445 Advances in Production Engineering & Management ISSN 1854-6250 Volume 20 | Number 4 | December 2025 | pp 445–457 Journal home: apem-journal.org https://doi.org/10.14743/apem2025.4.551 Original scientific paper Measuring the coordinated development of the advanced manufacturing cluster based on patent data: A composite system approach Zhang, H.K. a , Yang, J.N. a , Buchmeister, B. b , Ojstersek, R. b,* a Business School, Beijing Technology and Business University, Beijing, P.R. China b Faculty of Mechanical Engineering, University of Maribor, Maribor, Slovenia A B S T R A C T A R T I C L E I N F O The advanced manufacturing industry serves as a key driver of high-quality economic development, with patent data widely used to assess regional tech- nological innovation and synergy. Evaluating coordination within advanced in- dustrial clusters offers practical insights into industrial upgrading. Focusing on the Beijing–Tianjin–Hebei advanced manufacturing cluster in China, the com- posite system is divided into four subsystems: quantity, quality, efficiency, and value. Using patent data from 2013 to 2023, a composite system synergy model was built to measure the degree of synergy. A coupling coordination degree model and an obstacle degree model were applied together to examine coordi- nation states and limiting factors. The findings show a generally rising synergy degree in the cluster, indicating increased organizational coherence over time. Coupling coordination displayed short-term fluctuations but exhibited a posi- tive long-term trend, achieving high-quality coupling by 2023. Collaborative development is dynamically influenced by multidimensional obstacles, calling for timely and tailored measures to improve synergy efficiency. This study of- fers an empirical basis for optimizing collaboration in the Beijing–Tianjin–He- bei cluster and provides a theoretical reference for understanding synergy mechanisms in similar industrial clusters. Keywords: Advanced manufacturing cluster; Patent data; Composite system synergy; Coupling coordination degree; Obstacle degree; Regional industrial coordination; Innovation synergy *Corresponding author: robert.ojstersek@um.si (Ojstersek, R.) Article history: Received 23 October 2025 Revised 29 November 2025 Accepted 3 December 2025 Content from this work may be used under the terms of the Creative Commons Attribution 4.0 International Licence (CC BY 4.0). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction Driven by breakthroughs and integration in information technology, automation, and related fields, the manufacturing industry is undergoing a profound shift toward intelligent, green, and service-oriented development [1]. To secure future industrial leadership, countries have launched supportive policies such as the U.S. “re-industrialization” strategy, Germany’s “Industry 4.0”, and China’s “Made in China 2025”. A central aim is to cultivate globally competitive advanced manu- facturing clusters that integrate innovative resources and enhance overall industrial efficiency [2,3]. However, differences in resource endowments and development stages among entities within a cluster often lead to insufficient systemic coordination, limiting further competitiveness gains [4,5]. Scientifically diagnosing the synergy status and identifying key obstacles thus be- comes essential to promote high-quality cluster development [6]. Zhang, Yang, Buchmeister, Ojstersek 446 Advances in Production Engineering & Management 20(4) 2025 In order to objectively describe the synergy level of the cluster system, this study introduces patent data as the core observation index. As the key carrier of the legalization and marketization of technological achievements, patent has become an important indicator for measuring the pro- gress of technological innovation and the advanced manufacturing industry because of its objec- tivity and quantifiable characteristics, which provides a scientific perspective for insight into the collaborative innovation efficiency of regional industrial clusters. Against the global backdrop of patent-driven innovation, the Beijing–Tianjin–Hebei advanced manufacturing cluster is examined as a composite system. By constructing a synergy measure- ment model, the development synchronization and synergy degree of the composite system are quantitatively evaluated [7]. This approach helps reveal dynamic patterns and key obstacles in the cluster’s coordinated development, offering both empirical support for strategic decision- making in the Beijing–Tianjin–Hebei region and a theoretical and practical reference for similar industrial clusters worldwide to optimize resource allocation and improve collaborative innova- tion efficiency. 2. Literature review 2.1 Coordinated development approach In terms of measuring the overall synergy of the system, there are many methodological paths, mainly including the DEA method, distance synergy model, Haken model, composite system syn- ergy model, coupling coordination degree, among others. Khare et al. [8] used the DEA model to quantitatively evaluate the synergistic relationship between bus rapid transit system and land use under 16 public transit-oriented development models in Bhopal, India. Li et al. [9] used the distance synergy model to quantitatively analyse the panel data of Beijing- Tianjin-Hebei from 1995 to 2014, reflecting the regional coordinated development. Luo and He [10] combined the similarity measure of the Vague set with the distance collaborative model, used the grey correlation analysis to calculate the grey correlation degree between the supply and de- mand subsystems, and established a comprehensive collaborative development model of regional high-tech industry technology supply and demand composite system based on Vague set distance and grey correlation. Ouyang and Yang [11] constructed a new measurement method of regional coordinated devel- opment based on the Haken model and simultaneously constructed an index system to examine the situation of regional coordinated development in China from five dimensions. Based on the panel data of 13 cities from 2003 to 2020, Deng et al. [12] established an evaluation index system for high-quality economic development of Beijing–Tianjin–Hebei urban agglomeration, and used the spatial Durbin model to examine the impact of collaborative innovation on high-quality eco- nomic development. Leydesdorff and Fritsch [13] used the three sub-powers of economic exchange, technological innovation and institutional control to construct a triple helix model to measure the level of col- laborative innovation in German manufacturing. Fang et al. [14] measured the performance level of coordinated development in a certain region, and found that the level of coordinated develop- ment in the region showed a slow growth trend over time. Zhao and Zhang [15] used the composite system synergy model to measure the ecological syn- ergy level of Beijing, Tianjin and Hebei, and found that the level of ecological synergy development in Beijing, Tianjin and Hebei increased year by year, but the development of subsystems within the region was uneven. Sun et al. [16] used the order degree model and the composite system synergy model to measure both the subsystem order degree and the composite system synergy degree of CSHDME, based on the data of 11 coastal provinces in China from 2010 to 2020. Based on the coordination degree model of composite system, Shen et al. [17] constructed the coordination degree model of aviation logistics industry in Beijing–Tianjin–Hebei region, and an- alysed the ordered variables reflecting the development of aviation logistics industry in the region from 2005 to 2014, and dynamically depicted its collaborative evolution process. Ren et al. [18] constructed a four-level comprehensive evaluation index system based on the DPSIR framework, Measuring the coordinated development of the advanced manufacturing cluster based on patent data: A composite … Advances in Production Engineering & Management 20(4) 2025 447 used the entropy weight method to determine the index weight, and used the coupling coordina- tion degree model to analyse the coordination relationship between the systems. Xu and Shi [19] constructed an evaluation index system including four major industrial sub- systems such as network economy, and used the coupling coordination degree model to statically measure the level of collaborative innovation of strategic emerging industries in China and prov- inces from 2012 to 2022. Zheng and Gu [20] divided the Beijing–Tianjin–Hebei carbon emission reduction system into four subsystems such as policy support. Based on the synergy degree of the composite system, the coupling coordination degree model was used to analyse the synergistic governance effect of regional carbon emissions. 2.2 Analysis model of influencing factors At present, the academic research on the analysis methods of influencing factors is not uniform, mainly including the DEMATEL method, factor analysis, principal component analysis, obstacle degree model and so on. Shi et al. [21] used the DEMATEL method to identify 11 key risk factors from the perspective of causality, and revealed the differences between causal factors and out- come factors by analysing their centrality and causality. Kang et al. [22] used geographical detector to conduct factor analysis, and found that natural factors dominated the spatial and temporal distribution of GPP in a province. Hou et al. [23] used the grey correlation model to comprehensively evaluate the quality of medicinal materials by calcu- lating the relative correlation between each batch of samples and the optimal reference sequence. Some researchers used structural equation modeling to investigate household tourism con- sumption, and discussed the influencing factors of potential variables, such as income structure, household and regional characteristics of consumption structure [24, 25]. Gao et al. [26] used the principal component analysis method to reduce the dimension of the five dimensions of produc- tion efficiency, overall prosperity, coordinated development, achievement sharing and ecological welfare, so as to extract the main factors affecting common prosperity. Yang et al. [27] used the obstacle degree model to identify the main obstacle factors affecting the coordinated development of the regional innovation environment-resource-output-benefit composite system by calculating the product of the deviation degree and weight of each index, so as to provide a basis for policy optimization. Wang et al. [28] used the obstacle degree model to identify the main obstacle dimensions affecting the development of new quality productivity of water conservancy, among which the high-tech dimension had the largest obstacle, followed by high-quality, green and efficient dimensions [29]. 2.3 Synthesis and research gap Existing research offers a diverse methodological foundation for assessing coordinated develop- ment. However, integration between dynamic and static analysis remains limited, with most stud- ies focusing on level evaluation rather than diagnosing development bottlenecks. To address this, a tiered analytical framework is constructed. First, the synergy degree of the composite system and the coupling coordination degree are comprehensively applied to reveal the coordinated de- velopment mechanism of the Beijing–Tianjin–Hebei advanced manufacturing cluster from a com- bined dynamic-static perspective. Then, compared with other approaches, the obstacle degree model is introduced to more accurately quantify key constraints on coordination. This framework provides a clear basis for formulating targeted governance strategies. 3. Methodology Based on the constructed index system and weights determined by the coefficient of variation method, the system coordination level is first measured using the composite system synergy de- gree model. These results then serve as direct input into the coupling coordination degree model to evaluate the system’s overall coordination state. Finally, key constraints are identified through the obstacle degree model, thereby completing a full analytical cycle from diagnosis to solution. Method Framework is shown in Fig. 1. Zhang, Yang, Buchmeister, Ojstersek 448 Advances in Production Engineering & Management 20(4) 2025 Fig. 1 Methodological framework 3.1 Evaluation index system construction The coordinated development process of Beijing–Tianjin–Hebei advanced manufacturing cluster can be regarded as a complex system. In order to comprehensively and objectively reflect the co- ordinated development of the industrial cluster, on the basis of the availability of advanced man- ufacturing data, constructs the quantity, quality, efficiency and value as the four subsystems of the coordinated development of the cluster, and selects the order parameters that can reflect the dom- inant role of each subsystem, and constructs the evaluation index system of the synergy degree of the composite system of the coordinated development of the cluster, which is shown in Table 1. Table 1 Evaluation index system for the coordinated development of the advanced manufacturing cluster Subsystem Numbering Order parameter Attribute Quantity A1 Quality requested Positive A2 Growth of invention patent applications Positive A3 Authorized amount Positive A4 Invention patent authorization Positive Quality B1 The average number of families Positive B2 Average number of weights Positive B3 Average number of IPC categories Positive B4 The average number of the same family of invention patents Positive B5 The average number of invention patent rights Positive B6 The average IPC classification number of invention patents Positive Efficiency C1 Average review duration Negative C2 The average examination time of invention patents Negative C3 The proportion of review duration meeting the threshold value Positive C4 The proportion of invention patent examination time meeting the threshold value Positive Value D1 Average annual DPI patent comprehensive value Positive D2 Average annual DPI patent comprehensive value of invention patents Positive 3.2 Data sources and processing Data processing primarily involves retrieving, downloading, deleting duplicates, filtering, and merging data. The data processing flowchart is shown in Fig. 2. Patents were searched using the Dawei Innojoy patent search engine, yielding 180,689 results. Duplicates were then deleted, re- taining the latest by disclosure time, resulting in 152,168 patents. Subsequently, data were filtered based on conditions including application year, authorization year, and legal status announce- ments indicating dismissal or withdrawal, with the filtering period set to 2013-2023. Finally, data merging produced a valid dataset of 146,765 patents. Measuring the coordinated development of the advanced manufacturing cluster based on patent data: A composite … Advances in Production Engineering & Management 20(4) 2025 449 Fig. 2 Data processing flowchart 3.3 Evaluation index weight calculation The weight of evaluation indexes significantly impacts overall evaluation outcomes. Empower- ment is achieved using the coefficient of variation (CV) method, a statistical approach measuring data dispersion through index CV values. Higher CV values correspond to greater weights. Calcu- lation steps follow: (1) Raw data collection: Suppose that there are 𝑚𝑚 indicators in a set of data, 𝑛𝑛 samples to be eval- uated, that is, an 𝑛𝑛 ∗ 𝑚𝑚 matrix, let it be 𝑋𝑋 . Where 𝑥𝑥 𝑖𝑖𝑖𝑖 denotes the data of row 𝑖𝑖 and column 𝑗𝑗 . 𝑋𝑋 = � 𝑥𝑥 11 ⋯ 𝑥𝑥 1 𝑚𝑚 ⋮ ⋱ ⋮ 𝑥𝑥 𝑛𝑛 1 ⋯ 𝑥𝑥 𝑛𝑛 𝑚𝑚 � (1) (2) Positive index data: convert all indicators into positive index (that is, the larger the value, the better). For the positive index, keep its original data unchanged: 𝑥𝑥 𝑖𝑖𝑖𝑖 ′ = 𝑥𝑥 𝑖𝑖𝑖𝑖 (2) For the negative index, the following methods are used: 𝑥𝑥 𝑖𝑖𝑖𝑖 ′ = 1 𝑘𝑘 + 𝑚𝑚 𝑚𝑚𝑥𝑥 � 𝑥𝑥 𝑖𝑖 � + 𝑥𝑥 𝑖𝑖𝑖𝑖 (3) (3) Data standardization: its purpose is to eliminate the influence of the unit, so that all data can be calculated in the same way. The normalized data matrix is 𝑅𝑅 : 𝑟𝑟 𝑖𝑖𝑖𝑖 = 𝑥𝑥 𝑖𝑖𝑖𝑖 ′ � ∑ 𝑥𝑥 𝑖𝑖𝑖𝑖 ′2 𝑛𝑛 𝑖𝑖 = 1 (4) (4) Calculate the coefficient of variation: Eqs. 5 to 7 are the calculation of the mean, standard de- viation (mean square error) and coefficient of variation of each index. 𝐴𝐴 𝑖𝑖 = 1 𝑛𝑛 � 𝑟𝑟 𝑖𝑖𝑖𝑖 𝑛𝑛 𝑖𝑖 = 1 (5) 𝑆𝑆 𝑖𝑖 = � 1 𝑛𝑛 � ( 𝑟𝑟 𝑖𝑖𝑖𝑖 − 𝐴𝐴 𝑖𝑖 ) 2 𝑛𝑛 𝑖𝑖 = 1 (6) 𝑉𝑉 𝑖𝑖 = 𝑆𝑆 𝑖𝑖 𝐴𝐴 𝑖𝑖 (7) (5) Calculate the weight: 𝑤𝑤 𝑖𝑖 = 𝑉𝑉 𝑖𝑖 ∑ 𝑉𝑉 𝑖𝑖 𝑛𝑛 𝑖𝑖 = 1 (8) Zhang, Yang, Buchmeister, Ojstersek 450 Advances in Production Engineering & Management 20(4) 2025 3.4 Composite system coordination degree 3.4.1 Subsystem order degree measurement model A composite system 𝑆𝑆 for Beijing–Tianjin–Hebei advanced manufacturing cluster coordinated de- velopment is established. This system comprises 𝑛𝑛 subsystems, denoted as 𝑆𝑆 𝑖𝑖 ( 𝑖𝑖 = 1, 2, . . . . , 𝑛𝑛 ). For each subsystem 𝑆𝑆 𝑖𝑖 , the order parameter 𝑆𝑆 𝑖𝑖𝑖𝑖 ( 𝑗𝑗 = 1, 2, . . . , 𝑚𝑚 ; 𝑚𝑚 ≥ 1) is defined, where 𝑚𝑚 repre- sents the number of order parameters. The lower and upper bounds of 𝑆𝑆 𝑖𝑖𝑖𝑖 are 𝛼𝛼 𝑖𝑖𝑖𝑖 and 𝛽𝛽 𝑖𝑖𝑖𝑖 respec- tively, with 𝛼𝛼 𝑖𝑖𝑖𝑖 ≤ 𝑆𝑆 𝑖𝑖𝑖𝑖 ≤ 𝛽𝛽 𝑖𝑖𝑖𝑖 . Based on composite system synergy degree theory, the order parame- ter order degree is calculated as follows: 𝜇𝜇 𝑖𝑖 � 𝑆𝑆 𝑖𝑖𝑖𝑖 � = ⎩ ⎪ ⎨ ⎪ ⎧ 𝑆𝑆 𝑖𝑖𝑖𝑖 − 𝛼𝛼 𝑖𝑖𝑖𝑖 𝛽𝛽 𝑖𝑖𝑖𝑖 − 𝛼𝛼 𝑖𝑖𝑖𝑖 , 𝑗𝑗 ∈ [1, 𝑘𝑘 ] 𝛽𝛽 𝑖𝑖𝑖𝑖 − 𝑆𝑆 𝑖𝑖𝑖𝑖 𝛽𝛽 𝑖𝑖𝑖𝑖 − 𝛼𝛼 𝑖𝑖𝑖𝑖 , 𝑗𝑗 ∈ [ 𝑘𝑘 + 1, 𝑚𝑚 ] (9) In the equation, the order parameters 𝑆𝑆 𝑖𝑖 1 , 𝑆𝑆 𝑖𝑖 2 , … , 𝑆𝑆 𝑖𝑖𝑖𝑖 are positive indicators. The larger the value, the higher the order degree of the subsystem, where 𝑘𝑘 𝑖𝑖 𝜖𝜖 [1, 𝑚𝑚 𝑖𝑖 ]; the order parameters 𝑆𝑆 𝑖𝑖𝑖𝑖 + 1 , 𝑆𝑆 𝑖𝑖𝑖𝑖 + 2 , … , 𝑆𝑆 𝑖𝑖 𝑚𝑚 are negative indicators. The smaller the value is, the lower the order degree of the subsystem is 𝜇𝜇 𝑖𝑖𝑖𝑖 𝜖𝜖 [0, 1], and the larger the value, the greater the contribution of the order pa- rameter 𝑆𝑆 𝑖𝑖𝑖𝑖 to the subsystem. The contribution of each order parameter 𝑆𝑆 𝑖𝑖𝑖𝑖 to the subsystem is also related to its combina- tion form and their respective weights. The linear weighted average method (LWA, Eq. 10) and geometric average method (GA, Eq. 11) are used to calculate the order degree 𝜇𝜇 𝑖𝑖 ( 𝑆𝑆 𝑖𝑖𝑖𝑖 ) of the sub- system. 𝜇𝜇 𝑖𝑖 ( 𝑆𝑆 𝑖𝑖 ) = � 𝑤𝑤 𝑖𝑖𝑖𝑖 𝑚𝑚 𝑖𝑖 = 1 𝜇𝜇 𝑖𝑖𝑖𝑖 � 𝑆𝑆 𝑖𝑖𝑖𝑖 � , 𝑤𝑤 𝑖𝑖𝑖𝑖 ≥ 0 ∧ � 𝑤𝑤 𝑖𝑖𝑖𝑖 𝑚𝑚 𝑖𝑖 = 1 = 1, ∀ 𝑖𝑖 ∈ (1, 2, ⋯ , 𝑛𝑛 ) (10) 𝜇𝜇 𝑖𝑖 ( 𝑆𝑆 𝑖𝑖 ) = � � 𝜇𝜇 𝑖𝑖 � 𝑆𝑆 𝑖𝑖𝑖𝑖 � 𝑚𝑚 𝑖𝑖 = 1 𝑛𝑛 (11) In the equation 𝜇𝜇 𝑖𝑖 ( 𝑆𝑆 𝑖𝑖 ) 𝜖𝜖 [0, 1], and the larger the value, the higher the order degree of the sub- system 𝑆𝑆 𝑖𝑖 , and vice versa. 𝑤𝑤 𝑖𝑖𝑖𝑖 represents the weight of the order parameter 𝑆𝑆 𝑖𝑖𝑖𝑖 in its subsystems, which can be obtained by the coefficient of variation method. 3.4.2 Composite system synergy model Assuming that at a given initial time 𝑡𝑡 0 , the order degree of the subsystems of the advanced man- ufacturing cluster collaborative development composite system is 𝜇𝜇 𝑖𝑖 0 ( 𝑆𝑆 𝑖𝑖 ), respectively. At another time 𝑡𝑡 1 in the collaborative development process, the order degree of each subsystem is 𝜇𝜇 𝑖𝑖 1 ( 𝑆𝑆 𝑖𝑖 ), respectively, where, 𝑖𝑖 = 1, 2, . . . , 𝑛𝑛 . Through the linear weighted average method and the geomet- ric average method, the synergy degree of the advanced manufacturing cluster collaborative de- velopment composite system in the region can be calculated as follows: 𝐷𝐷 = 𝜃𝜃 � 𝑤𝑤 𝑖𝑖 𝑛𝑛 𝑖𝑖 = 1 � � 𝜇𝜇 𝑖𝑖 1 ( 𝑆𝑆 𝑖𝑖 ) − 𝜇𝜇 𝑖𝑖 0 ( 𝑆𝑆 𝑖𝑖 ) � � , 𝑤𝑤 𝑖𝑖𝑖𝑖 ≥ 0 ∧ � 𝑤𝑤 𝑖𝑖 𝑛𝑛 𝑖𝑖 = 1 = 1 (12) 𝐷𝐷 = 𝜃𝜃 � � � 𝜇𝜇 𝑖𝑖 1 ( 𝑆𝑆 𝑖𝑖 ) − 𝜇𝜇 𝑖𝑖 0 ( 𝑆𝑆 𝑖𝑖 ) � 𝑛𝑛 𝑖𝑖 = 1 𝑛𝑛 (13) In Eq. 13, 𝜃𝜃 = 𝑚𝑚 𝑖𝑖𝑛𝑛 𝑖𝑖 [ 𝜇𝜇 𝑖𝑖 1 ( 𝑆𝑆 𝑖𝑖 ) − 𝜇𝜇 𝑖𝑖 0 ( 𝑆𝑆 𝑖𝑖 ) ≠ 0] � 𝑚𝑚 𝑖𝑖𝑛𝑛 𝑖𝑖 [ 𝜇𝜇 𝑖𝑖 1 ( 𝑆𝑆 𝑖𝑖 ) − 𝜇𝜇 𝑖𝑖 0 ( 𝑆𝑆 𝑖𝑖 ) ≠ 0] � , 𝜔𝜔 𝑖𝑖 represents the weight of the subsystem, which can be obtained by the coefficient of variation method. Measuring the coordinated development of the advanced manufacturing cluster based on patent data: A composite … Advances in Production Engineering & Management 20(4) 2025 451 The value of the synergy degree 𝐷𝐷 of the composite system can directly reflect the synergy de- gree of the composite system. It can be seen from the above equation that the larger the value of 𝐷𝐷 , the higher the synergy degree of the composite system. The parameter 𝜃𝜃 is used to determine the stability of the synergy degree of the composite system. When 𝜃𝜃 = 1, the synergy degree 𝐷𝐷 is positive, indicating that the development of advanced manufacturing cluster in the region is in a coordinated and orderly development stage. The larger the value is, the higher the synergy degree of the system is. When 𝜃𝜃 = −1, the degree of synergy 𝐷𝐷 is negative, which means that the ad- vanced manufacturing cluster in the region has not achieved coordinated development. 3.5 Coupling coordination degree model The coupling degree refers to the dynamic relationship of mutual influence between multiple sys- tems or elements. The coupling coordination degree is formed based on this relationship and is used to analyse the coordinated development level of things. The calculation method is as follows: 𝐷𝐷 = √ 𝐶𝐶 × 𝑇𝑇 (14) 𝐶𝐶 ( 𝑈𝑈 1 , 𝑈𝑈 2 , ⋯ , 𝑈𝑈 𝑛𝑛 ) = 𝑛𝑛 × � 𝑈𝑈 1 𝑈𝑈 2 ⋯ 𝑈𝑈 𝑛𝑛 ( 𝑈𝑈 1 + 𝑈𝑈 2 + ⋯ + 𝑈𝑈 𝑛𝑛 ) 𝑛𝑛 � 1 𝑛𝑛 (15) 𝑇𝑇 = 𝑤𝑤 1 𝑈𝑈 1 + 𝑤𝑤 2 𝑈𝑈 2 + ⋯ + 𝑤𝑤 𝑛𝑛 𝑈𝑈 𝑛𝑛 (16) In the equation, 𝐷𝐷 is the coupling coordination degree; 𝑐𝑐 is the coupling degree; 𝑡𝑡 is the compre- hensive evaluation value; 𝑈𝑈 1 , 𝑈𝑈 2 , . . . 𝑈𝑈 𝑛𝑛 are the internal coordination degree of each region; 𝑤𝑤 1 , 𝑤𝑤 2 . . . , 𝑤𝑤 𝑛𝑛 are weights. The coordinated development of Beijing, Tianjin and Hebei affects each other and has the same status. Therefore, 𝑤𝑤 1 = 𝑤𝑤 2 = 𝑤𝑤 3 = 1 3 in the three systems and 𝑤𝑤 1 = 𝑤𝑤 2 = 1 2 in the two systems. 3.6 Obstacle degree model In order to explore the factors that hinder the coordinated development of advanced manufactur- ing cluster in Beijing–Tianjin–Hebei, after evaluating its synergy level, the obstacle degree model is introduced. Taking 16 indicators in the evaluation system as obstacle factors, the obstacle de- gree of each index is calculated, and the main reasons that restrict the improvement of synergy degree are analysed, which lays a foundation for putting forward the guarantee mechanism. The specific steps are as follows: 𝑂𝑂 𝑖𝑖𝑖𝑖 = 𝑤𝑤 𝑖𝑖 × � 1 − 𝑟𝑟 𝑖𝑖𝑖𝑖 � ∑ 𝑤𝑤 𝑖𝑖 × � 1 − 𝑟𝑟 𝑖𝑖𝑖𝑖 � 𝑚𝑚 𝑖𝑖 = 1 (17) In Eq. 17, 𝑤𝑤 𝑖𝑖 is the index weight of the 𝑗𝑗 -th index to the synergy of advanced manufacturing cluster, 𝑚𝑚 = 16 is the total number of evaluation indexes, 𝑟𝑟 𝑖𝑖𝑖𝑖 is the index value of the 𝑗𝑗 -th index in the 𝑖𝑖 -th year after standardization, where the standardization processing adopts the above Eq. 4, (1 − 𝑟𝑟 𝑖𝑖𝑖𝑖 ) represents the gap between the 𝑗𝑗 -th index and the ideal value in the 𝑖𝑖 -th year, and 𝑂𝑂 𝑖𝑖𝑖𝑖 represents the influence degree of the 𝑗𝑗 -th index on the synergy of advanced manufacturing clus- ter in Beijing–Tianjin–Hebei region in the 𝑖𝑖 -th year. The greater the 𝑂𝑂 𝑖𝑖𝑖𝑖 , the higher the degree of hindrance to the coordinated development. 4. Authentic proof analysis 4.1 Weight The importance of each index in the comprehensive evaluation can be scientifically determined by analysing the relative variation degree through the coefficient of variation method. Based on the calculation results of the coefficient of variation, the weight of each index is assigned as follows. Zhang, Yang, Buchmeister, Ojstersek 452 Advances in Production Engineering & Management 20(4) 2025 Table 2 Weight of indicators Subsystem Numbering Order parameter weight Subsystem weight LWA GA Quantity A1 0.2304 0.2835 0.2988 A2 0.2403 A3 0.2527 A4 0.2765 Quality B1 0.0785 0.1204 0.1186 B2 0.2162 B3 0.1860 B4 0.1172 B5 0.2225 B6 0.1795 Efficiency C1 0.0676 0.2177 0.2046 C2 0.0611 C3 0.2411 C4 0.6302 Value D1 0.5560 0.3785 0.3780 D2 0.4440 4.2 Coordinated development of Beijing–Tianjin–Hebei advanced manufacturing cluster 4.2.1 Composite system synergy model The maximum value of each order parameter from 2013 to 2023 is increased by 10 %, and the minimum value is reduced by 10 %, so as to determine the upper and lower limits of the order parameters. Then, the linear weighted average method and the geometric average method are used to calculate the order degree of the subsystem and the synergy degree of the composite sys- tem respectively. Linear weighted average method The linear weighted average method is a method of calculating the average value after giving dif- ferent weights to different data points. It considers the difference of the overall contribution of each data point. The advantage of this method is that it can be adjusted according to the im- portance of the data. Subjectivity in the weight calculation process is eliminated by using the co- efficient of variation method to calculate the order parameter weight. Substituting the patent data and weights of the advanced manufacturing cluster in the Beijing– Tianjin–Hebei region into Eq. 10 and Eq. 12, the order degree of the Beijing–Tianjin–Hebei sub- system and the synergy degree of the composite system are shown in Table 3. According to the data of Table 3, although the subsystem order degree of the Beijing–Tianjin– Hebei cluster fluctuates in some years, the overall trend shows a gradual upward trend, reflecting the overall progress of the Beijing–Tianjin–Hebei advanced manufacturing cluster. Overall, in the comparison of the same base period, the overall synergy degree of the cluster shows an overall upward trend from 2013 to 2023, which shows that in the long run, its development level contin- ues to improve, and the overall synergy development ability continues to increase. In the compar- ison of adjacent base periods, the synergy degree shows a trend of fluctuating development, which indicates that the synergy effect or cooperation level between subsystems has changed in a con- tinuous period of time. Table 3 The subsystem order degree and composite system synergy degree of linear weighted average method Year Subsystem order degree Composite system coordination degree Quantity Quality Efficiency Value Same base period Adjacent base period 2013 0.3420 0.3640 0.2120 0.0137 - - 2014 0.3779 0.4033 0.2051 0.0238 -0.0202 -0.0202 2015 0.4508 0.3974 0.1913 0.0557 -0.0553 -0.0365 2016 0.4836 0.4094 0.1848 0.0737 -0.0742 -0.0190 2017 0.4637 0.4023 0.1295 0.0945 -0.0877 -0.0264 2018 0.5276 0.4149 0.1692 0.1130 -0.1056 0.0353 2019 0.5892 0.4949 0.2100 0.1453 -0.1361 0.0482 2020 0.7603 0.5496 0.3250 0.1993 0.2358 0.1006 2021 0.8118 0.6060 0.4391 0.2739 0.3103 0.0745 2022 0.8330 0.6079 0.6708 0.4244 0.4239 0.1136 2023 0.8218 0.6337 0.8220 0.8413 0.6145 -0.1970 Measuring the coordinated development of the advanced manufacturing cluster based on patent data: A composite … Advances in Production Engineering & Management 20(4) 2025 453 Geometric average method The geometric average method is to calculate the average value by multiplying all data points and opening the corresponding root mean square, which is suitable for processing data with ratio or exponential growth. The advantage of the geometric mean method is that it is not sensitive to extreme values and is suitable for processing proportional data, but the disadvantage is that it cannot be used when there are zero or negative numbers in the data points. Substituting the patent data and various weights of the advanced manufacturing cluster in the Beijing-Tianjin-Hebei region into Eq. 11 and Eq. 13, the order degree of the Beijing–Tianjin–Hebei subsystem and the synergy degree of the composite system are shown in Table 4. According to the data of Table 4, the trend of the synergy degree of the composite system calcu- lated by the geometric average method is basically the same as that of the linear weighted average method. Compared with 2013, the order degree of each subsystem is improved as a whole, and the coordination degree of the same base period shows a significant upward trend, which indicates that the coordinated development of advanced manufacturing cluster in Beijing–Tianjin–Hebei region is increasing. The coordination degree of adjacent base periods fluctuates, which indicates that the level of cooperation between subsystems is not stable in a continuous period of time. Through the comprehensive analysis of the calculation results of the two methods, although the linear weighted average method and the geometric average method are different in the mean and data fluctuation characteristics, they are highly consistent in the trend description of the over- all synergy degree. Table 4 The subsystem order degree and composite system synergy degree of geometric average method Year Subsystem order degree Composite system coordination degree Quantity Quality Efficiency Value Same base period Adjacent base period 2013 0.3353 0.3447 0.2375 0.0137 - - 2014 0.3708 0.3836 0.1962 0.0230 -0.0270 -0.0270 2015 0.4424 0.3771 0.1924 0.0558 -0.0507 -0.0155 2016 0.4717 0.3716 0.2080 0.0735 -0.0504 -0.0145 2017 0.4563 0.3703 0.1674 0.0945 -0.0647 -0.0114 2018 0.5272 0.3882 0.2300 0.1137 -0.0500 0.0351 2019 0.5873 0.4571 0.2478 0.1462 0.0789 0.0393 2020 0.7599 0.5155 0.3265 0.2004 0.1863 0.0810 2021 0.8108 0.5626 0.4680 0.2756 0.2812 0.0711 2022 0.8287 0.5396 0.7118 0.4269 0.3705 -0.0624 2023 0.8143 0.5661 0.8029 0.8451 0.4725 -0.0617 4.2.2 Comparative analysis Adjacent-period analysis of Beijing–Tianjin–Hebei advanced manufacturing cluster coordinated development is susceptible to short-term factors like external environments and policy adjust- ments, thus affecting judgments of the real coordination state. To eliminate interference, enhance reliability, and reveal long-term evolution trends of regional synergy degree, the same base-period synergy degree is selected for in-depth analysis. In order to compare the difference between the linear weighted average method and the geo- metric average method in measuring the synergy degree of the advanced manufacturing cluster composite system in Beijing, Tianjin and Hebei, this part presents the statistical analysis results of the two methods in different regions in the form of box line diagram (Fig. 3), and visually shows the performance of the two methods in terms of mean value and dispersion degree through the box line diagram. It can be seen from Fig. 3 that the mean value of the linear weighted average method is close to the mean value of the geometric average method, indicating that the difference between the two methods is reduced from the perspective of the region as a whole. In terms of dispersion, the linear weighted average method has a longer box and a large span of whiskers, indicating that when the linear weighted average method is used, the fluctuation of the synergy degree data is more obvious, while the geometric average method data is relatively concentrated, showing better stability. Zhang, Yang, Buchmeister, Ojstersek 454 Advances in Production Engineering & Management 20(4) 2025 Fig. 3 Box line diagram analysis 4.2.3 Coupling coordination degree model Combined with the previous comparative analysis of the linear weighted average method and the geometric average method, because the geometric average method performs better in data stabil- ity and can more reliably reflect the real trend of coordinated development, according to the above-mentioned Beijing–Tianjin–Hebei advanced manufacturing cluster. The calculation results of the synergy degree of the composite system, based on the synergy degree under the geometric average method, are substituted into Eqs. 14 to 16 to calculate the coupling degree of the cluster's coordinated development, and Fig. 4 is obtained. Fig. 4 Beijing-Tianjin-Hebei coupling coordination degree In Fig. 4, the coupling coordination degree of the Beijing–Tianjin–Hebei cluster presents a 'V- shaped' development trajectory: in 2014-2020, the coupling coordination degree of the cluster was in a state of imbalance, and fell to a low point in 2019. In 2021-2023, the coupling coordina- tion degree rises rapidly, and finally reaches the development level of high-quality coupling in 2023. In general, although there are fluctuations in individual periods, from the long-term trend since 2013, the coordinated development of advanced manufacturing clusters in Beijing–Tianjin– Hebei has shown an upward trend, reflecting the gradual optimization of its resource allocation and cooperation mechanism, and the level of coordinated development of Beijing–Tianjin–Hebei clusters has been continuously improved [30]. Measuring the coordinated development of the advanced manufacturing cluster based on patent data: A composite … Advances in Production Engineering & Management 20(4) 2025 455 4.3 Obstacle degree model The 16 indicators are selected in the evaluation index system constructed above to calculate the obstacle degree, and calculates the obstacle factors that hinder the coordination of advanced man- ufacturing clusters in Beijing–Tianjin–Hebei through Eq. 17. The results are shown in Fig. 5. Fig. 5 Trend of obstacle degree of each dimension Based on the calculation results, in the development process of Beijing–Tianjin–Hebei ad- vanced manufacturing cluster, the obstacle degree of different dimensions shows different trends at different stages. • The obstacle degree for the quantitative dimension showed a declining trend from 2013 to 2021, reflecting a weakening hindrance to coordinated development. However, from 2022 to 2023, it increased sharply, becoming the primary obstacle to synergy within the Beijing– Tianjin–Hebei advanced manufacturing cluster by 2023. This suggests that the quantitative dimension has emerged as a new development bottleneck. • A trend similar to the quantitative dimension is observed, with the obstacle degree rising volatility from 0.1132 in 2013 to 0.4222 in 2023. This indicates that declining patent quality increasingly constrains regional synergy. • The efficiency obstacle degree increased consistently from 2013 to 2020, pointing to a growing constraint from the efficiency dimension. After 2020, however, sustained improve- ment was observed. • From 2013 to 2022, the value obstacle degree fluctuated upward and consistently repre- sented the largest proportion among all dimensions, highlighting the value dimension as the principal shortfall in cluster coordination. Optimization began after 2022, and by 2023 its obstacle degree had dropped to the lowest level. In summary, the coordinated development of advanced manufacturing clusters in Beijing– Tianjin–Hebei from 2013 to 2023 is affected by the interweaving of multi-dimensional obstacles. The dynamic changes of the obstacles in each dimension reflect the complex challenges faced by the coordinated development of the industry. In the follow-up, it is necessary to accurately imple- ment policies to break the bottleneck and promote the coordinated high-quality development of the cluster according to the stage characteristics of the obstacles in different dimensions. Zhang, Yang, Buchmeister, Ojstersek 456 Advances in Production Engineering & Management 20(4) 2025 5. Conclusions and implications The integration of patent indices with the synergy degree and coupling coordination degree mod- els is confirmed to provide a universal analytical framework for assessing the collaborative evo- lution of advanced manufacturing systems. This approach systematically captures dynamic pat- terns and development bottlenecks in regional industrial cluster synergy. Empirical analysis of the Beijing–Tianjin–Hebei advanced manufacturing cluster reveals long-term improvement in co- ordination despite short-term fluctuations, reflecting a transition from disordered to ordered de- velopment, with internal subsystems achieving high-quality coordination. However, key constraints on synergy are identified through the obstacle degree model, includ- ing structural risks in innovation momentum, inadequate mechanisms for patent value realiza- tion, and the need to strengthen resource allocation and transformation efficiency. The cluster should focus on building an integrated patent collaboration ecosystem. This can be achieved by implementing patent navigation and high-value patent portfolios, establishing regional patent sharing and transformation platforms, and developing joint patent navigation programs to opti- mize the allocation of innovation resources. The proposed framework demonstrates high universality and transferability, enabling appli- cation to industrial regions and advanced manufacturing clusters globally for evaluating innova- tion-driven manufacturing progress and diagnosing synergy barriers. Adopting this patent-based perspective supports the establishment of patent cooperation mechanisms, fostering shared tech- nical standards and patent pools. These measures are expected to advance technology integration and collaborative development within and between clusters, as well as on a global scale. Acknowledgement The authors would like to acknowledge all support received for this study. References [1] Yin, L. (2025). Coordination degree between international business and ecosystem under the background of global supply chain reconstruction, Tehnički vjesnik, Vol. 32, No. 1, 1-8, doi: 10.17559/TV-20240318001409. [2] Wei, Z.H., Yan, L., Yan, X. (2024). Optimizing production with deep reinforcement learning, International Journal of Simulation Modelling, Vol. 23, No. 4, 692-703, doi: 10.2507/ijsimm23-4-CO17. [3] Ojstersek, R., Javernik, A., Buchmeister , B. (2024). Integrating simulation modelling for sustainable, human-centred Industry 5.0: ESG-based evaluation in collaborative workplaces, Advances in Production Engineering & Management, Vol. 19, No. 4, 527-538, doi: 10.14743/apem2024.4.522. [4] Zhang, L., Mu, R., Hu, S., Yu, J., Zhang, J. (2022). Industrial coagglomeration, technological innovation, and environmental pollution in China: Life-cycle perspective of coagglomeration, Journal of Cleaner Production, Vol. 362, Article No. 132280, doi: 10.1016/j.jclepro.2022.132280. [5] Liang, P .P ., Jili, H.H., Lv, Y.Q., Xu, Y. (2025). Con�iguring supply chain governance and digital capabilities for resilience: Evidence from the manufacturing sector, Advances in Production Engineering & Management, Vol. 20, No. 1, 43-60, doi: 10.14743/apem2025.1.526. [6] Liu, X.L. (2025). Research on the development path of new-type industrialization driven by new quality productive forces-A case study of Quanzhou, Fujian, China, Journal of Commerce, Vol. 34, No. 18, 132-135, doi: 10.19699/j.cnki.issn2096-0298.2025.18.132. [7] Nowakowska, M., Pajecki, M. (2021). Latent class analysis for identi�ication of occupational accident casualty pro�iles in the selected Polish manufacturing sector, Advances in Production Engineering & Management, Vol. 16, No. 4, 485-499, doi: 10.14743/apem2021.4.415. [8] Khare, R., Villuri, V.G.K., Chaurasia, D. (2021). Urban sustainability assessment: The evaluation of coordinated relationship between BRTS and land use in transit-oriented development mode using DEA model, Ain Shams Engineering Journal, Vol. 12, No. 1, 107-117, doi: 10.1016/j.asej.2020.08.012. [9] Li, J., Fan, C.G., Yuan, Q.M. (2017). Synergetic development degree evaluation of Beijing-Tianjin-Hebei region based on distance collaborative model, Science, Technology and Management Research, Vol. 37, No. 18, 45-50, doi: 10.3969/j.issn.1000-7695.2017.18.008. [10] Luo, Q., He, X.J. (2021). Research on the synergy development of regional industrial technology supply and demand based on vague sets distance and grey incidence theory, Operations Research and Management Science, Vol. 30, No. 5, 193-199. [11] Ouyang, H., Yang, G.L. (2019). Measurement method of regional synergetic development based on Haken model, Statistics & Decision, Vol. 35, No. 12, 9-13. Measuring the coordinated development of the advanced manufacturing cluster based on patent data: A composite … Advances in Production Engineering & Management 20(4) 2025 457 [12] Deng, J., Chen, T., Zhang, Y. (2023). Effect of collaborative innovation on high-quality economic development in Beijing–Tianjin–Hebei urban agglomeration—An empirical analysis based on the spatial Durbin model, Mathematics, Vol. 11, No. 8, Article No. 1909, doi: 10.3390/math11081909. [13] Leydesdorff, L., Fritsch, M. (2006). Measuring the knowledge base of regional innovation systems in Germany in terms of a Triple Helix dynamics, Research Policy, Vol. 35, No. 10, 1538-1553, doi: 10.1016/j.respol.2006.09.027. [14] Fang, C., Luo, K., Kong, Y., Lin, H., Ren, Y. (2018). Evaluating performance and elucidating the mechanisms of collaborative development within the Beijing–Tianjin–Hebei region, China, Sustainability, Vol. 10, No. 2, Article No. 471, doi: 10.3390/su10020471. [15] Zhao, L., Zhang, G. (2021). Evaluation and welfare effect of coordinated ecological development of the Beijing– Tianjin–Hebei region, Chinese Journal of Population, Resources and Environment, Vol. 19, No. 3, 283-290, doi: 10.1016/j.cjpre.2021.12.031. [16] Sun, Z., Guan, H., Zhao, A. (2023). Research on the synergistic effect of the composite system for high-quality development of the marine economy in China, Systems, Vol. 11, No. 6, Article No. 282, doi: 10.3390/systems11060282. [17] Shen, D., Li, S., Wang, M. (2019). Evaluation on the coordinated development of air logistics in Beijing-Tianjin-Hebei, Transportation Research Interdisciplinary Perspectives, Vol. 1, Article No. 100034, doi: 10.1016/j.trip.2019.100034. [18] Ren, D., Hu, Z., Cao, A. (2024). Evaluation and prediction of the coordination degree of coupling water-energy-food- land systems in typical arid areas, Sustainability, Vol. 16, No. 16, Article No. 6996, doi: 10.3390/su16166996. [19] Xu, C.W., Shi, Y.L. (2025). The collaborative innovation development level, regional differences and spatial convergence characteristics of strategic emerging industries, Statistics & Decision, Vol. 41, No. 9, 134-138, doi: 10.13546/j.cnki.tjyjc.2025.09.023. [20] Zheng, H., Gu, R.N. (2025). Characteristics of carbon emission network and evaluation of emission reduction synergy in the Beijing–Tianjin–Hebei urban agglomeration, Chinese Journal of Environmental Science, Vol. 46, No. 10, 6133-6141, doi: 10.13227/j.hjkx.202408062. [21] Shi, J., Liu, Z., Feng, Y., Wang, X., Zhu, H., Yang, Z., Wang, J., Wang, H. (2024). Evolutionary model and risk analysis of ship collision accidents based on complex networks and DEMATEL, Ocean Engineering, Vol. 305, Article No. 117965, doi: 10.1016/j.oceaneng.2024.117965. [22] Kang, Z., Fang, S., Qiang, Q., Bai, D. (2025). Analysis of the spatial and temporal distribution pattern and driving factors of GPP in Gansu Province, Journal of Gansu Sciences, Vol. 37, No. 5, 56-64, doi: 10.16468/j.cnki.issn1004- 0366.2025.05.008. [23] Hou, C.W., Zhu, J., Shi, Y.H., Li, X.L., Liu, Z.X., Gong, X.J., Liu, X.D., Tong, Q.Z. (2025). Study on the quality of Rhizoma Polygonati produced from Hunan based on one-way ANOVA and grey relational analysis, Journal of Li-Shizhen Traditional Chinese Medicine, Vol. 36, No. 17, 3299-3304. [24] Zhang, P ., Cao, K. (2022). Analysis of the impact of household tourism consumption based on multilevel structural equation model, Mobile Information Systems, Vol. 2022, No. 1, Article No. 7141837, doi: 10.1155/2022/7141837. [25] C h ai , Y . , L i , Q . (2 02 1). R es ear c h o n i n �l u en c i n g f ac t o r s o f k n o w l edg e s h ar i n g i n s u ppl y c h ai n e n t er pr i s es u n de r blockchain environment, Tehnički vjesnik, Vol. 28, No. 5, 1553-1559, doi: 10.17559/TV-20201116085005. [26] Gao, H.P., Song, Y.B., Qiu, B., Lu, J.T. (2024). Measurement of common prosperity level and analysis of spatiotemporal evolution pattern, Statistics & Decision, Vol. 40, No. 13, 16-21, doi: 10.13546/j.cnki.tjyic. 2024.13.003. [27] Yang, Y., Wang, Y., Zhang, Y., Liu, C. (2022). Data-driven coupling coordination development of regional innovation EROB composite system: An integrated model perspective, Mathematics, Vol. 10, No. 13, Article No. 2246, doi: 10.3390/math10132246. [28] Wang, W., Li, A., Li, Y., Zhou, X., Yang, Y. (2025). Spatiotemporal distribution and obstacle factors of new-quality productivity in water conservancy in China based on RAGA-PP and obstacle degree model, Sustainability, Vol. 17, No. 10, Article No. 4534, doi: 10.3390/su17104534. [29] Zaidi, M., Hasan, S.M. (2022). Supply chain risk prioritization using AHP and framework development: A perspective of the automotive industry, International Journal of Industrial Engineering and Management, Vol. 13, No. 4, 283-293, doi: 10.24867/IJIEM-2022-4-319. [30] Wang, S.L., Zhang, X. (2024). Optimization strategies and simulation of integrated management in supply chains, International Journal of Simulation Modelling, Vol. 23, No. 3, 543-554, doi: 10.2507/ijsimm23-3-CO15.