NK model simulation study of intelligent manufacturing transformation path selection in pharmaceutical manufacturing enterprises

Abstract

Under the wave of Industry 4.0, traditional pharmaceutical manufacturing enterprises are in urgent need of intelligent manufacturing transformation and upgrading, and exploring the optimal realization path of intelligent manufacturing transformation can help accelerate the transformation process of pharmaceutical manufacturing enterprises. This paper uses grounded theory research methods to conduct a multi-case study to summarize six influencing factors of the intelligent manufacturing transformation of Chinese pharmaceutical manufacturing enterprises; and then analyzes the correlation between the intelligent manufacturing influencing factors based on the DEMATEL method and the NK model, and identifies the key influencing factors according to this, and then draws the fitness landscape map of the intelligent manufacturing transformation, and finally arrives at the optimal path selection of the intelligent manufacturing transformation of pharmaceutical manufacturing enterprises. The study enriches and extends the research paradigm of intelligent manufacturing transformation, and provides lessons for pharmaceutical manufacturing enterprises to realize intelligent manufacturing transformation and upgrading.

Introduction

With the continuous evolution of technology and industry globally, traditional enterprises are seeking intelligent manufacturing transformation paths to improve their ability to cope with uncertainty, enhance their competitive advantages, and further acquire more resources¹. Intelligent manufacturing transformation pathways clarify the process of creating new value with new logic and resource utilization by enterprises, which in essence help manufacturing enterprises choose a path that best suits the long-term development of the enterprise in the process of optimizing and restructuring resources^2,3. In this selection process, enterprises must clarify their interconnections with intelligent manufacturing transformation and the key elements for achieving success in intelligent manufacturing transformation⁴, and then gain sustained competitive advantage by choosing the transformation path that best suits the long-term development of the enterprise^5,6. Therefore, it is particularly important for traditional manufacturing enterprises to explore intelligent manufacturing paths and methods that are suitable for their own long-term development.

The intelligent manufacturing transformation of the pharmaceutical manufacturing industry, as an important industry concerning the national economy and people's livelihoods, is particularly urgent⁷. In particular, the outbreak of COVID-19 has drawn attention to the medical industry and caused the state and society to pay more attention to the pharmaceutical manufacturing industry^8,9. At the same time, with the aging of the population and the improvement of health awareness, the market demand for pharmaceutical products continue to grow, while the traditional pharmaceutical production equipment and processes have been difficult to meet market demand, restricting the further development of the industry. Intelligent manufacturing transformation can bring significant advantages to pharmaceutical companies, such as increasing production efficiency, reducing production costs, improving product quality and safety, and helping companies achieve digital management and fine-tune operations¹⁰. For example, by introducing intelligent manufacturing technology, CR Jiangzhong Pharmaceutical has improved its productivity by 25% and successfully reduced its operating costs by 60%, while effectively reducing its comprehensive energy consumption and carbon emission intensity, and achieved a significant advantage in market competition. Therefore, pharmaceutical manufacturing enterprises must actively explore the road of intelligent manufacturing transformation to be invincible in the future competition.

The issue of intelligent manufacturing transformation path selection in pharmaceutical enterprises refers to the process of selecting the most suitable path for the long-term development of the enterprise in the transformation process, and this selection process describes optimal implementation steps for the enterprise to create new value with new logic. In short, choosing a suitable intelligent manufacturing transformation path can guide enterprises to gain profit and value in a more effective way¹¹. With the advent of the artificial intelligence era, more and more pharmaceutical companies are trying to gain sustainable competitive advantage through intelligent manufacturing transformation¹². Whether an enterprise can find an intelligent manufacturing transformation path suitable for its development is crucial for it to gain a competitive advantage in the fierce market competition^6,13. As a result, how to adopt a feasible decision analysis method to solve the problem of intelligent manufacturing transformation path selection is a research problem worthy of attention in the academic and business communities.

In addition, a review of the literature reveals that while scholars have attempted to identify the factors influencing enterprise transformation¹⁴ and explore the inherent relationship between information technology and intelligent manufacturing transformation and upgrading^15,16, few have adopted a feasible decision analysis methodology to address the issue of intelligent manufacturing transformation path selection. Existing studies have not yet effectively integrated the internal and external influences of intelligent manufacturing in enterprises. They have also ignored the role of path creation evolution in complex systems¹⁷ and failed to accurately reflect the actual results of enterprises’ use of technological tools to achieve intelligent manufacturing.

In view of this, this study selects Chinese pharmaceutical manufacturing enterprises as the research object. Based on the review and summarization of the research literature on the intelligent manufacturing transformation of prior enterprises, this study considers intelligent manufacturing transformation as a complex adaptive system. It explores the influencing factors of the intelligent manufacturing transformation of pharmaceutical manufacturing enterprises through grounded theory methods of multiple case studies. Then, drawing on the NK model theory proposed by Kauffman¹⁸, this study adopts a computer simulation method to give a selection method for the intelligent manufacturing transformation path for pharmaceutical manufacturing enterprises.

Path selection framework

Intelligent manufacturing transformation

Intelligent manufacturing transformation is a hot area in the research of enterprise digital transformation. American scholars Wright and Bourne¹⁹ were the first to put forward the concept of “intelligent manufacturing,” which is defined as the process of small-batch production without human intervention, driven by knowledge engineering, manufacturing software systems, and robot vision. Since then, scholars have defined intelligent manufacturing in terms of intelligent design²⁰, intelligent production²¹, intelligent management, intelligent service, and integrated perspectives²². With the acceleration of the new technological revolution of digitalisation and intelligence, the definition of intelligent manufacturing has been given a new connotation: not only automated and unmanned production but also the ability to help enterprises transition from mass production to customised production through the mechanism of prosumption^2,23, thereby improving production efficiency and optimising resource allocation.

With the wide application of AI technology, the manufacturing industry is transforming from traditional manufacturing to intelligent manufacturing, facilitating enterprises to climb up the global value chain^6,12. Currently, research related to the intelligent manufacturing transformation of enterprises is in its infancy, and theoretical explorations remain rare²⁴. Existing literature has explored issues such as the driving force and development mode of intelligent manufacturing and platform building in manufacturing from industrial fields such as home appliance manufacturing⁴, mechanical components²¹, and equipment manufacturing²⁵, revealing the role of digitalisation technologies²⁶, as well as in the areas of digitalisation strategies, innovation and entrepreneurship, and platforms⁴. However, less research has been conducted on traditional pharmaceutical manufacturing enterprises¹⁰, and there is a lack of understanding of the transformation process and its intrinsic mechanisms, which fails to address the difficulties faced by pharmaceutical enterprises in their intelligent manufacturing transformation.

In addition, although there have been studies that have provided valuable insights for intelligent manufacturing transformation^10,13, the research on how to choose the intelligent manufacturing transformation path has produced few results. It should be noted that, in the few specific studies on intelligent manufacturing transformation, most of them emphasise the single-factor impact of intelligent manufacturing transformation^6,16, lack systematic thinking about it, and there are fewer studies that use pharmaceutical manufacturing enterprises as specific research objects. Therefore, this study explores a simulation method that can effectively analyse the problem of intelligent manufacturing transformation path selection for pharmaceutical manufacturing enterprises from the perspective of complex systems and with reference to the NK model, to better guide traditional pharmaceutical manufacturing enterprises to achieve high-quality intelligent transformation.

NK model

The NK model is a structured simulation method for systems based on the theory of evolutionary biology, which can be used to study how modular systems reach an optimal state at a faster rate through adaptive search²⁷. It is particularly suitable for exploring the effects of interactions between elements within a system on the overall adaptability²⁸, and is now mainly used to study the evolutionary laws of a system by analyzing the interactions between elements within the system^29,30.

In the NK model, N represents the number of elements of the system, and each element can have two states, 0 or 1, so there are 2^N possible states of the system, and each state corresponds to a fitness value. K represents the degree of interaction between the elements, and a mutation in the alleles of an element affects not only its own fitness value for the system, but also the fitness values determined by the K elements associated with it. Mapping these 2^N states and their corresponding fitness values into three-dimensional space constructs the fitness landscape, which is a three-dimensional graphic consisting of all possible states in the system, taking the form of peaks and valleys. The horizontal coordinates characterise the state combinations of a certain part of the elements in the system, while the vertical coordinates represent the state combinations of the remaining elements, and the height of each point reflects the fitness value of that state combination. The evolutionary process of a complex system can be viewed as a "climb" on the fitness landscape³¹, in which the elements change their states through mutation to find higher peaks, i.e., local optima. The NK model determines the number of local optima and the climbing route needed to reach the summit of the optimal fitness, thus helping to understand the evolutionary mechanism of adaptation in complex systems.

Since Levinthal³² first used the NK model to explore the relationship between self-organizing behavior and natural selection, the NK model has been widely used in the field of organizational and strategic management research due to its characteristics and advantages^33,34. For example, Hsu et al.³⁵ used the NK model to solve the integration problem between different departments within an enterprise in supply chain management; Joseph & Gaba³⁶ used the NK model to explore the interrelationships between organizational structure and performance in multinational corporations; Li et al.³⁷ applied the NK model to technological innovation; and Ma et al.³⁸ applied the NK model to the effect of member interactions on organizational performance.

However, this study is based on the NK model and the fitness landscape map, which analyzes the key elements and their interactions in the intelligent manufacturing transformation of an enterprise, and evaluates the fitness of the transformation decisions formed by the key elements to find out the intelligent manufacturing transformation with the highest fitness and to determine the optimal transformation paths to reach that state. The reasons for adopting the NK model in this paper are as follows: First, the computer simulation method can provide a more quantitative description of the influencing elements in the system by integrating massive amounts of data, especially for the intelligent manufacturing transformation, for which the conceptual definition and dimensional division of the dynamic system have not reached agreement; through data simulation, it can effectively deduce the order of the influencing elements of the system. Second, the NK model can simulate the complex relationships within enterprise organizational management, making it suitable for solving complex systems like intelligent manufacturing transformation and predicting its evolutionary patterns.

Path selection framework

Although academics have not yet formed a unified view on the definition of intelligent manufacturing transformation³⁹, most scholars believe that intelligent manufacturing is a whole composed of a number of interconnected, interacting elements⁴⁰, which have interrelated relationships, and that the transformation of one of the constituent elements induces synergistic transformations in the other elements, or can indirectly change the combinatorial relationship between the constituent elements of intelligent manufacturing, giving uniqueness and complexity to enterprise intelligent manufacturing⁴¹. However, the complexity that constitutes intelligent manufacturing is not the diversity of elemental forms, but the stochasticity, diversity, and nonlinearity of the interaction relationship between combinations of intelligent manufacturing components, that is, structural complexity, which is very similar to the purpose of using the NK model theory, to judge the evolutionary laws of complex systems.

This study proposes an intelligent manufacturing transformation path selection method based on the NK model for the problem of intelligent manufacturing transformation path selection, whose core idea is to explore the constituent elements of intelligent manufacturing and the logical relationships between these constituent elements, so as to discover the direction and degree of change brought about by changes in the constituent elements of intelligent manufacturing to the complex system of intelligent manufacturing. To this end, this study gives a framework for intelligent manufacturing transformation path selection for pharmaceutical manufacturing enterprises based on the NK model, as shown in Fig. 1 below.

Fig. 1.

Intelligent manufacturing transformation path selection framework based on the NK model.

According to the intelligent manufacturing transformation path selection framework based on the NK model in Fig. 1, the optimal path for the intelligent manufacturing transformation of pharmaceutical manufacturing enterprises is derived through simulation. The specific steps are as follows: firstly, the influencing factors of intelligent manufacturing transformation in pharmaceutical manufacturing enterprises are identified by collecting multi-case data and using grounded theory methods; secondly, based on the DEMATEL method to identify the key influencing factors in the intelligent manufacturing transformation (determine the parameter N), and analyze the correlation between the key influencing factors (determine the parameter K), and then combine the different selection results of the influencing factors, which can be obtained as a set of intelligent manufacturing transformation decision options; thirdly, the random distribution function is used to assign fitness values randomly to the selection results of each transformation decision, and the fitness landscape map of intelligent manufacturing transformation is drawn based on the final set of intelligent manufacturing transformation decision options and their corresponding fitness values; fourthly, on this basis, according to the corresponding “climbing way,” the use of computer simulation technology for intelligent manufacturing transformation path selection is employed to get the intelligent manufacturing transformation path selection results.

Analysis of influential factors for intelligent manufacturing transformation

Case selection and data sources

Case selection

In this study, the selection of case samples is based on the criteria of typicality, availability, and matching with the research topic. The object of this paper is pharmaceutical manufacturing enterprises. Firstly, the enterprises within the pharmaceutical manufacturing industry that have at least one high-tech enterprise certification are selected; secondly, non-innovative enterprises that have been established for more than 15 years and have more detailed information disclosure are prioritized; thirdly, to take into account the accessibility and authenticity of the research, this study takes listed companies as the main source of the samples, so that it can be easier to access the sample data and can improve the credibility of the study.

Finally, this study has selected 12 representative pharmaceutical manufacturing enterprises as research samples, which are all leading enterprises in their localities and ranked high in the list of China's Top 100 pharmaceutical industry enterprises. These pharmaceutical manufacturing enterprises have been at the forefront of production intelligence in the national round of concentrated technological transformation of pharmaceutical enterprises since the “12th Five-Year Plan.” Case selection details are provided in Table 1. Drawing on Shaughnessy's⁴² “1/3 principle,” the 12 enterprises were divided into two groups: 2/3 (8) enterprises were used as the modeling group to conduct grounded theory methods and construct theoretical models, and 1/3 (4) enterprises were used as the testing group to conduct theoretical saturation tests. To avoid the interference of subjective factors, 8 enterprises such as China Resources (CR) Jiangzhong were randomly selected from them as modeling samples.

Table 1.

Case selection.

Number	Enterprise name	Stock code (China)	Location	Established
1	CR Jiangzhong	600750	Nanchang, Jiangxi	1969
2	Gan & Lee Pharmaceutical	603087	Beijing	1998
3	Dong-E-E-Jiao	000423	Liaocheng, Shandong	1994
4	Wolwo Bio-Pharmaceutical	300357	Huzhou, Zhejiang	2002
5	Livzon Pharmaceutical	000513	Zhuhai, Guangdong	1985
6	Autobio Diagnostics	603658	Zhengzhou, Henan	1998
7	Dashenlin	603233	Guangzhou, Guangdong	1999
8	King-Friend Biochemical	603707	Nanjing, Jiangsu	2000
9	Kangmei Pharmaceutical	600518	Puning, Guangdong	1997
10	Jiuzhou Pharmaceutical	603456	Taizhou, Zhejiang	1973
11	Baiyunshan	600332	Guangzhou, Guangdong	1973
12	New Harmony Union (NHU)	002001	Shaoxing, Zhejiang	1999

Data sources

According to the triangulation method proposed by Madihally & Matthew⁴³, data and case information collection were conducted through multiple ways:

1. Searching for information about the development history of the enterprise, main products, enterprise news dynamics, and other materials related to the transformation and upgrading of intelligent manufacturing through the official website of the enterprise.

2. Searching for expert comments, interview records of core enterprise personnel, feature stories, and other relevant information reported by news media on the transformation and upgrading of the enterprise's intelligent manufacturing.

3. Searching for information about enterprises through information disclosure websites such as Juchao.com, Shenzhen Stock Exchange (SZSE), and Shanghai Stock Exchange (SSE).

4. Collecting and organizing the data in the process of production and operation of enterprises through digital resources such as Cathay Pacific Database (CSMAR) and China Research Data Service Platform (CNRDS).

The original case information collected were coded. After coding each of the eight case enterprises, if there was a coded content that could not be agreed upon or was significantly different, the remaining four case enterprises were utilized for testing and supplementation; if the test samples were able to support the coded content, the corresponding additions and modifications were made, otherwise the coded content was abandoned.

Case study and model construction

Open coding

Coding is the analytical process of transforming collected semi-structured interviews and secondary data from simple descriptions into quantitative data⁴⁴. The core element of open coding is to obtain initial categories by summarizing all raw data multiple times through the processes of labeling, defining phenomena, conceptualization, and categorization to extract concepts with the same or similar meanings and categories. In this study, 62 initial categories are finally formed through labeling and refining. Due to space limitations, only CR Jiangzhong is used as a representative for illustration. See Table 2.

Table 2.

CR Jiangzhong open coding.

Initial category	Example of a primitive statement
A1 ERP System	The earliest introduction of ERP system pharmaceutical enterprises, the system can be used to help daily operation and maintenance of operations and management
A2 MES System	Introduced MES to meet the demand for continuous production of large quantities of single products and improve production efficiency
A3 Self-developed	Establishing a software development department to independently research and develop the software required for the work of various departments in the enterprise and develop terminal sales management software
A4 Encouraging production innovation	The “Tenth Five-Year Plan” of the national pharmaceutical industry puts forward that the status of the pharmaceutical industry in the national economy has been further improved, and the main task of the industry during the “Tenth Five-Year Plan” period is to actively promote the modernization of traditional Chinese medicine
A5 Upgrading opponents' remodeling	In the face of market competition, traditional enterprises have gained more competitive advantages in the market by accelerating the upgrading and transformation of manufacturing and production at the provincial level
A6 Organizational learning	Going to other enterprises to research and study, formulate enterprise production development planning, overall planning program of the whole situation
A7 Equity reform	Jiangzhong Pharmaceutical Co., Ltd. replacement of Jiangxi Dongfeng Pharmaceutical Co., Ltd., equity reform enterprises to promote the technical engineering transformation of the production line of stomach-healthy elimination tablets
A8 Market product demand	Employee Work Facilitation Needs Urgent, Interdepartmental Cooperation Needs Information Connectivity, Market Demand for Products Unmet
A19 Internet of Things	Utilizing Internet of Things technology, accelerating technological innovation and application, and promoting green production standards
A20 Cloud Computing	Utilizing cloud computing and other new-generation information technology, enterprises subvert the original business model and service model
A21 Internet service platform	Developing and releasing an automated APP platform for sales software, a sales management platform covering more than 200,000 terminals, and developing an Internet service platform
A22 E-commerce platform sales	The State Food and Drug Administration proposed to liberalize the sales of prescription drugs in e-commerce channels, and the General Office of the State Council issued the Opinions on Promoting the Development of “Internet + Healthcare”
A23 Counterparty Service Enhancement	Influenced by the Internet, mobile terminals and other emerging media, pharmaceutical companies are moving towards e-commerce to provide differentiated services
A24 Capacity Building Enhancement	With the development of the Internet of Things, the enterprises redeployed to establish an e-commerce team and set up an e-commerce subsidiary
A25 Resource Redistribution	The team develops systems including sales, including production, and then finance, and efficiently allocates resources through the Internet
A26 Demand servitization	The production process lacks the placement of traceability information, quality traceability is difficult, there are information islands
A39 Industrial Internet + Big Data	For enterprises with such a large volume as CRC, there are many application scenarios for the establishment of an industrial Internet, which can be used to effectively deploy resources through the industrial Internet, as well as including laboratories and quality control equipment
A40 Digital Twin	Build a digital twin of equipment, through the digital twin platform to realize the remote operation and maintenance and repair of equipment, and quickly improve the ability of Jiangzhong employees to accurately operate, deeply understand, and quickly diagnose the equipment
A41 Intelligent Inspector System	Self-developed AI intelligent supervision system and AI technical inspection system
A42 Intelligent Manufacturing Enabling Policies	Made in China 2025 clearly states from the national strategic level that intelligent manufacturing should be the main direction of attack to move into the ranks of manufacturing powers
A43 Industry Push	Manufacturing enterprises achieve transformation and upgrading through intelligent manufacturing, supporting personalized customization as well as flexible manufacturing and lean production based on industrial chain collaboration
A44 Innovative and Creative Thinking	Building intelligent customer service capabilities based on artificial intelligence means and intelligent new product capabilities based on innovative technologies
A45 Resource Synergy and Policy Protection	Effective resource allocation through the Industrial Internet is implemented and tracked annually by CRG in terms of resource inputs and budget usage
A46 Multiple Demand Surge	As new technologies develop rapidly and external demands become diversified, the use of big data and web technologies to collect customer information enables rapid iteration of products and services to meet diversified demands

Axial coding

Successive comparisons of data segments are made to discover related ideas or themes, which in turn enables small-scale categorization to study causal relationships between categories⁴⁵. Through axial coding, this study categorized the categories based on the inter-category linkages, which were eventually grouped into 13 main categories, such as information technology use, as shown in Table 3.

Table 3.

Axial coding.

Main category	Brief description	Initial category
B1 Information Technology Usage	Utilizing EPR to improve work efficiency to achieve information management	A1 ERP System
		A2 MES System
		A3 Self-development
B2 Government tax subsidies	Through government tax subsidies, enterprises receive financial support to promote economic development	A4 Encourage production innovation
B3 Accelerated transformation and upgrading	Accelerated transformation and upgrading strategies to adapt to market changes	A5 Rivals remodeling and upgrading
		A43Industry promotion
B4 Increased market demand	Increase in market demand as the popularity of products or services rises	A8 Market Product Demand
		A46 Diversified demand surge
B5 Digital technology use	Utilizing digital technology to enhance business capabilities	A19 Internet of Things
		A20 Cloud Computing
		A21 Internet service platform
B6 Service differentiation	Satisfying the specific needs of different customers by providing unique services	A23 Rival Service Enhancement
B7 Capacity Reconfiguration	Undertake capability reconfiguration to adapt to market demands and changes	A24 Capacity building enhancement
		A6Organizational Learning
B8 Resource reallocation	Reallocation of resources and optimization of resource allocation to meet market demands more effectively	A25 Resource Redistribution
B9 Demand servitization	Satisfying customers' needs through a service-oriented approach, emphasizing the provision of all-around solutions	A26 Demand servitization
B10 Intelligent Technology Use	Utilizing intelligent technologies in production and marketing	A39 Industrial Internet + Big Data
		A40 Digital Twin
		A41 Intelligent Inspector System
B11 Preferential Policies	Utilizing preferential policies to gain tax or other economic advantages	A22 E-commerce platform sales
		A42 Intelligent Manufacturing Enabling Policies
B12 Win–win thinking	Enterprises adopt win–win thinking to maximize benefits for all parties	A44 Innovative Creative Thinking
B13 Resource Reconfiguration	Undertake resource reconfiguration to adjust and reorganize resources to adapt to market changes	A45 Resource synergy and policy guarantee
		A7 Equity Reform

Selective coding

The grounded theory proposed by Glaser et al.⁴⁶ uses data collection as a starting point for constructing initial categories and continually explores these to discover a core category that explains the others to improve practical perspectives and develop a formal theory. Based on this, data is analyzed so that the intrinsic connections between the established categories can be explored in depth. In this paper, the main categories are summarized, and then the core categories are identified and saturated through further theoretical sampling and data collection. Finally, technological innovation, pharmaceutical policy, industry competition, organizational management, resource heterogeneity, and market demand are obtained as the core categories, and the results of selective coding are shown in Table 4 below.

Table 4.

Selective coding results.

Core category	Content	Main category
Technology Innovation	Information, digital and intelligent technology use and own new technology research and development	Information Technology Usage
		Digital Technology
		Intelligent Technology Usage
Pharmaceutical Policy	Government guiding the development of the industry and subsidies and preferential policies	Government Tax Subsidies
		Preferential Policies
Industry Competition	Helping enterprises to make correct and reasonable judgment on the market	Service Differentiation
		Accelerated Retrofitting and Upgrading
Organizational Management	Effective integration of internal and external resources to cope with external threats and identify and utilize external opportunities	Capacity Reconfiguration
		Win–Win Thinking
Resource Heterogeneity	Integration of resources is the basis of competitive advantage	Resource Reallocation
		Resource Reconfiguration
Market Demand	The sum of customers' purchases of willing products in a marketing environment	Increased Market Demand
		Demand Servicing

In the process of intelligent manufacturing transformation, technological innovation is the development of new technologies by enterprises or the development of new products and services based on existing technologies; pharmaceutical policy is the macroeconomic policy of the government to guide the development of the industry, adjust the imbalance of the industrial structure, build healthy market competition, and provide organizations with the necessary financial subsidies, tax exemptions, technological support, and other resources for industry development; industry competition is the degree of industry competition, which can be used as an indicator to measure the level of competition among enterprises and is also an important factor in regulating the relationship between economic policy uncertainty and enterprise investment behavior; organizational management is the ability of an enterprise to effectively integrate internal and external resources to cope with external threats and to identify and take advantage of external opportunities. It reflects the competitive advantage in a dynamic environment and plays an important role in the organization's strategy and normative behavior; resource heterogeneity encompasses all the things used by the organization to create value, including assets, knowledge, and various capabilities; market demand refers to the sum of the quantities of products or services that a particular customer is willing to purchase in a particular region, time, marketing environment, and marketing plan. Market demand is the main manifestation of the market's regulation of supply and demand, and the new demand generated by consumers can continuously promote the innovative activities of enterprises.

To test the theoretical saturation of the above findings, the core categories included in the theoretical model were tested using an additional four case studies. No new categories were generated, and the six existing categories were saturated with the available case studies, thus confirming that the model was theoretically saturated.

NK model construction for intelligent manufacturing transformation path selection

Symbolic description of model construction

To elucidate the construction method proposed in this study regarding the NK model, the following notations are relevant:

C = {C₁, C₂, …, C_n}: the set of intelligent manufacturing transformation elements, in which: C_i represents the ith constituent element, i ∈ {1, 2, …, n}.

P = {P₁, P₂, …, P_h}: the set of experts involved in the construction of the intelligent manufacturing transformation NK model, where: P_o represents the oth expert, o ∈ {1, 2, …, h}. Here the weight or importance of each expert is considered equal.

Z = {Z₁, Z₂, …, Z_u}: A set of linguistic evaluation phrases for evaluating the strength of association between intelligent manufacturing transformation elements, where: Z_r refers to the rth linguistic phrase in the linguistic phrase set Z, r ∈ {1, 2, …, r}. Here the set of language evaluation phrases is set as Z = {Z₀ = NO (no association), Z₁ = VL (very low), Z₂ = L (low), Z₃ = H (high), Z₄ = VH (very high)}.

F₀ = ${\left[f_{\text{ij}}^0\right]}_{\text{n}\times \text{n}}$: the association evaluation matrix between the constituent elements given by the expert Po based on the set of linguistic evaluation phrases Z, where: denotes that the expert P_o selects a linguistic phrase from the set of linguistic evaluation phrases Z as the evaluation value of the strength of the association effect between the element C_i and the element C_j, o ∈ {1, 2, …, h}, i, j ∈ {1, 2, …, n}. The correlation of the evaluation elements themselves is not considered here, so the main diagonal element of the matrix F_p is denoted as “one” and is considered as 0 in the operation.

B = {B₁, B₂, …, B_N}: the key elements of intelligent manufacturing transformation path selection extracted from the intelligent manufacturing transformation elements C_i, where: B_s denotes the sth key element, s ∈ {1, 2, …, N}.

${\phi }^s=\{{\phi }_1^s,{\phi }_2^s,\cdots ,{\phi }_q^s\}$: the set of alleles of the key elements B_s of intelligent manufacturing transformation, where: ${\phi }_1^s$ denotes the lth allele of the key elements B_s, l ∈ {1, 2, …, q}. Here, the number of alleles q = 2 can be considered based on the reality, that is, the set of alleles of key elements Bs is ${\phi }^s=\left\{\begin{array}{c}{\phi }_1^s=0\end{array}\right),{\phi }_2^s=1\}$, Where: 0 and 1 denote no element transition and element transition.

M = {M₁, M₂, …, M_a}: the set of allele combinations of key elements of intelligent manufacturing transformation, where: M_d denotes the dth allele combination of key elements of intelligent manufacturing, which can be expressed as$M_d={\phi }_{l_1}\prime {\phi }_{l_2}^2\cdots {\phi }_{l_N}^N,l_1,l_2,\cdots ,l_N\in \left\{\begin{array}{c}1,2\end{array}\right),a=q^N,\text{d}\in \left\{1,2,\cdots ,a\right\}$ .

$E={\left[e_s^d\right]}_{a\times N}$: matrix of adaptability of key elements of intelligent manufacturing transformation, where: $e_s^d$ denotes the effect of an allele of key element B_s in the allele combination M_d of key elements of intelligent manufacturing transformation on the adaptability of the system, that is, the adaptability value.

The aim of this study is to construct the NK model of intelligent manufacturing transformation based on the key elements B_s of intelligent manufacturing transformation, consider the participation of expert P_o, generate the fitness landscape map of intelligent manufacturing transformation, and select the path of intelligent manufacturing transformation through computer simulation.

Identification of key elements (identification of parameter N)

Firstly, according to the correspondence between the linguistic evaluation phrase and its subscript value, the linguistic evaluation phrase Z_r is transformed into its corresponding subscript value r, r ∈ {1, 2, …, u $\}$, and the arithmetic average method is applied, and the direct correlation evaluation matrix between the constituent elements given by all the experts, $F_0={\left[f_{\text{i}j}^0\right]}_{n\times n}$, is assembled into the constituent elements' direct correlation group evaluation matrix, $G={\left[g_{\mathit{ij}}\right]}_{n\times n}$, in which the formula for the calculation of the element $g_{\mathit{ij}}$ in the matrix is:

$$g_{\mathit{ij}}=\frac{1}{h}\sum _{p=1}^h{f}_{\mathit{ij}}^0,i,j\in \left\{1,2,\cdots ,n\right\}$$ 1

Secondly, using the DEMATEL method (Fontela and Gabus, 1976; Si et al., 2018), the constituent direct correlation group evaluation matrix $G={\left[g_{\mathit{ij}}\right]}_{n\times n}$ is normalised to obtain the normalised direct correlation group evaluation matrix $X={\left[x_{\mathit{ij}}\right]}_{n\times n}$, where the element $x_{\mathit{ij}}$ of the matrix is is calculated by the formula:

$$x_{\mathit{ij}}=\frac{g_{\mathit{ij}}}{\underset{0\le \text{i}\le 1}{\text{max}}\left\{{\sum }_{j=1}^n,g_{\mathit{ij}}\right\}},\text{i},j\in \left\{\text{1,2},,,\cdots ,,,n\right\}$$ 2

Based on Markov absorptivity^47,48, it is known that the DEMATEL method assumes that there is at least one i such that ${\sum }_{3=1}^n{q}_{\mathit{ij}}<\underset{0\le \text{i}\le n}{\text{max}}\left\{{\sum }_{j=1}^n,g_{\mathit{ij}}\right\}$ holds, and this assumption is reality is mostly satisfied. As a result, the matrix $x={\left[x_{\mathit{ij}}\right]}_{n\times n}$ satisfies the properties (1)${\mathit{lim}}_{\tau \to \infty }{\left(\text{X}\right)}^{\tau }=OZ$; (2) ${\mathit{lim}}_{\tau \to \infty }\left[1,+,X,+,{\left(X\right)}^2,+,\cdots ,{\left(\text{X}\right)}^{\tau }\right]={\left(1,-,\text{X}\right)}^{-1}Z$, where O and I are the zero and the unit matrices.

From the above properties, the indirect correlation evaluation matrix $Y={\left[Y_{\mathit{ij}}\right]}_{n\times n}$ can be constructed for intelligent manufacturing transformation elements respectively, and its calculation formula is:

$$Y={\mathit{lim}}_{\tau \to \infty }\left[{\left(X\right)}^2,+,{\left(X\right)}^3,+,\cdots ,{\left(\text{X}\right)}^{\tau }\right]={\left(X\right)}^2{\left(1,-,\text{X}\right)}^{-1}$$ 3

Further, a comprehensive association evaluation matrix $T={\left[{\gamma }_{\mathit{ij}}\right]}_{n\times n}$ is constructed for the intelligent manufacturing transformation elements, where ${\gamma }_{\mathit{ij}}$ denotes the sum of the degree of direct association and indirect association, that is, the degree of comprehensive association, of the elements $C_i$ and $C_j$, and $\text{i},j\in \{\text{1,2},\cdots ,n\}$. The formula for the element ${\gamma }_{\mathit{ij}}$ in the matrix T is:

$${\gamma }_{\mathit{ij}}=x_{\mathit{ij}}+y_{\mathit{ij}},\text{i},j\in \left\{\text{1,2},,,\cdots ,,,n\right\}$$ 4

Gathering the row and column elements in the matrix $T={\left[{\gamma }_{\mathit{ij}}\right]}_{n\times n}$ respectively, the centrality ${\alpha }_i$ of the constituent elements can be obtained, which is calculated as:

$${\alpha }_i=\sum _{j=1}^n{\gamma }_{\mathit{ij}}\sum _{j=1}^n{\gamma }_{\mathit{ji}},\text{i}\in \left\{\text{1,2},,,\cdots ,,,n\right\}$$ 5

In the formula, the centrality degree ${\alpha }_i$ denotes the size of the role played by the corresponding intelligent manufacturing transformation element in the set C of constituent elements. To extract the key elements of intelligent manufacturing transformation, that is, to determine the parameter N in the NK model, the centrality extraction threshold $\xi$ can be pre-determined, which is calculated as:

$$\xi =\chi ·\text{max}\left\{\left({\alpha }_i\right|,\text{i},\in ,N\right\}$$ 6

In the above formula, χ indicates the maximum centrality percentage of intelligent manufacturing transformation elements, 0 < $\chi$≤ 1, the value of which is determined by the decision maker based on subjective willingness, experience or historical data, the larger the $\chi$, the higher the centrality of the extracted key elements, and accordingly the smaller the number of the extracted key elements (namely elements to be subjected to transformation decisions).

When ${\alpha }_i\ge \xi$ is satisfied, the corresponding intelligent manufacturing constituents will be extracted, and finally N constituents will be extracted as the set of key elements for intelligent manufacturing transformation, that is, B = {B₁, B₂, …, B_N}, where B_s denotes the sth key element, that is, B_s = {$\left(C_i\right|{\alpha }_i\ge \xi ,$ $\text{i}\in \{\text{1,2},\cdots ,n\}$}, which determines the value of the parameter N. Clearly, B ⊂ C.

Complexity identification (determination of parameter K)

On the basis of the direct correlation group evaluation matrix $G={\left[g_{\mathit{ij}}\right]}_{n\times n}$ of the elements of intelligent manufacturing transformation, only the rows and columns where the key elements B_s of intelligent manufacturing transformation are located are retained, and then the direct correlation group evaluation matrix $\mathrm{\Psi }={\left[{\phi }_{\alpha \beta }\right]}_{N\times N},\alpha ,\beta \in \{\text{1,2},\cdots ,N\}$ of the key elements is obtained. By assembling the elements in $\mathrm{\Psi }$, the mean value $\phi$ of the direct correlation evaluation between the key elements can be obtained, which is calculated as following:

$$\overline{\phi }=\frac{1}{N\times N}\sum _{\alpha =1}^N\sum _{\beta =1}^N{\phi }_{\alpha \beta }$$ 7

On the basis of $\mathrm{\Psi }={\left[{\phi }_{\alpha \beta }\right]}_{N\times N}$, the mean value $\overline{\phi }$ is used as the threshold for judging whether or not there is an association between the elements, and the key element adjacency matrix $A={\left[a_{\alpha \beta }\right]}_{N\times N}$ is constructed, and the formula for calculating the element ${\alpha }_{\alpha \beta }$ in the matrix is:

$$a_{\alpha \beta }=\left\{\begin{array}{c}1,{\phi }_{a\beta }\ge \overline{\phi }\\ 0,{\phi }_{a\beta }<\overline{\phi }\end{array}\right),\alpha ,\beta \in \left\{1,2,\cdots ,N\right\}$$ 8

In the formula, $a_{\alpha \beta }=1$, indicates that there is an influence of key element $\beta$ on key element $\alpha$, i.e., the change of key element $\beta$ will change the adaptation value of key element $\alpha$; $a_{\alpha \beta }=0$, indicates that there is no influence of key element $\beta$ on key element $\alpha$, i.e., the change of key element $\beta$ will not change the adaptation value of key element $\alpha$ or the change is negligible.

Further, assembling the row elements in the matrix $A={\left[a_{\alpha \beta }\right]}_{N\times N}$ yields the key element correlation degree $k_{\chi }$, which is calculated by the formula:

$$k_{\chi }=\sum _{\beta =1}^N{a}_{\chi \beta },\chi \in \left\{\text{1,2},,,\cdots ,,,N\right\}$$ 9

In the formula, $k_{\chi }$ denotes the number of key elements that have influence on the key element χ and does not consider the influence of the key element on itself. According to the theory of NK model, the average value of the correlation degree $k_{\chi }$ of each key element is the parameter K, which is calculated by the following formula:

$$K=\frac{1}{N}\sum _{\chi =1}^N{k}_{\chi }$$ 10

In this equation, K represents the average number of times each key element in the system is affected by other elements. Obviously, the larger the value of K, the stronger the interaction between elements in the system and the more complex the system becomes. If the value of K is 0, there is no interaction between the elements in the system and the complexity is the lowest; on the contrary, if the value of K is equal to N minus 1, each element in the system will be affected by the other elements and the complexity will be the highest.

Fitness landscape map generation

Based on the NK model theory, the association relationship expressed in terms of the key element neighbourhood matrix $A={\left[a_{\alpha \beta }\right]}_{N\times N}$, when the alleles of the key elements Bs are mutated (namely element selection is performed) or the genes of the key elements that have a superordinate relationship with them are mutated, a random number is taken from the (0, 1) uniformly distributed random variable as the fitness value of the key elements Bs. that is:

$$e_s^d\sim U\left(0.1\right),s\in \left\{\text{1,2},,,\cdots ,,,N\right\},d\in \left\{\text{1,2},,,\cdots ,,,a\right\}$$ 11

Based on the constructed adaptation matrix $E={\left[e_s^d\right]}_{a\times N}$ for key elements of intelligent manufacturing transformation, the adaptation value ${\text{e}}^d$ of the allele combination $M_d$ of key elements of intelligent manufacturing transformation can be obtained, and its calculation formula is:

$${\text{e}}^d=\frac{1}{N}\sum _{s=1}^N{e}_s^d,d\in \left\{\text{1,2},,,\cdots ,,,a\right\}$$ 12

Combining the bit gene combinations of key elements of intelligent manufacturing transformation and mapping them to the three-dimensional space, the intelligent manufacturing transformation fitness landscape map is constructed, reflecting the interactions between key elements in the process of intelligent manufacturing transformation, as well as all the possible adaptive states of the complex system of intelligent manufacturing, as shown in Fig. 2 below.

Fig. 2.

Climbing schematics of fitness landscapes. Generated using Matlab R2021b software.

Optimal path selection

Based on the above analysis, it is known that the change exploration process of intelligent manufacturing involves selecting key elements (in general, the number of alleles is 2), and while the state of alleles changes from 0 to 1, the overall adaptive value of the intelligent manufacturing system will continue to increase, which is shown as a state of climbing in the fitness landscape map (Fig. 2). According to Fig. 2, starting from point a, after judging the adaptation value of the surrounding locations, it passes through points b, c, and d, and finally reaches the highest point, point e. Therefore, the process of starting from point a, comparing the overall fitness value of each point, and finally reaching point e after continuous searching is the search process for the transformation path of intelligent manufacturing, and the transformation path of intelligent manufacturing is a → b → c → d → e.

In the transformation process, according to the difference in the number of key elements of intelligent manufacturing transformation involved in each “climb,” the strategy for key elements of intelligent manufacturing transformation can be classified into single-element selection and multiple-element selection. Single-element selection refers to changing the allele values of only one key element at a time and is presented on the fitness landscape map as a process of climbing from one vertex to neighboring vertices, whereas multiple-element selection is a process that can change the allele values of more than one key element at a time and is presented as a process of climbing from a single vertex to more distant vertices. With the increase in the number of key elements for intelligent manufacturing transformation, the probability of finding the global optimal point may increase accordingly, but the risk borne by the enterprise will also increase, so the enterprise should choose properly according to the actual situation. Summarizing the previous section, we can obtain the calculation steps of the intelligent manufacturing transformation path selection method based on the NK model, as shown in Table 5.

Table 5.

Path selection steps for intelligent manufacturing transformation.

Steps	Details
1	Determination of intelligent manufacturing transformation constituent elements and acquisition of expert data
2	Calculate the centrality ai of the intelligent manufacturing constituent elements and the centrality threshold ξ according to Eqs. (1–6) to determine the parameter N, and extract the elements with centrality greater than the threshold as the key elements of intelligent manufacturing transformation
3	According to Eqs. (7–10), calculate the chi-square Kχ of the relevant key factors to determine the parameter K
4	According to Eqs. (11–12), construct the key factor adaptation matrix $E={\left[e_s^d\right]}_{a\times N}$, and calculate the adaptation value $e^d$ of the key factor allele combination $M_d$
5	By mapping the key element allele combination $M_d$ and its corresponding adaptation degree value to the three-dimensional space, the fitness landscape map of intelligent manufacturing transformation is constructed
6	Select the search method of intelligent manufacturing transformation path, and complete the selection of intelligent manufacturing transformation path through computer simulation

Simulation study of intelligent manufacturing transformation path selection

Based on the composition of the influencing factors of the intelligent manufacturing transformation of pharmaceutical enterprises, a questionnaire was designed to collect data on the direct influence relationship between the influencing factors. The DEMATEL method proposed by Fontela and Gabus⁴⁹ was then applied to determine the two key parameters N and K of the NK model. The NK model of adaptability to the intelligent manufacturing transformation was finally constructed, and computer simulation was used to determine the preferred intelligent manufacturing transformation path for pharmaceutical manufacturing enterprises, providing the corresponding selection results and analyses.

Data collection

To obtain the correlation evaluation data between the core elements, an expert committee was set up, consisting of five experts from universities and research institutes of different organizations and five experts from pharmaceutical manufacturing companies. They scored individually and anonymously, and the questionnaire was constructed as a comparative matrix of judgments between the six influencing factors in the form of a five-point Likert scale to evaluate the influence of the row elements on the column elements. The scores ranged from 1 to 5, corresponding to “no correlation” to “very high correlation”, with Z = {Z₀ = NO (no correlation), Z₁ = VL (very low), Z₂ = L (low), Z₃ = H (high), and Z₄ = VH (very high)}. The detailed questionnaire is shown in the appendix.

After evaluating the strengths and weaknesses of the correlations between the different elements mentioned above by the experts, 10 linguistic assessment phrase sets were obtained, and the final elemental correlation assessment matrix was compiled based on these evaluations as:

$$F_1=\left[\begin{array}{cccccc}-& H& L& L& \mathit{VH}& \mathit{VH}\\ VH& -& L& L& \mathit{VH}& \mathit{VH}\\ H& \mathit{VL}& -& H& H& H\\ VL& \mathit{VL}& L& -& L& L\\ H& H& \mathit{VL}& L& -& \mathit{VH}\\ L& H& \mathit{VL}& H& L& -\end{array}\right]$$

$$F_2=\left[\begin{array}{cccccc}-& \mathit{VH}& H& L& H& H\\ VH& -& \mathit{VH}& H& \mathit{VH}& L\\ H& L& -& H& H& \mathit{VH}\\ H& \mathit{VL}& H& -& H& L\\ H& L& L& H& -& H\\ L& H& L& L& H& -\end{array}\right]$$

$$F_3=\left[\begin{array}{cccccc}-& L& H& L& \mathit{VH}& \mathit{VH}\\ L& -& H& \mathit{VH}& \mathit{VH}& \mathit{VH}\\ H& H& -& H& \mathit{VH}& H\\ L& \mathit{VH}& H& -& \mathit{VH}& \mathit{VH}\\ VH& \mathit{VH}& \mathit{VH}& \mathit{VH}& -& \mathit{VH}\\ VH& \mathit{VH}& H& \mathit{VH}& \mathit{VH}& -\end{array}\right]$$

$$F_4=\left[\begin{array}{cccccc}-& H& H& \mathit{VH}& H& H\\ VH& -& \mathit{VH}& L& \mathit{VH}& H\\ H& \mathit{NO}& -& \mathit{NO}& H& H\\ H& \mathit{VL}& \mathit{VL}& -& \mathit{VH}& \mathit{VH}\\ H& \mathit{VL}& \mathit{VL}& \mathit{VL}& -& L\\ VH& \mathit{NO}& \mathit{VL}& \mathit{VL}& \mathit{VH}& -\end{array}\right]$$

$$F_5=\left[\begin{array}{cccccc}-& H& \mathit{VL}& L& L& L\\ L& -& L& L& L& L\\ VL& \mathit{VL}& -& H& L& \mathit{VL}\\ L& L& L& -& H& H\\ H& H& L& L& -& L\\ H& L& \mathit{VL}& L& L& -\end{array}\right]$$

$$F_6=\left[\begin{array}{cccccc}-& \mathit{VH}& \mathit{VH}& \mathit{NO}& \mathit{VH}& L\\ VH& -& \mathit{VH}& L& L& \mathit{VH}\\ VH& L& -& \mathit{VL}& \mathit{VH}& \mathit{VL}\\ NO& \mathit{NO}& L& -& \mathit{VH}& L\\ VH& \mathit{VH}& \mathit{VH}& L& -& \mathit{VH}\\ VH& \mathit{VH}& \mathit{VH}& L& \mathit{VH}& -\end{array}\right]$$

$$F_7=\left[\begin{array}{cccccc}-& L& H& L& L& \mathit{VH}\\ VL& -& H& \mathit{NO}& \mathit{VH}& \mathit{NO}\\ L& H& -& \mathit{NO}& H& \mathit{VL}\\ H& L& \mathit{VH}& -& L& L\\ H& L& H& L& -& \mathit{VL}\\ VH& \mathit{VL}& \mathit{VH}& H& H& -\end{array}\right]$$

$$F_8=\left[\begin{array}{cccccc}-& H& H& L& \mathit{VH}& \mathit{VH}\\ VH& -& L& \mathit{VL}& H& \mathit{VH}\\ VH& L& -& \mathit{NO}& L& L\\ H& L& \mathit{VL}& -& H& H\\ H& H& L& \mathit{VL}& -& \mathit{VH}\\ H& \mathit{VL}& \mathit{VL}& H& H& -\end{array}\right]$$

$$F_9=\left[\begin{array}{cccccc}-& \mathit{VL}& \mathit{VL}& \mathit{VL}& H& \mathit{VH}\\ VL& -& \mathit{VL}& \mathit{NO}& \mathit{VL}& \mathit{NO}\\ VL& \mathit{VL}& -& \mathit{VL}& H& L\\ H& \mathit{VL}& \mathit{VL}& -& \mathit{NO}& \mathit{NO}\\ VH& \mathit{VL}& L& L& -& H\\ VH& \mathit{VL}& H& \mathit{NO}& \mathit{VH}& -\end{array}\right]F_{10}=\left[\begin{array}{cccccc}-& H& \mathit{VH}& L& \mathit{NO}& \mathit{NO}\\ NO& -& L& L& \mathit{VH}& \mathit{VH}\\ NO& H& -& \mathit{VL}& H& \mathit{VL}\\ NO& \mathit{VH}& H& -& \mathit{VH}& L\\ VL& L& H& \mathit{VL}& -& \mathit{VH}\\ VL& H& \mathit{VH}& H& \mathit{VH}& -\end{array}\right]$$

Determination of key parameters

For the determination of parameter N, the linguistic phrase forms Z₀, Z₁, Z₂, Z₃, Z₄ provided by the experts are converted to their corresponding numerical values 0, 1, 2, 3, 4 based on the correspondence between the linguistic phrases and their subscripted values. In accordance with Eq. (1), the constituent element association evaluation matrices (F₁, F₂, F₃, F₄) provided by the experts are integrated to arrive at the intelligent manufacturing constituent elements The direct association group evaluation matrix G is:

$$G=\left[\begin{array}{cccccc}0.000& 2.900& 2.500& 2.000& 3.000& 3.100\\ 2.700& 0.000& 2.800& 1.900& 3.300& 2.800\\ 2.500& 1.900& 0.000& 1.600& 3.100& 2.200\\ 2.100& 1.900& 2.300& 0.000& 3.000& 2.500\\ 3.200& 2.600& 2.500& 2.100& 0.000& 3.200\\ 3.200& 2.300& 2.500& 2.400& 3.400& 0.000\end{array}\right]$$

Based on Eq. (2), the DEMATEL method is applied to normalise the constituent direct correlation group evaluation matrix G to obtain the normalised direct correlation group evaluation matrix X as follows:

$$X=\left[\begin{array}{cccccc}0.000& 0.210& 0.181& 0.145& 0.217& 0.225\\ 0.196& 0.000& 0.203& 0.138& 0.239& 0.203\\ 0.181& 0.138& 0.000& 0.116& 0.225& 0.159\\ 0.152& 0.138& 0.167& 0.000& 0.217& 0.181\\ 0.232& 0.188& 0.181& 0.152& 0.000& 0.232\\ 0.232& 0.167& 0.181& 0.174& 0.246& 0.000\end{array}\right]$$

Based on Markov absorptivity (Bosso et al., 1969; Atlaskin et al., 2021), according to matrix X and Eq. (3), the indirect correlation evaluation matrix Y of intelligent manufacturing transformation elements is constructed as follows:

$$Y=\left[\begin{array}{cccccc}0.780& 0.855& 0.831& 0.856& 0.800& 0.830\\ 0.818& 0.811& 0.840& 0.855& 0.810& 0.822\\ 0.685& 0.696& 0.669& 0.712& 0.682& 0.677\\ 0.706& 0.729& 0.727& 0.730& 0.707& 0.716\\ 0.839& 0.852& 0.838& 0.865& 0.759& 0.839\\ 0.851& 0.856& 0.851& 0.885& 0.831& 0.797\end{array}\right]$$

According to Eq. (4), the further construction of the comprehensive correlation matrix T of intelligent manufacturing components of pharmaceutical manufacturing enterprises is as follows:

$$T=\left[\begin{array}{cccccc}0.780& 1.065& 1.013& 1.001& 1.018& 1.055\\ 1.014& 0.811& 1.043& 0.992& 1.049& 1.024\\ 0.866& 0.834& 0.669& 0.828& 0.907& 0.837\\ 0.858& 0.867& 0.894& 0.730& 0.924& 0.897\\ 1.071& 1.041& 1.019& 1.017& 0.759& 1.071\\ 1.083& 1.023& 1.032& 1.059& 1.078& 0.797\end{array}\right]$$

Further, the centrality a_i of the constituent elements can be obtained by assembling the row and column elements in the matrix T. The centrality a_i indicates the size of the role played by the corresponding intelligent manufacturing transformation element in the set of constituent elements C. According to Eq. (5), the specific centrality calculation results are shown in Table 6 below.

Table 6.

Calculation results of centrality.

	C1	C2	C3	C4	C5	C6
Direct association	5.931	5.934	4.94	5.17	5.977	6.07
Indirect association	5.671	5.64	5.669	5.627	5.735	5.68
Centrality ai	11.602	11.574	10.609	10.797	11.712	11.750

To extract the key factors of intelligent manufacturing transformation, i.e. to determine the parameter N in the NK model, the centrality extraction threshold $\xi$ must be determined first, where the maximum centrality percentage $\chi$ of the intelligent manufacturing transformation factors takes the value of 0.75 according to the experience of previous studies (Kourtellis et al., 2013; Weng et al., 2020). According to formula (6) and Table 6, the centrality extraction threshold $\xi$ = 0.75*11.750 = 8.8125. from Table 6, the values of centrality a_i of key factors are greater than the threshold $\xi$, from which it can be obtained that the parameter N = 6.

In this paper, based on the direct correlation group evaluation matrix G, we continue to calculate and analyses the complexity parameter K, keeping the rows and columns where the key element B_S is located, and obtain the key element direct correlation group evaluation matrix $\psi$. This is shown as follows:

$$\psi =\left[\begin{array}{cccccc}0.000& 2.900& 2.500& 2.000& 3.000& 3.100\\ 2.700& 0.000& 2.800& 1.900& 3.300& 2.800\\ 2.500& 1.900& 0.000& 1.600& 3.100& 2.200\\ 2.100& 1.900& 2.300& 0.000& 3.000& 2.500\\ 3.200& 2.600& 2.500& 2.100& 0.000& 3.200\\ 3.200& 2.300& 2.500& 2.400& 3.400& 0.000\end{array}\right]$$

By assembling the elements in $\psi$, the mean value $\overline{\phi }$ of the direct correlation evaluation between their elements can be calculated, and according to Eq. (7), the mean value $\overline{\phi }$=77.5/36 = 2.153. Then, according to the threshold value of the mean value $\overline{\phi }$ for determining whether there exists a correlation between the elements, and according to Eq. (8), the element ${\phi }_{\alpha \beta }\ge$ the mean value $\overline{\phi }$ is assigned to be 1, and the element ${\phi }_{\alpha \beta }<$ the mean value $\overline{\phi }$ is assigned to be 0, and the resultant construction of the key element neighbourhood matrix A is shown below:

$$A=\left[\begin{array}{cccccc}0& 1& 1& 0& 1& 1\\ 1& 0& 1& 0& 1& 1\\ 1& 0& 0& 0& 1& 1\\ 0& 0& 1& 0& 1& 1\\ 1& 1& 0& 0& 0& 1\\ 1& 1& 0& 1& 1& 0\end{array}\right]$$

The number of key elements that have influence on the key element $\chi$, i.e., $k_{\chi }$, is obtained by aggregating the row elements in matrix A. According to Eq. (9), it is calculated that: $k_{\chi }$=21; based on the theory of NK model, the average value of the correlation $k_{\chi }$ of each key element is used as the parameter K. K represents the average number of key elements in the system that are affected by the other elements, and according to the Eq. (10), it is calculated that: K = 3.5.

NK model analysis

The six key factors that may affect intelligent manufacturing transformation are abstracted into six entities, and the entity modeling approach is applied to explore the interactions between the factors and their impact on intelligent manufacturing transformation. Intelligent manufacturing transformation path selection based on the NK model refers to the decision-making process regarding the key elements of intelligent manufacturing transformation during morphological change. This process involves randomly selecting numbers from the interval (0, 1) as the adaptability values, based on the correlation relationship between the key elements as shown by the key elements' neighborhood matrix A. Therefore, based on the state coding of each element of the NK model and the formula for calculating the degree of adaptation, the Matlab R2021b software (version number: R2021b, URL: https://www.mathworks.com/products/matlab.html) is used to simulate the change process of the degree of adaptation of the factors affecting intelligent manufacturing transformation, as shown in Table 7. The results are plotted as a fitness landscape map of the intelligent manufacturing transformation, in which the climbing process can be clearly seen, as shown in Fig. 3.

Table 7.

Fitness changes of core elements of intelligent manufacturing transformation.

a1	a2	a3	a4	a5	a6	e1	e2	e3	e4	e5	e6	E	Optimum
0	0	0	0	0	0	0.192	0.197	0.501	0.495	0.388	0.030	0.301	a
0	0	0	0	0	1	0.842	0.081	0.906	0.326	0.757	0.160	0.512
0	0	0	0	1	0	0.680	0.154	0.757	0.702	0.610	0.327	0.538	b
0	0	0	1	0	0	0.009	0.837	0.523	0.927	0.205	0.412	0.486
0	0	1	0	0	0	0.421	0.555	0.741	0.291	0.259	0.647	0.485
0	1	0	0	0	0	0.487	0.889	0.586	0.409	0.341	0.329	0.507
1	0	0	0	0	0	0.533	0.504	0.273	0.532	0.782	0.343	0.494
1	1	0	0	0	0	0.770	0.657	0.297	0.492	0.073	0.974	0.544
1	0	1	0	0	0	0.354	0.918	0.998	0.444	0.234	0.032	0.497
1	0	0	1	0	0	0.404	0.679	0.292	0.120	0.637	0.465	0.433
1	0	0	0	1	0	0.627	0.640	0.370	0.090	0.298	0.469	0.416
1	0	0	0	0	1	0.118	0.756	0.471	0.783	0.364	0.813	0.551
0	1	1	0	0	0	0.401	0.389	0.559	0.253	0.998	0.747	0.558
0	1	0	1	0	0	0.286	0.253	0.693	0.558	1.000	0.296	0.514
0	1	0	0	1	0	0.765	0.096	0.475	0.610	0.697	0.801	0.574	c
0	1	0	0	0	1	0.395	0.681	0.430	0.014	0.594	0.320	0.406
0	0	1	1	0	0	0.089	0.893	0.901	0.127	0.425	0.883	0.553
0	0	1	0	1	0	0.881	0.245	0.273	0.884	0.666	0.234	0.531
0	0	1	0	0	1	0.254	0.261	0.352	0.481	0.479	0.940	0.461
0	0	0	1	1	0	0.440	0.424	0.857	0.004	0.303	0.050	0.346
0	0	0	1	0	1	0.629	0.105	0.760	0.394	0.371	0.884	0.524
0	0	0	0	1	1	0.778	0.143	0.715	0.215	0.569	0.277	0.449
1	1	1	0	0	0	0.935	0.336	0.106	0.978	0.171	0.516	0.507
1	1	0	1	0	0	0.560	0.960	0.100	0.497	0.853	0.104	0.512
1	1	0	0	1	0	0.639	0.780	0.930	0.466	0.539	0.140	0.582
1	1	0	0	0	1	0.528	0.342	0.341	0.572	0.423	0.706	0.485
1	0	1	1	0	0	0.511	0.215	0.521	0.842	0.716	0.212	0.503
1	0	1	0	1	0	0.883	0.532	0.596	0.239	0.010	0.670	0.488
1	0	1	0	0	1	0.944	0.591	0.760	0.067	0.016	0.348	0.454
1	0	0	1	1	0	0.647	0.378	0.892	0.423	0.609	0.298	0.541
1	0	0	1	0	1	0.640	0.522	0.658	0.812	0.144	0.123	0.483
1	0	0	0	1	1	0.901	0.509	0.827	0.438	0.580	0.008	0.544
0	1	1	1	0	0	0.565	0.866	0.356	0.701	0.486	0.059	0.506
0	1	1	0	1	0	0.760	0.667	0.373	0.821	0.716	0.437	0.629	d
0	1	1	0	0	1	0.776	0.434	0.200	0.882	0.712	0.043	0.508
0	1	0	1	1	0	0.378	0.823	0.006	0.634	0.399	0.223	0.411
0	1	0	1	0	1	0.323	0.731	0.353	0.993	0.258	0.615	0.545
0	1	0	0	1	1	0.873	0.539	0.408	0.729	0.634	0.068	0.542
0	0	1	1	1	0	0.974	0.295	0.257	0.921	0.769	0.216	0.572
0	0	1	1	0	1	0.501	0.107	0.567	0.950	0.338	0.305	0.461
0	0	1	0	1	1	0.813	0.169	0.635	0.578	0.787	0.097	0.513
0	0	0	1	1	1	0.070	0.624	0.278	0.775	0.908	0.205	0.477
1	1	1	1	0	0	0.708	0.754	0.649	0.423	0.662	0.250	0.574
1	1	1	0	1	0	0.627	0.875	0.141	0.555	0.764	0.778	0.623
1	1	1	0	0	1	0.041	0.672	0.776	0.598	0.795	0.767	0.608
1	1	0	1	1	0	0.337	0.916	0.130	0.832	0.708	0.644	0.594
1	1	0	1	0	1	0.939	0.763	0.935	0.415	0.302	0.252	0.601
1	1	0	0	1	1	0.539	0.870	0.618	0.905	0.261	0.728	0.653
1	0	1	1	0	1	0.994	0.638	0.504	0.036	0.894	0.008	0.512
1	0	1	0	1	1	0.417	0.219	0.963	0.442	0.993	0.672	0.618
1	0	0	1	1	1	0.806	0.472	0.145	0.739	0.358	0.931	0.575
0	1	1	1	1	0	0.687	0.365	0.598	0.875	0.283	0.771	0.596
0	1	1	1	0	1	0.342	0.903	0.117	0.453	0.600	0.689	0.517
0	1	1	0	1	1	0.401	0.767	0.993	0.910	0.555	0.448	0.679	e
0	1	0	1	1	1	0.898	0.317	0.137	0.517	0.964	0.714	0.591
0	0	1	1	1	1	0.915	0.708	0.879	0.173	0.337	0.631	0.607
0	1	1	1	1	1	0.853	0.741	0.586	0.520	0.512	0.714	0.654
1	0	1	1	1	1	0.649	0.952	0.612	0.715	0.268	0.978	0.696
1	1	0	1	1	1	0.540	0.934	0.706	0.331	0.526	0.690	0.621
1	1	1	0	1	1	0.641	0.742	0.788	0.897	0.556	0.770	0.732	f
1	1	1	1	0	1	0.894	0.717	0.903	0.749	0.531	0.336	0.688
1	1	1	1	1	0	0.570	0.742	0.724	0.931	0.813	0.208	0.665
1	1	1	1	1	1	0.719	0.729	0.973	0.654	0.896	0.963	0.822	g

Fig. 3.

Fitness landscape and climbing process of intelligent manufacturing transformation in pharmaceutical manufacturing enterprises. Generated using Matlab R2021b software.

To ensure the stability and reliability of the results of the software simulation and analysis, this study simulated the fitness matrix and performed local search 100,000 times. Statistical analyses revealed that: first, the importance of key element B5 in achieving the allele combination of intelligent manufacturing transformation exceeds that of other key elements. Therefore, the first stage of the transformation needs to consider key element B5, namely pharmaceutical policy, which is manifested as a climb from position {000000} to position {000010} on the fitness landscape map, as shown in Fig. 3a; second, the importance of key element B2 in achieving the B5 allele combination exceeds that of other key elements. Thus, the second stage of the transformation needs to consider key factor B2, namely technological innovation, which is manifested as a process of climbing from position {000010} to position {000010} on the fitness landscape map, as shown in Fig. 3b; and so on. The results of the simulation and analysis suggest that the intelligent manufacturing transformation of pharmaceutical manufacturing enterprises should be carried out in the order of pharmaceutical policy, technological innovation, organizational management, resource heterogeneity, market demand, and industry competition. In other words, the path of the transformation system adaptability is B5 → B2 → B3 → B6 → B1 → B4.

In the search for the optimal path, the two-dimensional fitness landscape map expression is clearer and more intuitive. Drawing the two-dimensional landscape map, as shown in Fig. 4 below, the first line of numbers in the box represents the combination state of the factors affecting intelligent manufacturing. The second line of numbers represents the fitness value corresponding to the combination state of the factors. Among them, the mountain peaks represent the states of the combinations that have a high fitness value. The line with an arrow is the path of the landscape map, indicating the path from the lower fitness combination state to the neighboring higher fitness combination state. The optimal path of the process of migration of each layer of the factor combination to the point of the global optimal state is shown as a dashed arrow route, which is shown as follows:

Fig. 4.

Two-dimensional fitness landscape map.

000000 → 000010 → 010010 → 011010 → 011011 → 111,011 → 111,111.

Based on Figs. 3 and 4, the intelligent manufacturing transformation path climbing diagram of pharmaceutical manufacturing enterprises shows the climbing process of the influential elements of intelligent manufacturing transformation from point 00000 to 111,111. This indicates that pharmaceutical manufacturing enterprises in the process of intelligent manufacturing transformation should first consider the pharmaceutical policy, which is the government's macroeconomic policy to guide the development of the industry, adjust the imbalance of the industrial structure, and build a well-ordered market competition. Secondly, they should consider technological innovation, meaning the enterprise develops new technologies or new products and services based on existing technologies. Thirdly, organizational management should be considered, which is the enterprise's ability to effectively integrate internal and external resources to cope with external threats and identify and exploit external opportunities, reflecting its competitive advantage in a dynamic environment. Fourthly, the heterogeneity of resources should be considered, encompassing all the things used by the organization to create value, including assets, knowledge, and various capabilities. Fifthly, market demand should be considered, referring to the quantity of products or services that a particular customer is willing to buy in a particular region, time, marketing environment, and marketing plan. Finally, industry competition should be considered, representing the level of competition in an industry that measures the degree of competition among enterprises, which is a key factor in adjusting for economic policy uncertainty and the investment behaviors of enterprises.

Conclusion and discussion

Conclusions

Each industrial revolution promotes the transformation of organizational forms, bringing new phenomena and issues for business transformation, which contribute to the continuous development of organizational theory knowledge systems^15,40. Some scholars have studied the influencing factors and patterns of intelligent manufacturing transformation from different perspectives^3,4,10. However, few consider intelligent manufacturing transformation as a complex adaptive system to explore the influencing factors of pharmaceutical manufacturing enterprises and the optimal path choice for intelligent manufacturing transformation. In this study, based on the NK model theory, we use computer simulation technology for the path selection of intelligent manufacturing transformation.

The findings of the study can be summarized as follows: First, the study identified the main factors affecting the intelligent manufacturing transformation of pharmaceutical manufacturing enterprises as technological innovation, pharmaceutical policy, industry competition, organizational management, resource heterogeneity, and market demand. Understanding the interactions between these factors can help pharmaceutical enterprises develop more effective intelligent manufacturing transformation strategies and avoid potential risks. Second, in the context of intelligent transformation, pharmaceutical manufacturing enterprises should consider improving the degree of adaptability of the intelligent manufacturing transformation system in the order of “Pharmaceutical policy → Technological innovation → Organizational management → Resource heterogeneity → Market demand → Industry competition.” The proposed path determines the implementation order of core elements of intelligent manufacturing under a complex dynamic system, which can help pharmaceutical manufacturing enterprises prioritize the development of more critical transformation elements and enhance the performance of intelligent manufacturing transformation under the background of intelligentization.

Theoretical contributions

Firstly, exploring the intelligent manufacturing transformation issue from the perspective of complex systems helps to systematically understand the internal logic and evolutionary process of intelligent manufacturing. Although previous studies have explored the issue of intelligent manufacturing transformation a great deal based on the perspectives of technology, resources, and markets^14,23, they have focused on the enumeration among individual elements and neglected the synergistic effects among the constituent elements of intelligent manufacturing^4,25, the lack of systematic thinking on intelligent manufacturing transformation, but also did not explain the order in which these elements can form the optimal path of intelligent manufacturing transformation, cannot give the real ‘path’. Based on the theory of complex adaptive systems, this paper considers the interactions between elements, explores the synergistic influence of multiple intelligent manufacturing elements on the overall transformation through the perspective of complex systems, and selects the optimal path of intelligent manufacturing transformation with the sequential order of the elements by using the simulation method of the NK model, which expands the research on the path of intelligent manufacturing transformation, and provides new references for the systematic understanding of intelligent manufacturing transformation.

Secondly, based on the NK model theory, the sequence of elements in the optimal path of intelligent manufacturing transformation of pharmaceutical manufacturing enterprises is clearly delineated. Since the application of NK model theory to organisational management, scholars have mainly applied it to the relationship between enterprises^34,35, the organisational performance of enterprises^30,33, and very few researches have paid attention to the selection of paths for intelligent manufacturing transformation in pharmaceutical manufacturing enterprises. In contrast, this paper explores the optimal path selection for intelligent manufacturing transformation of pharmaceutical manufacturing enterprises based on the NK model theory and through the Matlab computer simulation method, which clearly delineates the sequential order of the elements that form the optimal path for intelligent manufacturing transformation of pharmaceutical manufacturing enterprises and expands the application field of the NK model theory.

Practical implications

First, for manufacturing enterprises, the interrelationship between influencing factors should be comprehensively considered in the process of intelligent manufacturing transformation, and systematic thinking should be adopted to formulate transformation strategies. At the same time, manufacturing enterprises are at different stages and levels in the process of intelligent manufacturing transformation. For manufacturing enterprises in different conditions or states, they can draw on simulation methods to select the path most suitable for their development by systematically analyzing and evaluating the advantages and disadvantages of different transformation paths. This method can help enterprises better understand the various possible influencing factors and provide a scientific basis for transformation decisions.

Second, for the government, it should formulate more precise and targeted policies to support the intelligent manufacturing transformation of pharmaceutical manufacturing enterprises. At the same time, the government can build an industry-university-research cooperation platform to promote cooperation and exchanges between pharmaceutical manufacturing enterprises and scientific research institutions, universities, and other relevant units.

Research limitations and prospects

Although this paper explains the path selection of intelligent manufacturing transformation of pharmaceutical manufacturing enterprises and forms some valuable findings, there are still some shortcomings: first, this paper only explores the intelligent manufacturing transformation influencing factors and path selection of pharmaceutical manufacturing enterprises, and there may be different evolutionary development path in other traditional manufacturing industries, which is worthy of in-depth exploration in the future. It is possible to further extend the study to other industries, and explore the commonalities and differences of intelligent manufacturing transformation paths in different industries to form a more general theoretical framework. Secondly, intelligent manufacturing transformation of manufacturing enterprises has gradually become a necessary path for the transformation and upgrading development of enterprises, but the relevant research is still insufficient, and the path selection can be tested in the future using empirical and other research methods. Empirical data can be further collected to validate the path selection framework proposed in this paper and explore more effective path selection methods.

Ethical approval

This study involved human participants. Informed consent was obtained from all subjects. The participants were fully informed about the purpose and procedure of the study, and their participation was entirely voluntary. Data collection was conducted anonymously to protect the participants' privacy.

Author contributions

W.Z: Conceptualization, methodology, data curation, writing-review and editing; P.O: methodology, validation, writing-review and editing; X.K: investigation, data curation; S.Q: Data collection, data analysis, Supervision; S.L: Data collection, data analysis; Z.J: Data collection, data analysis.All authors reviewed the manuscript.

Funding

This research was supported by the National Social Science Foundation of China (23BGL313); National Natural Science Foundation of China (72362025); Jiangxi University Humanities and Social Sciences Research Project (GL23133); Jiangxi Province Vocational Education Teaching Reform Research Project (JXJG-23-49-11); Jiangxi Provincial Social Science Foundation Project (24GL65D).

Data availability

All data generated or analysed during this study are included in this published article.

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-024-70502-7.