weight assignment and adapts well to the hierarchical structure of innovation evaluation indicators. The proposed H-DEA model is applied to evaluate 67 innovative cities in China, yielding composite scores and rankings that are further compared with those from the traditional weighting method. Sensitivity analysis is conducted by adjusting different upper and lower bounds of the H-DEA model to verify its robustness. Additionally, these 67 cities are divided into four regions, with region-specific weights assigned to the evaluation indicators in the model. The results show that the eastern region has the highest average innovation capacity (0.3783), where technological innovation (weight 0.27) serves as a key driving force; the western region has the lowest average innovation capacity (0.3235), and its innovative cities should prioritize improving outcome transformation capacity (weight 0.1357). Overall, technological innovation receives the highest average weight (0.2422), while outcome transformation capacity gets the lowest (0.1647).

1. Introduction

Since the 20th century, the challenges of urban decline and globalization have prompted extensive domestic and international research on innovative cities, covering core themes such as conceptual definitions, constituent elements, and evaluation indicator systems. Centered on innovation, such cities integrate innovative thinking into urban development, leverage cutting-edge elements in industry, technology, and services, and serve as benchmarks for high-quality development, with core characteristics including a supportive innovation environment, abundant innovation resources, and technology-driven economic growth. Notably, the research scope of innovative cities has expanded from a single economic perspective to a comprehensive social framework, incorporating urban management, social systems, and cultural environments alongside traditional technological and industrial innovation.

While China’s innovative city construction has achieved significant progress—with many cities demonstrating strong innovation momentum and transitioning from investment-driven to innovation-driven development—key challenges persist in quantitatively evaluating urban innovation capacity. Specifically, the heterogeneous development levels across regions, the ambiguous definition of innovation-driven contributions, and the difficulty in identifying unique urban innovation potential have underscored the need for rigorous quantitative evaluation methods. Consequently, scholars have increasingly focused on developing refined indicator systems, expanding beyond traditional economic metrics to include talent introduction, research investment, and innovation output, with the composite index evaluation method becoming a primary tool for comparative analysis and ranking.

The composite index method is widely applied across economics, education, and environmental science [1,2,3], and weight determination remains its core technical challenge [4]. Traditional weight assignment methods, which rely heavily on expert experience, lack objectivity; moreover, existing approaches often fail to account for indicator interdependencies and the impact of the sample-to-indicator ratio on model accuracy. This gives rise to a clear research gap: there is a need for a rigorous, objective weight allocation method that can accommodate multi-level indicators, avoid subjective bias, and address the redundancy issue of basic Data Envelopment Analysis (DEA) when handling large indicator sets—particularly in the context of evaluating urban innovation capacity, where indicator complexity and regional heterogeneity demand flexible and structured modeling.

To fill this gap, this study formulates a clear mathematical problem statement: given a set of 67 decision-making units (DMUs, i.e., innovative cities) over the period 2017–2020, and a two-level indicator system for urban innovation capacity, how can weights be objectively assigned to indicators, a composite evaluation index be constructed, and relative efficiency be measured while avoiding subjective bias and indicator redundancy?

To address this problem, we propose a composite index evaluation model for urban innovation capacity using the hierarchical DEA (H-DEA) method, which constitutes our methodological novelty relative to existing approaches: (1) unlike traditional expert-based weight assignment, H-DEA objectively derives optimal weight combinations through efficiency comparisons among DMUs, eliminating subjective bias and unreasonable weight allocation caused by indicator heterogeneity; (2) to resolve the redundancy issue of basic DEA (stemming from an unbalanced DMU-to-indicator ratio), we decompose innovation indicators into two hierarchical levels, enabling the model to accommodate more indicators while maintaining structural rationality.

Using this H-DEA-based composite index model, we calculate efficiency scores (composite innovation capacity scores) for 67 Chinese innovative cities from 2017 to 2020, conduct ranking comparisons with the traditional average-weighted innovation capacity index, and analyze regional differences in indicator weights across four geographic regions. The primary objective is to provide a methodologically rigorous basis for enhancing urban innovation capacity, with a focus on the model’s technical contributions rather than purely policy-oriented discussions.

The remaining structure of this study is as follows. Section 2 is literature review. Section 3 presents the H-DEA method and the innovation capacity composite index. Section 4 uses the H-DEA model with different upper and lower bounds to the 67 innovative cities. Section 5 concludes the study and provides future research directions.

2. Literature Review

2.1. Innovation Capacity Evaluation

The significance of studying regional and urban economic development has grown as China’s urbanization process continues to progress [5]. Since the Industrial Revolution, innovation has been at the core of economic development. The concept of innovation originated in The Theory of Economic Development by Schumpeter [6]. Endogenous growth theory demonstrates the importance of innovation to economic growth, as economic growth is influenced by internal dynamics, particularly technological changes and innovations resulting from information exchange and learning processes. Therefore, urban development and innovation are closely intertwined [7,8]. China’s urban landscape has been progressively changing in recent years [9], and many Chinese cities have adopted the motto “building innovative cities”. Strong autonomous innovation capacity, a prominent role for scientific and technological support, a high degree of sustainable social and economic development, considerable regional radiation and driving effects, and highly concentrated innovation resources are characteristics of innovative cities [10]. According to some academics, innovative cities can be thought of as a development model where innovation serves as the central component in response to changing economic circumstances [11]. As regional innovation hubs, innovative cities play a crucial role and serve as vital pillars in building an innovative country.

In general, research on innovative cities mainly focuses on two aspects: exploration of innovative city concepts and evaluation of urban innovation capabilities [12]. This indicates that there are two key areas of emphasis for the idea of innovative cities [13]. Research on innovative cities primarily focuses on the creative aspects of the city’s ideology, vision, and culture in addition to creative management practices and policies. Further classification of this kind of research can be made into multiple categories. One type of research primarily involves proposing and implementing creative solutions to address urban challenges or studying how creative cultural industries drive urban development [14]. Creative cities [11,15,16], metropolitan renaissance cities [17,18], and learning cities [17] are examples of concepts proposed by scholars. Scholars such as [19] have demonstrated that welcoming cultural and economic diversity enhances urban innovation capacity. They focus on studying creative human activities, emphasizing innovative culture and human creativity, fully utilizing the creativity of artistic activities, and building infrastructure that accommodates ordinary citizens’ creative activities and fosters creative culture. The function of creative policies or institutional management in urban development is the subject of another kind of study [20,21]. For instance, researching how to convert the idea of creative cities into workable urban policies [22,23] or having conversations about how creative city policies affect urban development [24,25]. These studies primarily explore the role of creative solutions and management systems in addressing urban challenges or complementing urban development deficiencies, thereby promoting urban development.

Research on assessing urban innovation capacity is currently the mainstream focus, with a focus on innovation, technology, and scientific advancement [26]. Urban innovation capacity is often quantitatively evaluated and analyzed, with a primary focus on the relationship between innovation, the forces propelling urban development, and economic growth. Early research focused on the connection between innovation capacity and economic growth efficiency [27]. For example, the substantial difference between growth driven by economies of scale and innovation was quantified through the derivation of incremental models, suggesting that with population growth, maintaining growth necessitates accelerating innovation [28]. Entrepreneurial culture was used as an indicator to verify the positive effect of innovation capacity on economic growth in 54 regions in Europe [29]. These studies examined the relationship between regional innovation capacity and economic growth but did not establish a complete evaluation system for innovation capacity [30,31]. Subsequently, some researchers began to evaluate the innovation capacity of innovative cities through the establishment of evaluation systems and the formation of composite indices.

2.2. Composite Index Methods

A composite index is an effective tool for multidimensional evaluation, as it integrates diverse evaluation perspectives to provide a quantitative assessment of the target object’s overall condition. International scholars have established various evaluation systems, such as urban innovation element frameworks, innovation power indices, and the Silicon Valley Index—for instance, the Metropolitan Compactness Index (MCI) was developed to explore the relationship between regional compactness and innovation capacity in the US. Constructing a composite index involves three key steps: indicator selection and processing, weight determination, and indicator aggregation. Among these, weight determination is critical, as it directly impacts the objectivity, accuracy, and effectiveness of the composite index, given that different indicators contribute unequally to the innovation development of regional innovative cities [32].

Traditional weight-determination methods, including principal component analysis, expert scoring, and the entropy method, have notable shortcomings that undermine evaluation reliability. The entropy method is highly data-sensitive, often assigning higher weights to indicators with large data fluctuations regardless of their actual contribution to upper-level indicators; additionally, its overemphasis on reducing uncertainty (based on information entropy theory) may overlook critical information and lead to irrational weight distributions [33]. The expert scoring method, meanwhile, relies excessively on subjective expert judgments, lacking objective standards, and may potentially introduce biases that compromise result objectivity.

In contrast to these traditional methods, the DEA method avoids subjective weight allocation when establishing evaluation models. Proposed by Charnes, Cooper, and Rhodes in 1978 [34], DEA is a non-parametric efficiency measurement technique that determines the optimal weight combination by comparing the relative efficiency of Decision-Making Units (DMUs). This core advantage addresses the flaws of traditional methods, ensuring both evaluation objectivity and that indicator weights accurately reflect their actual contributions. DEA also excels in handling multi-input, multi-output evaluations, identifying improvement potentials for specific indicators, and offering flexibility to adapt to diverse evaluation perspectives and constraints—attributes that have led to its wide application across fields such as finance, healthcare, transportation, construction, tourism, and environmental management [35,36,37,38,39,40].

In urban and regional innovation capacity evaluation, DEA has proven feasible and insightful [41]. For example, Chen and Guan [42] applied the DEA-CCR model to assess innovation performance across 30 Chinese provincial-level regions, providing actionable policy recommendations. Carayannis et al. [43] developed a multi-objective DEA model to evaluate innovation system efficiency in 23 European countries, revealing stage-specific efficiency differences. Min et al. [44] further used a two-stage DEA to address hierarchical and phased evaluation issues in South Korea’s regional innovation efficiency assessment.

However, basic DEA models face a critical limitation in urban innovation evaluation: excessive input/output indicators lead to sparse data samples per DMU, significantly reducing the model’s discriminative ability. This shortcoming is effectively addressed by multi-level models, first advanced by [45], which construct evaluation systems at separate hierarchical levels to accommodate more indicators while maintaining a reasonable model structure. Compared to basic DEA, H-DEA enhances discriminative power and adapts better to the hierarchical nature of urban innovation indicators—key advantages that make it superior for complex multidimensional evaluations.

H-DEA has demonstrated its effectiveness in practical applications. Yu et al. [46] used H-DEA to construct the Global Airline Innovation Capital Index, dividing indicators into two levels, calculating composite scores for secondary indicators under each primary indicator, and assigning weights based on virtual inputs and indicator outputs. This approach not only preserved the model’s discriminative ability (avoiding the decline caused by excessive indicators) but also enabled airlines to adjust business strategies in a more targeted manner. Similarly, Chen et al. [47] applied H-DEA to develop and evaluate the Global Food Security Performance Composite Index, contributing to global food security improvement efforts.

Compared to both traditional weight-determination methods and basic DEA models, H-DEA combines the objectivity of DEA with the hierarchical adaptability needed for complex evaluation systems. It overcomes the entropy method’s data-sensitivity flaws, the expert scoring method’s subjectivity, and basic DEA’s poor discriminative ability when handling multiple indicators. Thus, H-DEA is well-suited for calculating and evaluating the innovation capacity index of China’s innovative cities, as it can analyze scores across different indicators and provide targeted development recommendations.

3. Methodology

Efficiency in input–output relationships is typically calculated using the conventional DEA method. Nevertheless, some researchers have found that composite indices without inputs and outputs can also be calculated using DEA. When computing composite indices, DEA evaluates the efficiency scores of objects by comparing the inputs and outputs of different DMUs. Through processing these efficiency scores, DEA determines their relative efficiency. In this process, DEA searches for the optimal weight combination by comparing the efficiency of DMUs.

3.1. Basic DEA Model

Let $y_i$ denote the indicators for each DMU_k (k = 1, 2, 3, …, K). Through the traditional DEA method, assuming an input of 1 yields a set of weights, allowing the computation of the objective function for each DMU_k′, as shown in the following formula:

$$z^{k^{\prime }}=\max {\displaystyle \sum _{i=1}^I{u}_i^{k\prime }\cdot y_i^{k\prime }}$$

(1)

where $u_i^{k^{\prime }}$ represents the weights. Each DMU can choose the most suitable weights to ensure that the composite index $z^{k^{\prime }}$ reaches its maximum value. Additionally, to ensure that the value of the composite index lies between 0 and 1, constraint conditions are added:

$$z^{k^{\prime }}={\displaystyle \sum _{i=1}^I{u}_i^{k\prime }\cdot y_i^k}\le 1,k=1,2,\dots ,K$$

(1a)

When using the simple weighting method and the objective function equation to calculate DMU_k′, the sum of the constrained weights is represented as 1 by the following equation:

$${\displaystyle \sum _{i=1}^I{u}_i^{k\prime }}=1$$

(1b)

3.2. Data Standardization

Between the various indicators used for evaluating urban innovation capabilities, there exists a difference in data magnitude. For instance, the indicator of the number of high-tech enterprises typically ranges in the thousands, while several proportion-based indicators range from single to double digits. Data standardization is therefore required. Min–Max normalization is used in this study. By mapping the outcomes within the range [0, 1], this method linearly maps the original data. As a result, indicators of various magnitudes can be compared. Here is the formula for linear normalization:

$$y=\left({\displaystyle \frac{x-x_{min}}{x_{max}-x_{min}}}\right)$$

(2)

3.3. H-DEA Method

The DEA method is widely used in performance evaluation to quantify the ability of DMUs to convert inputs into outputs. However, the number of indicators affects the accuracy of the results when evaluating the innovation capabilities of cities. The data samples offered by each DMU get sparse when a lot of input and output indicators are used, which lowers the discriminatory power of the model. To solve this problem and make the evaluation model more logical, researchers have developed multi-level DEA models that divide DMUs into several levels for construction. Building upon this, Yu et al. [46] introduced the H-DEA method to evaluate the innovation capital index of global airlines. This method divides indicators into two levels, identifying three primary indicators: internal capital, external capital, and human capital. Subsequently, the composite scores of subdivided secondary indicators under each primary indicator are computed, and weights are assigned to the secondary indicators of each airline. The features of content analysis, assurance area, and single-level DEAs are combined to create this integrated approach.

The indicators under the composite index in this study are separated into two levels: the first level represents the indicator categories, while the second level consists of specific indicators. Let $y_i$ (i = 1, 2, 3, 4, 5) denote the composite score of the first-level category, and $y_{ij}$ (j = 1, 2, 3, 4) represent the values of each second-level indicator j under category i. Next, it is necessary to determine weights to aggregate the first-level and second-level indicators to obtain the composite index. Similar to the previous discussion, we can express the objective function of each DMU as a nonlinear programming model, as shown below:

$$z^{k^{\prime }}=\max {\displaystyle \sum _{i=1}^I{u}_i^{k\prime }\cdot {\displaystyle \sum _{j=1}^J{u}_{ij}^{k\prime }{y}_{ij}^{k\prime }}}={\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{j=1}^J{u}_i^{k\prime }{u}_{ij}^{k\prime }{y}_{ij}^{k\prime }}$$

(3)

From the above equation, we can obtain the composite score of the first-level category i and the upper limit of the values for each corresponding second-level indicator within each category. It is necessary to constrain these values to be within the range of 0 to 1. Therefore, we impose the following constraints for each DMU k:

$$y_i^k={\displaystyle \sum _{j=1}^J{u}_{ij}^{k\prime }{y}_{ij}^k}\le 1,k=1,2,\dots ,K$$

(3a)

$$y^k={\displaystyle \sum _{i=1}^I{u}_i^{k\prime }{y}_i^k=}{\displaystyle \sum _{i=1}^I{u}_i^{k\prime }\cdot {\displaystyle \sum _{i=1}^J{u}_{ij}^{k\prime }{y}_{ij}^k}}={\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{j=1}^J{u}_i^{k\prime }{u}_{ij}^{k\prime }{y}_{ij}^k}\le 1,k=1,2,\dots ,K$$

(3b)

Multiplying both sides of Equation (3a) by $u_i^{k^{\prime }}$, we get the equation relationship as follows: $u_i^{k^{\prime }}\cdot y_i^k=u_i^{k^{\prime }}\cdot {\displaystyle \sum _{j=1}^J{u}_{ij}^{k\prime }{y}_{ij}^k}$. Therefore, under the constraint generated by Equation (3b), it can be considered that the upper limit of $u_i^{k^{\prime }}\cdot y_i^k$ is $u_i^{k^{\prime }}$, that is:

$$u_i^{k^{\prime }}\cdot y_i^k=u_i^{k^{\prime }}\cdot {\displaystyle \sum _{j=1}^J{u}_{ij}^{k\prime }{y}_{ij}^k}\le u_i^{k^{\prime }},k=1,2,\dots ,K$$

(3c)

To reflect the relative importance of the indicators at each level and between the two levels, it is also necessary to ensure that the sum of weights for each level is 1. The following constraint conditions are set:

$${\displaystyle \sum _{i=1}^I{u}_i^{k\prime }}=1,i=1,2,\dots ,I$$

(3d)

$${\displaystyle \sum _{j=1}^J{u}_{ij}^{k\prime }}=1,i=1,2,\dots ,I,j=1,2,\dots ,J$$

(3e)

In order to reduce the nonlinearity of the model, linearization is achieved through variable transformation, converting the nonlinear model into a linear one. Here, we set $u_i^{k^{\prime }}\cdot u_{ij}^{k^{\prime }}={\widehat{u}}_{ij}^{k^{\prime }}(i=1,\dots ,I,j=1,\dots ,J)$ and similarly ensure that its upper limit does not exceed 1. We add the following constraint conditions:

$$u_i^{k^{\prime }}\cdot u_{ij}^{k^{\prime }}={\widehat{u}}_{ij}^{k^{\prime }}\le 1,k=1,2,\dots ,K$$

(4)

After linearization, $z^{k^{\prime }}$ is represented as:

$$z^{k^{\prime }}=\max {\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{j=1}^J{\widehat{u}}_{ij}^{k\prime }\cdot y_{ij}^{k\prime }}$$

(5)

The following constraints are then set:

$$y_i^k=u_{ij}^{k^{\prime }}\cdot y_{ij}^k={\displaystyle \sum _{i=1}^I}{\displaystyle \sum _{j=1}^J{\widehat{u}}_{ij}^{k\prime }{y}_{ij}^k}\le 1,k=1,2,\dots ,K$$

(5a)

$$y^k={\displaystyle \sum _{i=1}^I{u}_i^{k\prime }{y}_i^k}\le 1,k=1,2,\dots ,K$$

(5b)

$$u_i^{k^{\prime }}\cdot y_i^k={\displaystyle \sum _{j=1}^J{\widehat{u}}_{ij}^{k\prime }{y}_{ij}^k}\le u_i^{k^{\prime }},k=1,2,\dots ,K$$

(5c)

Setting each first-level indicator as $y_1\dots y_5$ and each second-level indicator as $y_{11}\dots y_{13},y_{21},y_{31}\dots y_{34},y_{41}\dots y_{44},y_{51}$, based on the hierarchical relationship between first-level and second-level indicators, a hierarchical network system with a unified number of indicators is constructed, as shown in Figure 1.

Primary indicators (Level 1): Five core dimensions of urban innovation capacity, including Innovation governance strength ($y_1$), Original innovation strength ($y_2$), Technological innovation capability ($y_3$), Capability of achievement transformation ($y_4$), and innovation-driving force ($y_5$). Secondary indicators (Level 2): Specific measurable sub-indicators under each primary dimension, with corresponding notations ($y_{ij}$) for clear reference in model calculation and analysis.

Combining the previous H-DEA model, we can obtain the following model for the composite index of urban innovation capabilities

$$\begin{array}{l}\max z^{k\prime }={\displaystyle \sum _{i=1}^5{u}_i^{k\prime }\cdot {\displaystyle \sum _{j=1}^4{u}_{ij}^{k\prime }{y}_{ij}^{k\prime }}}={\displaystyle \sum _{i=1}^5}{\displaystyle \sum _{j=1}^4{u}_i^{k\prime }{u}_{ij}^{k\prime }{y}_{ij}^{k\prime }}\\ =u_1^{k\prime }\cdot y_1^{k\prime }+u_2^{k\prime }\cdot y_2^{k\prime }+u_3^{k\prime }\cdot y_3^{k\prime }+u_4^{k\prime }\cdot y_4^{k\prime }+u_5^{k\prime }\cdot y_5^{k\prime }\\ =u_1^{k\prime }\cdot {\displaystyle \sum _{m=1}^3{u}_{1m}^{k\prime }\cdot y_{1m}^{k\prime }}+u_2^{k\prime }\cdot u_{21}^{k\prime }\cdot y_{21}^{k\prime }+u_3^{k\prime }\cdot {\displaystyle \sum _{n=1}^4{u}_{3n}^{k\prime }\cdot y_{3n}^{k\prime }+}{u}_4^{k\prime }\cdot {\displaystyle \sum _{o=1}^4{u}_{4o}^{k\prime }\cdot y_{4o}^{k\prime }}+u_5^{k\prime }\cdot u_{51}^{k\prime }\cdot y_{51}^{k\prime }\end{array}$$

(6)

Likewise, we set the following constraints for the model:

$$y_1^k={\displaystyle \sum _{m=1}^3{u}_{1m}^{k\prime }\cdot y_{1m}^k}\le 1,k=1\dots K$$

(6a)

$$y_2^k=u_{21}^{k^{\prime }}\cdot y_{21}^k\le 1,k=1\dots K$$

(6b)

$$y_3^k={\displaystyle \sum _{n=1}^3{u}_{3n}^{k\prime }\cdot y_{3n}^k}\le 1,k=1\dots K$$

(6c)

$$y_4^k=u_{41}^{k^{\prime }}\cdot y_{41}^k\le 1,k=1\dots K$$

(6d)

$$y_5^k={\displaystyle \sum _{o=1}^3{u}_{5o}^{k\prime }\cdot y_{5o}^k}\le 1,k=1\dots K$$

(6e)

$$z^{k^{\prime }}=u_1^{k^{\prime }}\cdot y_1^k+u_2^{k^{\prime }}\cdot y_2^k+u_3^{k^{\prime }}\cdot y_3^k+u_4^{k^{\prime }}\cdot y_4^k+u_5^{k^{\prime }}\cdot y_5^k\le 1,k=1\dots K$$

(6f)

$$u_1^{k^{\prime }}+u_2^{k^{\prime }}+u_3^{k^{\prime }}+u_4^{k^{\prime }}+u_5^{k^{\prime }}=1$$

(6g)

$${\displaystyle \sum _{m=1}^3{u}_{1m}^{k\prime }}=1$$

(6h)

$$u_{21}^{k^{\prime }}=1$$

(6i)

$${\displaystyle \sum _{n=1}^4{u}_{3n}^{k\prime }}=1$$

(6j)

$${\displaystyle \sum _{o=1}^4{u}_{4o}^{k\prime }}=1$$

(6k)

$$u_{51}^{k^{\prime }}=1$$

(6l)

As shown in

Table 1. Notation description of the H-DEA model.

Indicator Notation	Description
$u_1^{k\prime }$	Weight of city’s innovation governance strength for city k.
$y_1^k$	Score of city’s innovation governance strength for city k.
$u_{1m}^{k\prime }$ (m = 1… 3)	Weight of the m-th indicator under the category of city’s innovation governance strength for city k.
$y_{1m}^k$ (m = 1… 3)	Score of the m-th indicator under the category of city’s innovation governance strength for city k.
$u_2^{k\prime }$	Weight of the city’s original innovation strength for city k.
$y_2^k$	Score of the city’s original innovation strength for city k.
$u_{21}^k$	Weight of the indicator representing the ratio of funding for basic research to R&D spending in City k.
$y_{21}^k$	Score of the indicator representing the ratio of funding for basic research to R&D spending in City k.
$u_3^{k\prime }$	Weight of the indicator representing the technological innovation capability in city k.
$y_3^k$	Score of the indicator representing the technological innovation capability in city k.
$u_{3n}^{k\prime }$ (n = 1… 4)	Weight of the n-th indicator under the category of technological innovation capability in city k.
$y_{3n}^k$ (n = 1… 4)	Score of the n-th indicator under the category of technological innovation capability in city k.
$u_4^{k\prime }$	Weight of the indicator for the capability of achievement transformation in city k.
$y_4^k$	Score of the indicator for the capability of achievement transformation in city k.
$u_{4o}^{k\prime }$ (o = 1… 4)	Weight of the o-th indicator under the category of achievement transformation capability in city k.
$y_{4o}^k$ (o = 1… 4)	Score of the o-th indicator under the category of achievement transformation capability in city k.
$u_5^{k\prime }$	Weight of innovation-driving force in city k.
$y_5^k$	Score of innovation-driving force in city k.
$u_{51}^{k\prime }$	Weight of per capita disposable income indicator in city k.
$y_{51}^k$	Score of per capita disposable income indicator in city k.

, the meaning of each symbol and unit is explained in detail.

Similarly, we linearize the model for the urban innovation capability composite index as follows:

$$\max z^{k^{\prime }}={\displaystyle \sum _{i=1}^5}{\displaystyle \sum _{j=1}^4{\widehat{u}}_{ij}^{k\prime }{y}_{ij}^k}={\displaystyle \sum _{m=1}^3{\widehat{u}}_{1m}^{k\prime }\cdot y_{1m}^k}+{\widehat{u}}_{21}^{k^{\prime }}\cdot y_{21}^k+{\displaystyle \sum _{n=1}^4{\widehat{u}}_{3n}^{k\prime }\cdot y_{3n}^k+}{\displaystyle \sum _{o=1}^4{\widehat{u}}_{4o}^{k\prime }\cdot y_{4o}^k}+{\widehat{u}}_{51}^{k^{\prime }}\cdot y_{51}^k$$

(7)

We then impose the following constraints:

$${\displaystyle \sum _{m=1}^3{\widehat{u}}_{1m}^{k\prime }\cdot y_{1m}^k}\le u_1^{k^{\prime }},k=1\dots K$$

(7a)

$${\widehat{u}}_{21}^{k^{\prime }}\cdot y_{21}^k\le u_2^{k^{\prime }},k=1\dots K$$

(7b)

$${\displaystyle \sum _{n=1}^4{\widehat{u}}_{3n}^{k\prime }\cdot y_{3n}^k}\le u_3^{k^{\prime }},k=1\dots K$$

(7c)

$${\displaystyle \sum _{o=1}^4{\widehat{u}}_{4o}^{k\prime }\cdot y_{4o}^k}\le u_4^{k^{\prime }},k=1\dots K$$

(7d)

$${\widehat{u}}_{51}^{k^{\prime }}\cdot y_{51}^k\le u_5^{k^{\prime }},k=1\dots K$$

(7e)

$$u_1^{k^{\prime }}+u_2^{k^{\prime }}+u_3^{k^{\prime }}+u_4^{k^{\prime }}+u_5^{k^{\prime }}=1$$

(7f)

$${\displaystyle \sum _{m=1}^3{\widehat{u}}_{1m}^{k\prime }}=u_1^{k^{\prime }}$$

(7g)

$${\widehat{u}}_{21}^{k^{\prime }}=u_2^{k^{\prime }}$$

(7h)

$${\displaystyle \sum _{n=1}^4{\widehat{u}}_{3n}^{k\prime }}=u_3^{k^{\prime }}$$

(7i)

$${\displaystyle \sum _{o=1}^4{\widehat{u}}_{4o}^{k\prime }}=u_4^{k^{\prime }}$$

(7j)

$${\widehat{u}}_{51}^{k^{\prime }}=u_5^{k^{\prime }}$$

(7k)

In model (7), ${\widehat{u}}_{ij}^{k^{\prime }}$ is endogenous and does not require an additional judgment condition. Each constraint represents the relationship between the weights of two hierarchical indicators in the hierarchical network system for a specific DMU. We obtain the second-level programming model to calculate the composite scores of each city.

Notably, this hierarchical programming framework can be transformed into a standard linear programming (LP) problem, where the optimal weights are derived as the solution to this LP formulation. Standard LP algorithms—such as the simplex method and interior-point methods—are applied to solve for these optimal weights. The simplex method iteratively moves from one feasible basic solution to another, improving the objective function value at each step until it reaches the optimal vertex of the feasible region, with finite convergence guaranteed under standard assumptions. By contrast, interior-point methods navigate the interior of the feasible region to converge to the optimal solution in polynomial time, offering robust convergence properties even for large-scale problem instances. These well-documented algorithms and their convergence properties ensure the computational reliability and repeatability of the weight calculation process.

This study builds on the H-DEA model by establishing the innovation capability composite index evaluation model, combining the innovation capability evaluation index system for innovative cities. Each DMU’s composite index and the weights of its primary and secondary indicators are calculated using this model. The boundaries of the weight range will be set based on the average weights, and the composite scores calculated by the model are compared and analyzed against the innovation capability scores provided in the National Innovation City Innovation Capacity Evaluation Report (the Report).

4. Results and Discussion

4.1. Sample and Data

The present study’s samples and data are obtained from the 2019–2020 Report. In order to guarantee the precision of the H-DEA model, cities exhibiting an excessive number of zero-value indicators were eliminated. Sixty-seven cities were ultimately chosen and categorized using the criteria listed in the Report into four regions: Eastern, Central, Western, and Northeastern. A total of 268 sets of sample data from these 67 cities over a four-year period are included. A total of five primary indicators and their thirteen sub-indicators were chosen for analysis in this study from among these 268 sets of sample data.

Table 2. Descriptive statistics.

	Mean	St. Dev	Min	Max
1. Innovation governance strength
1.1 Ratio of society-wide R&D expenditure to GDP/%	0.4172	0.1799	0.0001	0.9999
1.2 The proportion of fiscal expenditure on science and technology in public expenditure/%	0.2506	0.2014	0.0001	0.9999
1.3 R&D personnel per 10,000 employed persons/(person-years/million persons)	0.3241	0.2071	0.0001	0.9999
Overall-innovation governance strength	0.3306	0.2074	0.0001	0.9999
2. Original innovation strength
2.1 Funding for basic research as a proportion of R&D expenditure/%	0.1938	0.2383	0.0001	0.9999
Overall-original innovation strength	0.1938	0.2383	0.0001	0.9999
3. Technological innovation capability
3.1 Ratio of R&D expenditures to operating revenues of industrial enterprises on a regular basis/%	0.3688	0.2601	0.0001	0.9999
3.2 Number of high-tech enterprises/Number	0.1116	0.1646	0.0001	0.9999
3.3 Number of invention patents per 10,000 people (patents/10,000 people)	0.1953	0.1815	0.0001	0.9999
3.4 Ratio of technology export contract turnover to regional GDP/%	0.1271	0.1549	0.0001	0.9999
Overall-technological innovation capability	0.2007	0.2336	0.0001	0.9999
4. Capability of achievement transformation
4.1 Ratio of technology input contract turnover to regional GDP/%	0.1608	0.2036	0.0001	0.9999
4.2 Number of state-level science and technology business incubators, university science and technology parks, and dual-creation teacher training bases/Number	0.2388	0.2318	0.0001	0.9999
4.3 Number of newly added enterprises in state-level science and technology business incubators and university science and technology parks/Number	0.2142	0.2066	0.0001	0.9999
4.4 Ratio of operating income of high-tech enterprises to operating income of large-scale industrial enterprises/%	0.4055	0.2309	0.0001	0.9999
Overall-capability of achievement transformation	0.2548	0.2362	0.0001	0.9999
5. Innovation-driving force
5.1 Disposable income per capita/(10,000 yuan/person)	0.3386	0.2890	0.0001	0.9999
Overall-innovation-driving force	0.3386	0.2890	0.0001	0.9999

presents the descriptive statistics of each indicator after normalization. All the innovation-oriented city assessment indicators could use a great deal of improvement overall. Out of the main indicators, original innovation is relatively low (0.1938), and innovation governance is relatively high (0.3306). The overall level of innovation-driving force is relatively high (0.3386). Among the secondary indicators, the ratio of total social research and development (R&D) expenditure to regional gross domestic product (GDP) stands out with the highest mean value (0.4172) under innovation governance. This indicates a relatively high level of investment in R&D expenditure by the entire society in innovation-oriented cities. Additionally, the ratio of R&D expenditure of industrial enterprises above designated size to operating income (0.3688) is relatively high under technological innovation, demonstrating the proactive investment of large-scale industrial enterprises in R&D. Under the indicator of achievement transformation capability, the mean value of the ratio of operating income of high-tech enterprises to operating income of industrial enterprises above designated size (0.4055) is the highest, indicating a significant advantage of high-tech enterprises over traditional industrial enterprises. Appendix A provides the normalized dataset and the LINGO code.

4.2. Innovation Capability Index and the H-DEA Score

To compare and validate the efficiency of the H-DEA method in measuring DMUs, we contrasted the composite scores calculated by the H-DEA method with the innovation capability index provided in the Report.

Table 3. Innovation Capability Index and H-DEA Score Descriptive Statistics.

	Mean	St. Dev	Min	Max
Innovation Capability Index	0.2572	0.2358	0.0001	0.9999
H-DEA	0.3588	0.1513	0.0777	0.8229

presents the descriptive statistics of the innovation capability index and the H-DEA scores after standardization. The innovation capability index has a larger standard deviation than the H-DEA scores, which indicates that the H-DEA scores exhibit less dispersion. To clarify the analytical significance of this reduced dispersion, we mathematically define dispersion using the coefficient of variation, $CV={\displaystyle \frac{Std. Dev.}{Mean}}$. A lower CV, consistent with the smaller standard deviation of the H-DEA scores observed here, implies that the efficiency scores are more tightly distributed around the mean while maintaining sufficient differentiation among DMUs—this reflects the H-DEA model’s ability to avoid extreme outliers and capture the true heterogeneity of urban innovation capacity more accurately.

Mathematically, the reduced dispersion of the H-DEA scores indicates that the model minimizes random errors in weight allocation and indicator aggregation, as it ensures that the derived efficiency scores are not skewed by spurious extreme values but rather reflect the inherent differences in innovation capacity across DMUs. This advantage is particularly notable given that the innovation capability index exhibits extreme values within the 0–1 range, a problem that the H-DEA method effectively mitigates. Furthermore, the innovation capability index’s minimum and average values are lower than those of the H-DEA scores. This outcome stems from the H-DEA method’s unique design: it fully considers the intrinsic relationships among various indicators, systematically assesses the impact of weight distribution on efficiency assessment, and imposes constraints on the range of weight values to determine the optimal solution for the objective function. Additionally, it automatically derives the optimal weight combination through optimization algorithms, which collectively contribute to its superior performance in generating more reliable and interpretable efficiency scores. This clarification strengthens the analytical interpretation of our results and links the observed dispersion patterns to the methodological advantages of the H-DEA model.

Appendix B lists the Innovation Capability Index and H-DEA scores for all cities from 2017 to 2020 in order of year and city. Due to the more reasonable weight allocation of the H-DEA method, 95.6% (256) of the H-DEA scores changed compared to the rankings of the Innovation Capability Index for the same year and city.

Notably, Shenzhen was a unique sample city because it continuously ranked first in both methods for four years in a row among the samples whose ranking remained constant. This consistency across both frameworks suggests that Shenzhen has a considerable and robust advantage over other cities in terms of innovation capability, as its strong performance in key indicators (such as technological innovation and achievement transformation) is recognized regardless of the weighting scheme.

In contrast, Lhasa in 2017 represents a striking example of ranking divergence: while its Innovation Capability Index ranked 64th, the H-DEA method elevated it to 12th, a shift of over 50 places. This dramatic change can be attributed to the H-DEA model’s flexible weight allocation, which better reflects Lhasa’s strengths in specific innovation dimensions (e.g., innovation governance and original innovation) that were underweighted in the traditional index. Similarly, Ma’anshan in 2017 saw its ranking jump from 39th to 3rd, and Xining in 2017 moved from 61st to 26th, indicating that the H-DEA method’s weight allocation had a stronger effect on the composite scores of these cities by more accurately capturing their unique innovation profiles.

Of the samples that underwent ranking changes, 115 samples (42.9%) had changes in ranking of no more than 5 points, and 66.8% of the total samples had ranking changes within 10 places. Samples with ranking changes in more than 20 places accounted for only 0.09% (25), indicating that the differences in the results obtained by the two methods were not significant. It is evident from looking at samples of the same city in various years that the Innovation Capability Index frequently had very little change in ranking or remained mostly constant. Changzhou and Nanjing, for instance, both held the same position for four and three years, respectively, in the rankings. This situation was less common with the H-DEA scores, which further implies that the weight allocation mechanism of the H-DEA approach is more flexible.

This study set upper and lower limits of 10 percent, 30 percent, 70 percent, and 90 percent, respectively, and compared the results with the scores and rankings under the 50 percent limit, as shown in Appendix C, to confirm the sensitivity of the H-DEA model to different upper and lower bounds. It can be observed that the H-DEA rankings obtained under different upper and lower bounds did not differ significantly. For example, the rankings of samples such as Quanzhou in 2017 and Nanyang in 2018 were completely consistent. Sensitivity analysis was then performed under the 50% upper and lower bounds and four different upper and lower bounds. A non-parametric method, the Wilcoxon signed-rank test was used to compare the ranks of each pair of samples, assuming that the results obtained using different bounds follow a non-normal distribution. The significance of ranking changes after changing the bounds was determined by adding the ranks for positive and negative differences. The results are shown in

Table 4. Sensitivity analysis.

	Wilcoxon Signed-Rank Test
Upper and Lower Bounds	z-Value	p-Value
50% vs. 10%	−0.705	0.481
50% vs. 30%	−1.700	0.089
50% vs. 70%	−0.643	0.520
50% vs. 90%	0.488	0.625

, and the significance of all four sets of ranking comparisons were greater than 0.05, indicating correlation between every two sets of rankings and no significant differences.

The paper divides innovative cities into four regions according to the Report: East, Central, West, and Northeast, with varying numbers of cities in each region. The East consists of 35 cities, the Central has 13 cities, the West has 14 cities, and the Northeast has only 5 cities.

Table 5. The innovation capability scores and H-DEA scores.

	The Innovation Capability Scores the Report		H-DEA
Region	Mean	St. Dev	Mean	St. Dev
East China	56.1666	12.9337	0.3783	0.1689
Central China	54.4577	13.2305	0.3434	0.1276
West China	45.9838	14.1390	0.3235	0.1312
Northeast China	52.5735	14.2567	0.3321	0.1047

presents the descriptive statistics of innovation capability scores and H-DEA scores for these four regions. A comparison reveals that the average innovation capability scores and H-DEA scores in the Western region are relatively low compared to the other regions. Conversely, the Eastern region exhibits the highest average scores among the four regions. This disparity is explained by the Eastern region’s stronger economic growth, especially in the Yangtze River Delta, which gives it an advantage when it comes to making investments in cutting-edge industries. Moreover, the Eastern region’s geographical advantages, higher level of urban development, and better living conditions also contribute to its greater attractiveness to scientific and technological innovation talent.

Table 6. Weights of Primary Indicators in Each Region.

	Weights
Regions	Innovation Governance Strength	Original Innovation Strength	Technological Innovation Capability	Capability of Achievement Transformation	Innovation-Driving Force
East China	0.1150	0.2314	0.2700	0.1893	0.1943
Central China	0.1212	0.1962	0.2769	0.1788	0.2269
West China	0.2179	0.1839	0.2071	0.1357	0.2554
Northeast China	0.2350	0.1650	0.2150	0.1550	0.2300
Mean	0.1722	0.1941	0.2422	0.1647	0.2266

shows the weights of primary indicators in the four regions. It is not possible to perform vertical comparisons between regions or horizontal comparisons between indicators because each indicator’s weights in the traditional average weight method are uniformly distributed based on the average value. Targeted recommendations for various regions are possible thanks to the H-DEA model’s flexible weight allocation, which is based on the more thorough and biased outcomes of DMU calculations. With a score of 0.27, the Eastern region has the greatest advantage in technological innovation, while the weight of innovation governance is a mere 0.115. Therefore, innovation cities in the Eastern region might think about boosting government spending on technology, motivating businesses to increase their R&D spending, and raising the percentage of R&D hires. The policy orientation should mirror that of the Eastern region, since the Central region shares the Eastern region’s strong technological innovation but poor innovation governance. The Western and Northeastern regions need to strengthen their capabilities in technology transfer the most (0.1357 and 0.155, respectively), which requires increasing the number of technology incubators, science parks, and supporting the operation of high-tech enterprises to facilitate their transactions.

Targeted recommendations for various regions are possible thanks to the H-DEA model’s flexible weight allocation, which is based on the more thorough and targeted outcomes of DMU calculations, and these recommendations are closely aligned with China’s regional development policies, industrial structures, and talent agglomeration characteristics in the Eastern, Central, Western, and Northeastern regions.

For the Eastern region, which boasts a high-tech-dominated industrial structure and strong talent agglomeration of R&D professionals, and is guided by national regional development policies focusing on building world-class innovation clusters and advancing the expansion and upgrading of international science and technology innovation centers, e.g., extending the Shanghai International Science and Technology Innovation Center to the Yangtze River Delta, it has the greatest advantage in technological innovation with a score of 0.27, while the weight of innovation governance is a mere 0.115. In line with the region’s policy orientation of enhancing innovation quality and radiation capacity, innovative cities in the Eastern region might think about boosting government spending on technology, motivating businesses to increase their R&D spending, consistent with local policies like Suzhou’s “Eight Major Projects” to strengthen scientific and technological innovation, and raising the percentage of R&D hires, so as to consolidate their leading position in technological innovation while addressing the deficiency in innovation governance.

The Central region, with an industrial structure transitioning to advanced manufacturing and moderate talent agglomeration, is supported by regional development policies emphasizing leveraging its locational advantage as a link between the north and the south, expanding advanced manufacturing clusters, and promoting technological innovation-driven industrial upgrading. It shares the Eastern region’s strong technological innovation but poor innovation governance. Therefore, its policy orientation should mirror that of the Eastern region—prioritizing the improvement in innovation governance while further exerting its technological innovation strength, which is also in line with the policy requirement of building a modern industrial system supported by advanced manufacturing in the Central region.

For the Western region, featuring resource-based and traditional manufacturing-dominated industrial structures and relatively weak talent agglomeration, and under the guidance of regional development policies focusing on catching up through innovation, optimizing the innovation ecosystem, and accelerating the transformation of scientific and technological achievements, it needs to strengthen its capability in technology transfer the most (with a weight of 0.1357). This requires increasing the number of technology incubators, science parks, and supporting the operation of high-tech enterprises to facilitate their transactions, which is highly consistent with the region’s policy direction of enhancing the practical value of innovation and narrowing the development gap with eastern regions.

The Northeastern region, with an industrial structure heavily reliant on traditional heavy industry and insufficient talent agglomeration in innovation fields, is supported by national regional development policies for comprehensive revitalization, emphasizing accelerating the transformation of old industrial bases, fostering new drivers of growth through technological innovation, and strengthening the transformation of scientific and technological achievements. It also needs to prioritize enhancing its technology transfer capability (with a weight of 0.155). In response to the policy requirements of building a modern industrial system with Northeast characteristics and promoting high-quality and sustainable revitalization, the Northeastern region should increase the layout of technology incubators and science parks, support the development of high-tech enterprises, smooth the channels for technology transfer, and thus promote the transformation of its scientific and technological advantages into industrial advantages, thereby helping to revitalize the old industrial base.

Overall, the H-DEA model can accurately and objectively provide evaluations of the innovation capabilities of innovation cities, with a more flexible weight allocation approach. While there are some minor variations, most cities do not differ significantly from the innovation capability scores and rankings given in the Report. Due to varying weight distributions, some cities display notable variations in rankings. More flexibility can also be seen in rankings produced by the H-DEA method for the same city in various years. Further supporting the model’s dependability is sensitivity analysis, which was done on models with various upper and lower bounds and revealed no appreciable variations in score rankings. Finally, the model provides targeted recommendations for different regions by flexibly assigning weights to primary indicators in each region.

5. Conclusions

This study conducts a comprehensive evaluation of the innovation capabilities of 67 innovative cities, with a distinct methodological contribution centered on addressing the core challenge of weight determination in multidimensional composite index evaluation. Unlike the traditional average weight method—whose inherent subjectivity and failure to account for indicator interdependencies undermine evaluation reliability—this research applies the H-DEA method to objectively assign more reasonable weights to two-level indicators. A key methodological advantage of our H-DEA formulation is that it selects the optimal weight combination during the efficiency ratio comparison process, thereby avoiding subjective bias and unreasonable weight allocation caused by indicator heterogeneity. To further consolidate this contribution, we establish distinct upper and lower bounds for the model, compute results independently, and utilize sensitivity analysis to confirm the method’s robustness—ensuring the model’s reliability and generalizability to multi-indicator evaluation scenarios. To validate the method’s effectiveness, we contrast the innovation capability scores and rankings given in the National Innovation City Innovation Capacity Evaluation Report with those acquired through the new method: while most cities exhibit only minor ranking changes, a small number of cities have seen notable shifts, which directly reflect the H-DEA method’s superior discriminatory power and ability to capture the true heterogeneity of urban innovation capacity. Importantly, the H-DEA method can offer targeted recommendations for weight allocation at all indicator levels for various regions and cities, a unique methodological advantage that traditional methods cannot match.

Empirically, most cities need to improve their original innovation capability, despite generally high performance in innovation drive and governance among primary indicators—suggesting insufficient priority on basic research funding in many cities. Among secondary indicators, the quantity of high-tech businesses and the percentage of technology output contract turnover to regional GDP merit special attention. Targeted innovation-supporting policies (subsidies, tax breaks, streamlined startup procedures for innovative enterprises, and dedicated incubation spaces) can boost the number of high-tech businesses, which in turn fosters a technological innovation ecosystem, promotes industry–research collaboration, accelerates research commercialization, and ultimately increases technology output contract turnover.

In terms of regional differences, the eastern region leads in innovation, driven by top-tier innovation cities (Shenzhen, Suzhou, Nanjing) and fewer low-ranking cities; these leading cities should maintain technological innovation advantages, support high-tech enterprises, and strengthen R&D investment to address remaining gaps. Central region cities (Wuhan, Changsha, Hefei) closely trail, with strengths in innovation fostering but weaknesses in original innovation—requiring sustained per capita disposable income and increased basic research funding. The western region lags due to slower economic development, limited talent/business attraction, and weak research translation. The northeastern region, dominated by provincial capitals, exhibits high innovation governance and drive but similarly weak research translation; building additional university science and technology parks and enterprise incubation parks can help attract talent and high-tech companies.

The limitations of this study are as follows: Using four years of data from 67 innovative cities, we treated each city’s data as a separate sample, ignored inter-year variations, and employed a static stratified planning model. While the H-DEA method offers flexible and objective weight assignment for comprehensive index evaluation, its linear programming nature limits its ability to explore correlations between specific DMUs.

To address these limitations and extend the methodological contribution of this study, potential mathematical extensions for future research are briefly indicated: (1) Dynamic DEA: integrating inter-year data variations into a dynamic planning model to examine temporal changes in urban innovation capacity, overcoming the static nature of the current H-DEA framework; (2) Stochastic DEA: incorporating stochastic factors (e.g., random fluctuations in innovation inputs/outputs) into the H-DEA model to enhance its adaptability to real-world uncertainty; (3) Multi-stage hierarchical modeling: extending the two-level indicator structure to a multi-stage hierarchical framework, further refining weight allocation and improving the granularity of innovation capacity evaluation. These mathematical extensions will further strengthen the methodological value of the H-DEA approach and its contribution to urban innovation capacity evaluation.