1 Introduction

Industry 4.0 has introduced a new production paradigm since its inception at the Hannover Messe fair in 2011 [1]. Often referred to as the "fourth industrial revolution," it underscores its transformative nature. Its adoption necessitates disruptive changes with profound impacts on all industrial and service processes across the supply chain due to end-to-end automation and digital integration, among other factors. This new approach to business operations leverages digital technologies to react more swiftly and effectively to rapidly changing markets [2]. Additionally, it empowers firms to offer more customized and sustainable products and services while concurrently increasing their efficiency and productivity [3].

A key aspect of Industry 4.0 is the comprehensive integration, both horizontally and vertically, of various business functions —product development, production, logistics, and marketing, among others— alongside suppliers and customers. This integration connects machines, sensors, facilities, and humans, facilitating the exchange of data among them [4, 5]. Therefore, interconnectedness stands as a pivotal feature of Industry 4.0 [6]. This interconnection occurs through a spectrum of innovative technologies [3, 7,8,9,10,11,12,13], whose adoption often evolves in tandem with enhancements in the company’s organizational structure, infrastructure, and culture.

In this context, embracing Industry 4.0 can guide organizations toward gaining competitive advantages, positively impacting society at large. However, its intricate nature presents a range of barriers and challenges, including scientific, technological, economic, cultural, and political ones [11]. Overcoming these challenges is essential for the successful implementation of Industry 4.0 to truly enhance the competitiveness and performance of firms. A key consideration is whether a company has the necessary resources and capabilities [14]. Notably, SMEs generally face different challenges and risks compared to larger companies [15,16,17]. Their modest scale and limited resources often magnify the risks posed by evolving business scenarios. For this reason, some authors argue that large enterprises are better prepared for Industry 4.0 than SMEs [18, 19]. However, lagging behind in this transition could limit SMEs' ability to fully realize the potential benefits arising from Industry 4.0 [20].

In these circumstances, several studies have investigated the challenges and inhibiting factors organizations face when endeavoring to adopt Industry 4.0 [21,22,23,24], revealing their multifaceted nature. Consequently, different fronts must be considered, often interrelated to varying extents. These challenges and barriers are particularly pronounced for SMEs.

The scenario described above was chosen by the authors to illustrate the utility and effectiveness of a hybrid evaluation model based on the integration of the Analytical Hierarchy Process (AHP) and Fuzzy Inference Systems (FISs) for companies undertaking transformative projects. This combination of methodologies allows companies to better handle the numerous complexities and uncertainties that these projects entail, aiding them in self-assessing their initial capabilities and crafting a strategic roadmap for success. The methodologies are integrated automatically and strengthened through Dong and Saaty’s consensus approach, eliminating the need for additional rounds of expert consultations.

While this work applies the hybrid evaluation model to facilitate the implementation of Industry 4.0 in SMEs, it is important to highlight that it has the potential to positively impact companies of various sizes and industries in various types of transformative journeys. In this sense, it establishes a generalizable instrumental framework for organizations to self-assess and support decision-making in improvement processes and projects of different natures.

Thus, our work offers both methodological contributions —showcasing the development of a decision support system for organizations undertaking transformative projects— and a deeper, empirically-grounded understanding of the implementation of Industry 4.0 in SMEs. This topic is particularly relevant today, given the potential of Industry 4.0 to elevate the performance of companies and their supply chains, and the importance of SMEs as the backbone of most economies in terms of contribution to employment and gross domestic product (GDP) [25,26,27]. Accordingly, prior works have emphasized the pressing need for further research on this matter [15, 28,29,30], especially from a practical standpoint [31].

In analyzing this specific problem and application scenario, this research revolves around the following two fundamental research questions (RQs):

  • RQ1: What are the most relevant barriers that SMEs need to surmount to enhance their ability to effectively implement Industry 4.0, and how can these barriers be structured?

  • RQ2: How can SMEs perform a self-assessment of their readiness to embrace the Industry 4.0 paradigm and, based on this assessment, how can they design, monitor, and control the progress of their implementation roadmap?

To address RQ1, this study constructs a three-level conceptual framework based on an exhaustive analysis of the relevant literature. This framework identifies and categorizes the primary barriers encountered by SMEs when implementing Industry 4.0 into factors and dimensions, facilitating a more comprehensive understanding of the problem. To address RQ2, and in line with prior discussions, this work integrates the AHP and FIS methodologies. Enhanced through Dong and Saaty’s consensus methodology [32], the AHP initially computes the importance weights of all identified barriers based on the judgments of a team of five experts. Next, the FIS utilizes this information to evaluate how well an SME is prepared to face the challenges of Industry 4.0. This approach interestingly circumvents the need for a new round of expert consultations, which can be overly cumbersome in practice. Upon obtaining this self-assessment, SMEs can strategize and oversee the progress of their Industry 4.0 implementation roadmap, aiming for heightened adaptation to this paradigm and increased resulting performance.

With these objectives, this paper is organized into six sections. Following this introductory section that establishes the context, aims, and contributions of our work, Section 2 reviews the literature concerning the barriers to Industry 4.0 and outlines the three-level framework. Subsequently, Section 3 describes the two-stage methodological approach developed. Moving forward, Section 4 engages in the analysis and discussion of the resulting surface maps. Next, Section 5 offers an illustrative example to showcase practical applicability and utility of the proposed model, demonstrating how it can function as a self-assessment tool while aiding in the planning and monitoring of Industry 4.0 adoption strategies. Finally, Section 6 presents the conclusions and suggests important avenues for future research.

2 Literature review and conceptual framework

Although the term Industry 4.0 has emerged relatively recently and the limits of its definition are not yet fully delineated [6], it has received significant attention in scientific literature, forming a substantial body of knowledge. The following two subsections describe the conducted literature review on the barriers of Industry 4.0 and the methodologies employed in their study, respectively.

2.1 The barriers to Industry 4.0

Understanding and addressing barriers is fundamental for the successful adoption of Industry 4.0 in all types of organizations. By identifying and mitigating these barriers, companies can strategically allocate resources to optimize their journey toward Industry 4.0 and maximize the benefits derived from its implementation. In these circumstances, our study focuses primarily on barriers due to their critical role in Industry 4.0 transformation.

To build the conceptual framework of our illustrative case, we examined works published from 2014 to the present that explicitly study the barriers faced during the implementation of Industry 4.0. Additionally, we focused on the specific context of SMEs. As mentioned earlier, this selection was motivated by the recognition that SMEs face greater limitations compared to larger corporations in fully capitalizing on the potential improvements offered by Industry 4.0 [29, 33, 34]. The causes behind these constraints are diverse, encompassing both internal factors within the SME itself and external factors beyond the company’s boundaries [14, 21].

The literature identified numerous barriers, the intricate relationships of which are complex to establish and unique to the Industry 4.0 transformation [30]. Prior studies have attempted to categorize these barriers. Below, we discuss some categorizations are especially relevant in light of the research questions posed in the previous section.

Orzes et al. [35] classified 19 barriers into six categories: economic / financial, cultural, competencies / resources, technical, legal, and implementation process. Luthra and Mangla [36] structured 18 challenges across four dimensions: organizational, legal and ethical issues, strategic, and technological. Elhusseiny and Crispim [37] adopted Luthra and Mangla’s four-dimensional model to analyze the manufacturing sector in both developed and developing countries. They concluded that developing countries face more barriers but also perceive more opportunities related to Industry 4.0. De Lucas et al. [38] proposed four dimensions in their study on LATAM and Spanish SMEs, distinct from Luthra and Mangla’s model. They classified barriers as technological, formative, economic, and contextual. Meanwhile, Rauch et al. [34] categorized 14 main limitations and barriers into six clusters: culture, implementation, people, resource management, security, and strategy. Antony et al. [39] conceptualized ten dimensions in their Industry 4.0 readiness model and evaluated their criticality across different sectors and types of organizations. On the other hand, Horváth and Szabó [18] identified nine barriers across five dimensions: human resources, financial resources and profitability, management reality, organizational factors, and technological and process integration and cooperation. They emphasized the significant impact of the first three on SMEs.

The sample of works mentioned above illustrates the lack of consensus regarding the number of barriers and the dimensions in which they can be structured. In our case, constructing our framework involved multiple iterations and regular meetings among the team of coauthors aiming to produce a robust and well-structured framework that accurately reflects the complexities of Industry 4.0 implementation barriers. Drawing from the groundwork laid by preceding research, we propose a six-dimensional classification encompassing the 17 most prominently mentioned barriers. Table 1 includes these barriers and dimensions, which we describe in the following paragraphs.

Table 1 Key dimensions and barriers to Industry 4.0
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An important dimension, which strongly interacts with the others, is related to economics and finance. Financial constraints have been identified as the most important barrier to successful adoption of Industry 4.0 [36]. The integration of various cutting-edge technologies and processes within and beyond the organization requires substantial investments. These investments are necessary not only in technological infrastructure but also in training employees to navigate the new digitally integrated business scenario [23, 46]. Meeting these investment requirements can be particularly challenging for SMEs due to their limited financial capacity and restricted access to funding mechanisms [18, 31, 55]. Additionally, reluctance to adopt Industry 4.0 may stem from an unclear vision of future benefits [22, 35].

A second key dimension emerges in the companies’ strategic and operational limitations. Several authors have highlighted the lack of a digital strategy and a clear vision towards Industry 4.0 as one of the most prominent barriers [14, 36]. On the operational side, the reconfiguration of businesses demands meticulous, long-term planning of operations, including the determination of necessary skills [23, 43]. Infrastructure must be adapted to facilitate the successful integration of new manufacturing technologies, processes, and business models. Special attention should be paid to ensuring connectivity, scalability, standardization, and data quality [2], enabling real-time decision-making synchronized with supply chain partners. In this context, the lack of technological infrastructure and poor-quality connectivity were identified as key barriers to address while implementing Industry 4.0 [23].

The human aspect is also critical, presenting additional challenges. Effective adoption of Industry 4.0 requires well-qualified and engaged top management, capable of establishing specific roadmaps to support implementation and daily operations [55, 56]. Managers also need to design training programs that address initial resistance to change among workers and equip them with the skills to succeed in the new environment [47, 50, 57]. The role of workers will evolve towards more responsible tasks, often transitioning from shop-floor operators to decision-makers [1]. Additionally, concerns may arise regarding the impact of Industry 4.0 on employment. While different sources offer varying viewpoints on these effects [58], there is often fear of job loss or displacement due to process automation and/or a lack of skills in new technologies, which managers should also consider [46, 49].

Several authors emphasize the lack of digital culture as a key dimension [35, 36, 40, 59], which has important implications. Many professionals still lack a clear understanding of the profound impact of Industry 4.0, particularly in certain industries, which limits their grasp of what the transformation entails [60]. This lack of understanding can lead to skepticism about the transition process and its consequences [42]. Such uncertainties may heighten perceptions of risks associated with adopting Industry 4.0, thereby discouraging necessary investments [28, 61]. Moreover, inadequate support from top management can exacerbate these challenges. Given the current limitations in research on this topic, there remains considerable ground to cover in transferring knowledge from scientists to practitioners [34].

Another important dimension concerns the lack of a legal and normative framework that businesses can rely on to counter unexpected threats or mitigate their effects [35]. Perceived risks associated with Industry 4.0 implementation, particularly in SMEs, can evoke feelings of vulnerability and demotivation [16]. Issues related to cybersecurity and data privacy also require careful consideration [3, 53, 62]. The absence of an adequate regulatory framework may lead to an inward look (silo-like) that hampers interoperability and information exchange; in contrast, a robust legal framework fosters trust and transparency [34]. Governments thus play a crucial role in enacting policies that address security issues and protect data privacy, thereby supporting Industry 4.0 adoption [21, 24].

The last dimension is related to collaboration in both internal and external spheres. In an Industry 4.0 scenario that integrates machines, sensors, facilities, and humans, complex interactions emerge. Cooperation assumes paramount importance within the organization, to foster unity across diverse organizational units, cultivate an organizational culture and achieve improved productivity [51]. Equally essential is coordination with all supply chain partners, including suppliers and customers [3, 4, 44, 63]. This collaboration is pivotal not only for harmonizing technological platforms but also for establishing global standards and data-sharing protocols, promoting transparency, and building trust in transactions [14, 54, 64,65,66].

In this sense, the hierarchical conceptual framework we propose is constructed with a total of 17 barriers situated at the lowest level (3rd hierarchy). These barriers are grouped into six dimensions that define the intermediate level (2nd hierarchy). Finally, these six dimensions are categorized based on their nature into two overarching types (1st hierarchy): intrinsic and extrinsic [21]. Among the six aforementioned dimensions, four (cultural, strategic/operational, human, and economic/financial) mostly pertain to the company's internal sphere, while the remaining two (legal, collaboration) extend beyond those boundaries.

2.2 Methodological approaches

In terms of the methodologies employed to examine the barriers to Industry 4.0 adoption, most papers used literature review techniques [30, 37] and empirically-grounded techniques, including surveys [16, 17, 29, 36, 48, 54, 60], case studies [18, 19, 49], and focus groups [35].

Other studies have employed multicriteria decision-making (MCDM) techniques to establish priority and dependence relations among barriers. Techniques used include DEMATEL (Decision MAking Trial and Evaluation Laboratory) [24, 42, 67, 68], BWM (Best–Worst Method) [23], and AHP (Analytic Hierarchy Process) [36, 69,70,71], as well as hybrid approaches. Hybrid examples are AHP – Fuzzy TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) [47, 72], AHP – ELECTRE (Elimination and Choice Expressing Reality) [73], Fuzzy AHP – DEMATEL [74], ISM (Interpretive Structural Modeling) –MICMAC (Cross-Impact Matrix Multiplication Applied to Classification) [22, 75], and COPRAS [76] among others.

From a methodological perspective, our work contributes to this body of knowledge by integrating Fuzzy Inference Systems (FISs) into the evaluation process. The incorporation of FISs offers significant advantages, such as handling uncertainty in quantifying variables and providing a flexible framework that accommodates various data types. Additionally, FIS evaluation avoids the 'reversal rank' issue common in Multicriteria Decision-Making (MCDM) methods, allowing individual assessments without affecting previous rankings. This integration facilitates a more comprehensive analysis of barriers to Industry 4.0 adoption and supports the development of a decision support tool. However, it also introduces potential constraints, including added complexity for users unfamiliar with fuzzy logic and subjectivity arising from linguistic variables and expert-based fuzzy rules.

Our work also presents a novel integration of AHP and FIS. In traditional FISs, knowledge generation typically relies on reaching a consensus among experts. This process involves defining rules for each potential combination of fuzzy labels on input variables, which can be cumbersome in practical applications. To address this challenge, we propose an innovative approach of automatically generating knowledge, as detailed in the next section. This method utilizes weights derived from AHP to formulate the rule bases of the FIS, eliminating the necessity for a second round of expert consultations and thereby saving time and costs.

3 Methods

Figure 1 provides an overview of the stages followed in the research process. Up to this point, we have defined the research problem, formulated the research questions, and proposed the conceptual framework based on the literature review. In this section, we describe the two-stage (AHP + FIS) methodological development of the tool that SMEs can use to assess their capabilities for successful Industry 4.0 implementation and support decision-making. The figure distinguishes clearly between the methodological development of the decision support tool vertically, and its periodic usage by the organization horizontally.

Fig. 1
figure 1

Overview of the research process

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In developing the decision support tool, we prioritized generality over deriving specific insights to ensure broad applicability and relevance. Therefore, the information gathering process was not restricted to any specific industry, geography, or technology, enabling a wider range of organizations to benefit from our research outcomes. By comprehensively detailing the model, our goal is to demonstrate how this methodological approach can be practically applied by organizations with diverse characteristics.

3.1 Analytical Hierarchy Process and consensus application

The hierarchical conceptual framework presented in the previous section, which is based on a three-level architecture and founded on comprehensive literature analysis, is visually depicted in Fig. 2.

Fig. 2
figure 2

Hierarchical structure of the factors, dimensions, and barriers to Industry 4.0

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After establishing the three hierarchical levels, the criteria (factors, dimensions, and barriers) considered at each one underwent both partial and global weighting. For this purpose, the AHP methodology [77, 78] was employed, which involves expert panels comparing the relative importance of each pair of criteria. Additionally, at each level, Dong and Saaty’s consensus methodology [32] was applied after aggregating the experts’ assessments to mitigate potential inconsistencies arising from diverse backgrounds and knowledge.

Five experts (1 to 5) from both academic and industrial domains participated in this study. They were strategically selected to form a well-balanced expert panel, with each contributing diverse perspectives on the integration of Industry 4.0 technologies and principles within companies. This approach ensured coverage of a range of fundamental aspects within Industry 4.0 ecosystems. Table 2 provides details about this panel of experts, including their expertise, professional roles, industries, and levels of experience.

Table 2 Anonymized description of the expert panel
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The weight assigned to each expert, denoted as ρi, was determined based on their professional experience (PEi), academic degree (ADi), and depth of knowledge in two relevant fields of knowledge (FK1i – Technology; and FK2i – Production), using the same rationale as in Sanchez-Garrido et al. (2022) [79]. Equation (1) formalizes this calculation in our case. These weights, derived from the individual values provided by the coauthors of the article for the five participating experts, are later normalized, denoted as ρi*, such that \({\sum }_{i}{\rho }_{i}^{*}=1\). The following normalized weights have been obtained after being rounded to one decimal: ρ1* = 0.1; ρ2* = 0.3; ρ3* = 0.1; ρ4* = 0.2; ρ5* = 0.3.

$${\rho }_{i}=\left(\begin{array}{c}\frac{{PE}_{i}}{\text{max}\{P{E}_{i}\}}+\frac{{AD}_{i}}{\text{max}\{{AD}_{i}\}}+\frac{{FK1}_{i}}{5}+\frac{{FK2}_{i}}{5}\end{array}\right)/4$$
(1)

The following paragraphs provide details on the procedure followed to compute the importance weights (priority vector) for the four dimensions included in the “Intrinsic” factor (C1: Economic / Financial, C2: Strategic / Operational, C3: Human, and C4: Cultural).

First, each expert provided their individual judgments by comparing the importance of each pair of criteria according to the standard Saaty scale (from 1 to 9 points). These evaluations aimed to obtain the reciprocal Pairwise Comparison Matrices (PCMs) specific to the hierarchical level under analysis. As an example, Fig. 3 showcases the judgments granted by Expert 1 for the four dimensions of the "Intrinsic" factor.

Fig. 3
figure 3

Comparative importance given by Expert 1 for the dimensions of the "Intrinsic" factor

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From the judgments of each expert, a PCM, \({x}_{i,j}^{k,t}\), is built, where \(k\in \{1,\text{ 2,3},\text{4,5}\}\) identifies the expert, \(i\) and \(j\) represent the barriers, and \(t\) refers to the iteration. To calculate the initial weights (Wi) of the criteria in a PCM, the eigenvector of the matrix's elements is determined. In our case, this is achieved by averaging the rows of the PCM normalized according to the sum of its columns. Subsequently, the consistency ratio (CR) is calculated to verify the consistency of the expert judgments. The CR is derived from the consistency index, which is determined using the eigenvalue of the initial PCM and the weighting vector \({W}_{i}\). It is then divided by the tabulated random index corresponding to the criteria being analyzed, as per Eq. (2). If the CR does not exceed the predefined threshold allowed for the given criteria, the consistency of PCM is considered acceptable; otherwise, it should be reformulated. This analysis for Expert 1 is illustrated in Fig. 4.

Fig. 4
figure 4

Calculation of the weighting vector (\({W}_{i}\)) of the four intrinsic dimensions using the AHP and the analysis of the consistency of the judgments of the first expert

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$$CR=\frac{{\uplambda }_{max}-n}{n-1}$$
(2)

After ensuring the consistency of the initial PCM of all experts (\({x}_{i,j}^{k,t})\), they were aggregated using the weighted geometric mean, according to Eq. (3). This mathematical approach was chosen because it allows for the incorporation of individual weights assigned by different experts. In our case, the aggregate matrix of the team of experts at time \(\mathrm t=0\), denoted as \({G}_{0}\), is illustrated in Fig. 5.

Fig. 5
figure 5

Aggregated-paired comparison matrix for the four intrinsic dimensions

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$${{G}_{t}=g}_{i,j}^{t}=\prod\nolimits_{k=1}^{m}{\left({x}_{i,j}^{k,t}\right)}^{{\rho }_{k}}$$
(3)

Next, the iterative consensus method developed by Dong and Saaty [30] was applied. Under this method, at each iteration, \(t\to t+1\), the expert who was farthest from the current aggregate matrix was prompted to adjust their individual PCM. The distance between the PCM of Expert \(k\) to the current aggregate matrix —referred to as the group compatibility index of Expert \(k\), \({GCI}_{k}\)—, was calculated using Eq. (4), where \(n\) represents the number of criteria.

$${GCI}_{k}{=d(x}_{i,j}^{k} , {g}_{i,j})=\frac{1}{{n}^{2}}\sum\nolimits_{i=1}^{n}\sum\nolimits_{j=1}^{n}{{g}_{i,j}\cdot x}_{j,i}^{k}$$
(4)

If the farthest expert, denoted by \(h\), accepted the modification, their updated PCM remained consistent and their components were calculated by Eq. (5), where \(\alpha\!\in\lbrack\text{0,1}\rbrack\) represents the willingness-to-change coefficient of Expert \(h\) in that iteration (the greater willingness, the closer \(\alpha\) is to zero). The determination of \(\alpha\) for each iteration was based on the variation between two consensus indices: that of the farthest expert [\(max\{{GCI}_{k}\)}] and the average value [\(mean\{{GCI}_{k}\)}]. Thus, higher percentage variations correspond to lower \(\alpha\) in the 0–1 range. The other experts, denoted by \(z\), kept their PCM unchanged. In this sense, the personal weight of Expert \(z\) remained constant, \(\left({\rho }_{z}^{t+1}={\rho }_{z}^{t}\right)\) and the new aggregate matrix (\({G}_{t+1}\)) was calculated using the weighted geometric mean applied at \(t+1\), allowing us to proceed to the next iteration.

$${x}_{i,j}^{h,t+1}={\left({x}_{i,j}^{h,t}\right)}^{\alpha }\cdot {\left({g}_{i,j}^{t}\right)}^{(1-\alpha )}$$
(5)

In case the farthest expert (\(h\)) rejected the modification, their personal weight was reduced according to Eq. (6), increasing that of the other experts (\(z\)) as per Eq. (7), where \(\beta\!\in [\text{0,1}]\) is the weight-adjustment coefficient proposed by the moderator in such iteration (the more penalizing for \(h\), the closer \(\beta\) is to zero). The value of \(\beta\) was determined based on the distance of Expert \(h\) with respect to the last final aggregate; higher percentage distances correspond to higher \(\beta\) in the 0–1 range. In this situation, the PCM of all experts, including \(h\), remained constant \(\left({x}_{i,j}^{h,t+1}={x}_{i,j}^{h,t}\right)\) and the new aggregate matrix (\({G}_{t+1}\)) was calculated according to the weighted geometric mean applied at time \(t+1\).

$${\rho }_{h}^{t+1}={\rho }_{h}^{t}\cdot \beta +{\rho }_{h}^{t}\cdot (1-\beta )\cdot {\rho }_{h}^{t}$$
(6)
$${\rho }_{z}^{t+1}={\rho }_{z}^{t}+{\rho }_{z}^{t}\cdot (1-\beta )\cdot {\rho }_{z}^{t}$$
(7)

In subsequent iterations, the expert whose PCM was farthest from the current aggregate matrix was prompted again to modify their PCM (excluding those previously rejected). The iterative process continued until one of the following three stopping conditions was met: (a) the distance of all experts from the previous aggregate matrix was less than 1.01 (1% variation); (b) the previously determined maximum number of iterations (in our case, 10) was reached; (c) all experts refused to change their opinion.

Figure 6 illustrates the calculations for Iterations 1 and 6. In Iteration 1, Expert 4 was identified as the farthest from the initial aggregate matrix and accepted the modification of his/her PCM, leading to the aggregation of matrix \({G}_{1}\). Note that \(\alpha\) is 0.3, indicating relatively low deviations from Expert 4’s PCM to the aggregated PCM. In Iteration 6, Expert 4 was once again identified as the farthest expert but declined to modify his/her judgment matrix. In this instance, both \(\alpha\) and \(\beta\) equal 0.6, indicating a moderate deviation from the aggregated PCM. Consequently, the individual weights of all experts were adjusted before calculating the aggregate matrix \({G}_{6}\).

Fig. 6
figure 6

Computations of Iterations 1 and 6 according to Dong and Saaty’s methodology [32]

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After meeting a stopping condition at Iteration 10, Fig. 7 displays the final aggregate matrix (\({G}_{9}\)), the weighting vector of the four dimensions (\({W}_{i}\); 0.51 for C1, 0.17 for C2, 0.19 for C3, and 0.13 for C4), and the corresponding consistency analysis.

Fig. 7
figure 7

Final aggregate matrix and the importance weights of the intrinsic dimensions

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Applying the same procedure to all levels within the hierarchical structure of the proposed framework, we derived the partial weights of criteria at each hierarchy. Using these partial weights, we computed the overall weights. These are reported in Table 3. The results of the consensus procedure have helped mitigate discrepancies in the judgments made by the experts, thereby contributing to the empirical validation of our findings.

Table 3 Weights of the different barriers, dimensions, and factors
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3.2 Fuzzy Inference System

FISs offer the capability to incorporate a knowledge base that guides the decision-making process of an evaluation model through linguistic concepts, closely resembling human reasoning. This trait makes them particularly valuable in contexts where evaluation involves a substantial degree of subjectivity, as is the case in this study. Defining this knowledge involves determining: (a) linguistic labels that define the partitions of input and output fuzzy variables; and (b) "if–then" rules that specify the label assigned to each fuzzy output variable based on the combination of labels assigned to the input fuzzy variables.

In this research work, inferences are drawn from the input data of an SME in the model, which includes crisp numerical assessments of the 17 indicators measuring the barriers at the lowest level of our hierarchical framework. The inference processes utilized follow the Mamdani approach, encompassing the following five steps: (1) fuzzification of crisp input variables; (2) application of appropriate logical operators to the antecedents of each rule; (3) implication of the antecedent applied to the consequent of each rule; (4) aggregation of consequents from all activated rules; and (5) defuzzification of the final aggregate [80].

In our study, the global FIS linked to the conceptual framework comprises nine interconnected inference subsystems: six evaluate dimension values from the 17 barriers (“B-D1.FIS” to “B-D6.FIS”), two assess factor values from the preceding six dimensions (“D-F1.FIS” to “D-F2.FIS”), and the final one derives the overall evaluation from the two preceding factors (“Eval.FIS”). This overall structure is depicted in Fig. 8. Within this framework, the evaluations generated for the output variables of inference subsystems at a specific hierarchy level subsequently serve as input values for the inference subsystems at the next higher level in the hierarchy. The relative importance of variables at each level was determined using the AHP results obtained in the previous subsection.

Fig. 8
figure 8

Structure of the global FIS formed by nine chained inference subsystems

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The fuzzy logic embedded in each FIS aims to formalize the reasoning process inherent in human language within its knowledge base. It is noteworthy that in our FIS, we retained all initially considered variables since none exhibited a high level of dependence, meaning no subsystem relied on more than four input variables. Therefore, we did not simplify any subsystem by removing variables, which would typically be advisable if there were significant interdependencies to prevent an overly complex set of rules.

3.2.1 Variable partition

Initially, designing the knowledge base required configuring the partitions of the fuzzy variables within each inference subsystem. Both the type and number of labels for dividing the traditional domain of each variable were established through a consensus approach based on input from the consulted experts.

For the type of labels, it was agreed to use trapezoidal labels for the extremes of each variable and triangular labels for those in between. These linear membership functions, commonly utilized in FISs, are deemed robust for representing the imprecision of linguistic evaluations across various types of information sources (e.g., Delgado et al., 1992). Regarding the number of labels, the consensus was to use:

  • three labels (L: low; M: medium; H: high) for all input variables,

  • five labels (VL: very low; L: low; M: medium; H: high; VH: very high) for the output variables of subsystems with two or three input variables, and.

  • seven labels (EL: extremely low; VL: very low; L: low; M: medium; H: high; VH: very high; EL: extremely high) for the output variables of subsystems that have four input variables.

To parameterize the size of the variables’ labels, an initial agreement was made on their core widths using the 2-tuples method [81]. Subsequently, their boundaries were determined with central symmetry [82] to establish robust partitions that enhance comprehensibility and maintain critical semantic properties [83]. For a visual representation, Fig. 9 illustrates the approved partitions of all model variables.

Fig. 9
figure 9

Approved partitions of all model variables

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3.2.2 Rule extraction

To complete the knowledge base, defining the conditional rules governing evaluations within each inference subsystem is essential. Typically, this involves consulting experts to determine the fuzzy label assigned to the output variable in each knowledge rule. However, our approach allowed for the automatic derivation of all FIS rules based on the local weights of its input variables, previously determined using the AHP. Thus, the integration of both methodologies occurs during the rule extraction process. The appropriate label for the output variable in each rule was determined by selecting the label (from the five or seven labels, as explained in the previous subsection) that had the minimum distance to the weighted trapezoid of the antecedent labels, in accordance with the AHP weights.

In this context, the rule extraction process consists of three steps for each rule. First, the weighted trapezoid for each rule was calculated as the weighted sum of its input variable labels according to their respective weights. Subsequently, the distances from each weighted trapezoid to all potential output variable labels in each rule (VL, L, M, H, and VH, in the case of five labels) were calculated using Eq. (8), where \(\left[{a}_{i},{b}_{i},{c}_{i},{d}_{i}\right]\) represents the weighted trapezoid of each rule, \(\left[{a}^{j},{b}^{j},{c}^{j},{d}^{j}\right]\) refers to each possible output label, and \({A}_{k}, k=\{1,\dots ,4\}\) denotes the assigned importance weights for the trapezoid apexes. Finally, the output label ultimately assigned to each rule corresponds to that associated with the minimum distance.

$$D\left(\left[{a}_{i},{b}_{i},{c}_{i},{d}_{i}\right]-\left[{a}^{j},{b}^{j},{c}^{j},{d}^{j}\right]\right)= \sqrt{{A}_{1}{\cdot ({a}_{i}-{a}^{j})}^{2}+{{A}_{2}\cdot ({b}_{i}-{b}^{j})}^{2}+{{A}_{3}\cdot ({c}_{i}-{c}^{j})}^{2}+{{A}_{4}\cdot ({d}_{i}-{d}^{j})}^{2}.}$$
(8)

To illustrate the rule extraction process, Fig. 10 provides complete detail of how the rules were derived for C6 (Collaboration) from C61 (Lack of standards/protocols) and C62 (Insufficient cooperation/trust). Each row (rule) includes the weighted trapezoids, the calculated distances, and the output label assigned based on the minimum distance.

Fig. 10
figure 10

Fuzzy rules extraction in subsystem C6

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This automatic derivation of FIS rules eliminates the need for a second round of expert involvement. However, it is important to note that this automatic rule extraction process is not rigid and could be subject to refinement or modification by the panel of experts if deemed necessary, although balancing the benefits of further refinement against associated resource expenditure is crucial.

Once the knowledge bases for the nine components of the global FIS were obtained, they were implemented in MATLAB’s Fuzzy Logic Toolbox to infer the overall capability of an SME to successfully implement Industry 4.0. The online version of this paper includes supplementary material consisting of the nine ".fis" files that comprise the global FIS.

3.2.3 Global assessment

To obtain the inference results for a specific company based on its assessment of barriers using MATLAB, the following script can be implemented. This script facilitates the generation of both local and global valuation inferences, allowing for subsequent representation and analysis of each subsystem’s behavior in the corresponding surface maps, which are discussed in the following section for our application.

figure a

4 Surface maps and discussion

Implementing the knowledge bases in MATLAB allows us to analyze the inference results of the FIS and the nine subsystems through their surface maps. These three-dimensional graphs provide information on the outcome and gradient of the output variable (z-axis) for any combination and variation of the values of two input variables (x- and y-axis), with the other input variables, if they exist, held constant (not shown in the map). This approach enables intuitive identification of patterns and trends, facilitating the understanding of the model's results.

The values in all three axes of each map were standardized on a [0–10] scale to enhance comparability, with higher scores indicating a better assessment for the SME in question. The linkage between the nine subsystems was established through MATLAB programming, following the structure previously outlined in Fig. 8. This setup enables users to evaluate the capability of an SME to effectively adopt Industry 4.0 based on the values of the lower-level input variables.

In the following paragraphs, the most relevant surface maps in each of the hierarchies are presented and discussed.

The upper hierarchy provides the overall evaluation of the SME’s capacity to effectively implement Industry 4.0 (I4) according to the values obtained for the intrinsic (F1) and extrinsic (F2) factors. Figure 11 displays the surface map corresponding to this subsystem. While AHP already established the higher significance of internal factors (\({W}_{F1}=0.83; {W}_{F2}=0.17\)), the surface map allows the visualization of this relative importance across all the solution space.

Fig. 11
figure 11

Surface map of subsystem I4 (vs. F1 and F2)

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Upon examining this map, it becomes apparent that regardless of the F2 level (which pertains to the organization's external environment), focusing on improvements in F1 yields substantial enhancements in the overall evaluation. These improvements in F1 may involve actions aimed at strengthening the economic and financial aspects of the transformation process, conducting systematic evaluations of strategic and operational objectives, giving greater attention to cultural factors embedded within the organization, and increasing awareness among human resources. Conversely, when the analysis is reversed, the gradient is less steep. In this context, while considering external factors does contribute to achieving higher levels of success in implementing Industry 4.0, efforts solely directed at improving them do not result in equally significant enhancements.

These results align with prior research, such as Moeuf et al. [84], who identified four primary risks in adopting Industry 4.0 in SMEs, all situated within the internal sphere of the company. Also, Horváth and Szabó [18] highlighted internal barriers as the most significant for SMEs when adopting Industry 4.0. Moreover, the recent study by Chauhan et al. [21] found that intrinsic barriers hold greater significance than extrinsic barriers in terms of their impact on the transformation journey toward Industry 4.0.

The second hierarchy establishes connections between factors and dimensions, resulting in two subsystems linked to intrinsic and extrinsic factors, respectively. We focus on the former (F1), given its higher importance, whose evaluation depends on four dimensions: economic / financial (C1), strategic / operational (C2), human (C3), and cultural (C4). Figure 12 illustrates the surface map of this subsystem, linking F1 with C1 and C4, the dimensions with the highest and lowest weights (\({W}_{C1}=0.51;\) \({W}_{C4}=0.13\)). The other input variables (C1 and C2) are held constant at three different levels: low (Fig. 12a), medium (Fig. 12b), and high (Fig. 12c).

Fig. 12
figure 12

Surface map of subsystem F1 (vs. C1 and C4, for three different combinations of constant C2 and C3 - low, medium and high respectively-)

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These surface maps illustrate the varying growth gradients of F1, emphasizing the stronger impact of C1 compared to C4, consistent with their respective weights. This underscores the critical importance of addressing economic and financial barriers when implementing Industry 4.0. An insightful observation from these scenarios is that enhancing F1 more effectively involves concurrent actions on both dimensions (highlighted by the red arrow) rather than maximizing them individually (white arrows).

Furthermore, dimensions C2 and C3 also contribute significantly to shaping the outcome, as indicated by surfaces for greater height (representing higher response values) with increasing values of these dimensions. It is noteworthy that all three maps exhibit a flat plateau in their lower zone, indicating poor F1 results for low values of C1 and C4. The height of this plateau increases as C2 and C3 values grow. Beyond this plateau threshold, the response tends to rise in line with increasing values of C1 and C4, especially when both dimensions increase simultaneously.

The lower hierarchy links the dimensions with their corresponding implementation barriers, resulting in six subsystems related to dimensions C1 to C6. Due to its high weight, we now consider the economic / financial dimension (C1), dependent on three underlying barriers: unclear vision regarding future economic benefits (C11), high investments required (C12), and limited financial capacity (C13). Among the various surface maps attainable through their combination, Fig. 13 showcases those corresponding to the barriers with higher weights: C11 (\({W}_{C11}=0.34\)) and C13 (\({W}_{C13}=0.37\)), in three scenarios of constant values of C12: low (Fig. 13a), medium (Fig. 13b), and high (Fig. 13c).

Fig. 13
figure 13

Surface map of subsystem C1 (vs. C11 and C13, for three different levels of constant C12 - low, medium and high respectively-)

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The displayed maps exhibit considerable symmetry, depicting comparable growth gradients in C1 along both the C11 and C13 axes. This level of symmetry underscores the similar influence of both barriers on the response, consistent with their designated weights. This aligns with prior works, such as [43, 85], who highlighted the considerable impact of SMEs' constrained financial capacity on the scale of prospective projects. In this fashion, companies are especially cautious with disruptive projects whose implementation process and future benefits are uncertain. In addition, C12 significantly modulates the result, leading to surfaces with greater elevation (indicating higher response values) when it increases. This reasonable finding shows the better adaptation of SMEs to the changes required and the faster benefit realization when they are able to undertake investment projects of a certain magnitude. This encompasses not only technology acquisition and infrastructure adaptation but also skill development within their human resources.

Similarly, to the previous analysis, a flat plateau is evident in the lower region of all three maps, indicating poor C1 results for low values of C11 and C13. The height of this plateau is influenced by C12. Once surpassing these plateau values, the response increases significantly with higher values of C11 and C13, particularly noticeable for medium and high levels of C12. Once again, optimizing C1 is more efficiently achieved by simultaneously increasing both C11 and C13 (as indicated by the red arrow) rather than individually (white arrows). This observation underscores that Industry 4.0 permeates the entire business ecosystem. Therefore, a comprehensive understanding and unified commitment across all involved aspects are essential to achieve successful implementation with a cohesive focus. Without this unified approach, a fragmented, localized strategy is unlikely to yield equally effective results in achieving desired outcomes.

5 Illustration of usage and managerial value

After discussing the surface maps that relate the output and input variables of the FIS, this section aims to demonstrate how the proposed model can benefit a specific SME by providing two valuable pieces of information. First, the SME gains the capability to analyze and assess its initial position concerning the effective implementation of Industry 4.0. Secondly, the model empowers the company to outline a strategic roadmap toward achieving greater alignment with the Industry 4.0 paradigm. It is important to note that regardless of the initial reference point, more advanced levels of successful implementation will necessitate more expensive and complex improvement strategies [55].

This potential is realized through the model’s ability to provide a comprehensive understanding of the SME’s initial state and to easily identify key areas for improvement. To demonstrate the application of the proposed decision support tool clearly, while ensuring privacy concerns are not compromised, we have constructed a simulated case study. However, the same process can be applied in a real-world scenario.

5.1 Evaluation of the initial position

To assess the initial positioning of an SME regarding the implementation of Industry 4.0, the user should assign values to each of the 17 barriers identified within our hierarchical conceptual model (see Fig. 2). The score for each barrier should range from 0 (worst) to 10 (best). In practice, this rating can be conducted either internally, through self-assessment by a dedicated team of employees well-versed in the company’s operations, or externally, by recruiting external professionals who bring an impartial and specialized perspective.

For illustrative purposes, let us assume the evaluation of a specific SME for these 17 barriers is represented in the "FIS Inputs (Assess_1)" row of Fig. 14. It is assumed that the company assigns moderate values to the barriers encompassed within C2 (strategic/operational), C3 (human), and C4 (cultural) dimensions, while assigning lower values to the barriers of the remaining dimensions, particularly C1 (economic/financial).

Fig. 14
figure 14

Illustration of initial evaluation through specific scoring of the 17 barriers in an SME

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Using the 17 scores for the barriers, the decision support tool automatically calculates the quantification for the six dimensions (intermediate level), the two factors (top level), and, finally, the overall evaluation of the SME’s potential to successfully embrace the Industry 4.0 paradigm. In this sense, with these 17 values, the upstream connected subsystems generate corresponding output values, as visualized in Fig. 14 for our example. In this specific instance, the final score is low, precisely 3.03 (out of 10). The low outcome can be attributed to the company’s low ratings across three dimensions: the highest-weight dimension, C1, with a score of 1.24/10, as well as both external dimensions: C5, with 3.00/10, and C6, with 2.71/10. Overall, Fig. 14 also reveals that the company receives a more favorable assessment of intrinsic factors (4.27/10) as opposed to extrinsic factors (1.24/10). For easier identification, shading in this figure indicates the criterion of highest importance within each subsystem.

5.2 Formulation of improvement strategies

The decision support tool provides significant value by guiding SMEs in their improvement endeavors towards better alignment with Industry 4.0. This guidance aims to reduce the probability of failure in the digital transformation process and leverage the considerable benefits of the new paradigm. In this context, it is feasible to analyze and propose the desired improvement trajectory, which can be visually depicted through surface maps. This trajectory depends on the distinct values assigned to barriers at the lowest level of the proposed model’s hierarchy.

In this example, it is strategic to initially focus improvement efforts on C1 (economic/financial), given its highest global weight (42.23%; see Table 3) and lowest rating (1.24/10). Assuming the company simultaneously addresses the three barriers of this dimension—C11, C12, and C13—the new assessments are shown in the “FIS Inputs (Assess_2)” row of Fig. 15, while the other assessments remain unchanged. Implementing improvement actions targeting the economic and financial barriers would have upstream effects on the FIS, resulting in an increased final evaluation of 5.18 (out of 10), which significantly surpasses the initial score (3.03/10), as depicted in Fig. 15.

Fig. 15
figure 15

Illustration of new evaluation after improvements in barriers C11, C12, and C13

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Let us now consider that, after addressing the financial and economic barriers, the SME shifts its focus to C51, which indicates the absence of a robust legal framework. Given its significant global rating, second only to those in dimension C1 (previously addressed), and its initial valuation of only 3/10, addressing this barrier is crucial. Proceeding in a similar manner, Fig. 16 portrays the updated evaluation of this barrier and the assessments after implementing further improvement actions targeting this barrier. Interestingly, this strategic enhancement raises the final evaluation to 6.40 (out of 10).

Fig. 16
figure 16

Illustration of final evaluation after additional improvements in barrier C51

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As evident from this illustrative example, the proposed model, with periodic updates of its input values, demonstrates remarkable responsiveness to the implementation of strategies and measures that SMEs might plan and execute to attain increased levels of adaptation to Industry 4.0. In this fashion, the model holds significant value for these organizations in two main ways. Firstly, it enables an SME to assess its status, making it possible to compare against objectives, benchmarks, or competition. Secondly, once the existing situation is understood, the model not only aids in directing the SME’s efforts towards strategic areas for improvement, but also streamlines the monitoring of key outputs from its inference systems. Therefore, organizations can design a roadmap to facilitate a successful implementation of Industry 4.0.

The roadmap and progression of the example company under study in this section can be visualized on the surface map presented in Fig. 17, where reference point (ref. p.) “1” signifies the initial position, ref. p. “2” reflects the enhanced position following improvements in C1, and ref. p. “3” showcases the final position after the enhancements in C51.

Fig. 17
figure 17

Example of improvement roadmap towards successful implementation of Industry 4.0

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Finally, it is essential to emphasize the iterative nature of improvement endeavors within the broad context of Industry 4.0. The journey toward operational excellence requires not only identifying weak points but also dynamically adapting strategies in response to fast-changing circumstances and technological advancements. Embracing this proactive approach to continuous improvement enables organizations to nimbly navigate the complexities and uncertainties of modern industrial landscapes, thereby fostering competitive advantages.

6 Conclusions

This research work proposes a hybrid evaluation model that integrates the Analytical Hierarchy Process (AHP) and Fuzzy Inference Systems (FISs) for companies involved in transformative projects. This unique combination of methodologies empowers companies to navigate the complexities and uncertainties inherent in such projects, enabling them to effectively self-assess their initial capabilities and develop strategic roadmaps for success.

To demonstrate the effectiveness of our proposed approach, we applied it to evaluate SMEs' capacity to tackle the primary challenges encountered during the adoption of Industry 4.0. Additionally, we illustrated how this methodology can be used to formulate improvement strategies, focusing on areas where SMEs show the most significant deficiencies. This case study was chosen because of the substantial potential of Industry 4.0 technologies and the critical role SMEs play in any economy. However, the hybrid methodology can be adapted for companies to conduct their own assessments and develop customized roadmaps to address various transformative challenges, whether strategic, operational, or technological in nature.

In this specific instance, this study contributes to enhancing both the theoretical understanding and practical implementation of Industry 4.0 by addressing two key research questions (RQs) posed in the introductory section. Our findings provide the following answers to these RQs:

  • RQ1: What are the most relevant barriers that SMEs need to overcome to enhance their ability to effectively implement Industry 4.0, and how can these barriers be structured?

Through an extensive examination of problem-specific literature, we identified 17 significant barriers that SMEs encounter when adopting Industry 4.0. These barriers form the basis of a hierarchical conceptual framework, grouped into six dimensions: (1) economic/financial; (2) strategic/operational; (3) human; (4) cultural; (5) legal; and (6) collaboration. The first four dimensions pertain to the internal sphere of the organization, while the remaining two dimensions extend beyond these boundaries.

We employed the AHP, complemented by the consensus methodology developed by Dong and Saaty [32], to assess the relative importance of these barriers using judgments from a panel of five experts. Economic and financial barriers emerged as the most significant, with constraints in financial capacity and unclear visions of future economic benefits being particularly critical. Human-related, strategic-operational, and legal barriers followed in importance.

  • RQ2: How can SMEs perform a self-assessment of their readiness to embrace the Industry 4.0 paradigm and, based on this assessment, how can they design, monitor, and control the progress of their implementation roadmap?

By identifying these barriers, SMEs can utilize the proposed integrated approach to conduct a comprehensive assessment of their capabilities for successful Industry 4.0 implementation. This enables organizations to understand their current status relative to the 17 barriers, identify strategic areas for improvement, and develop a customized roadmap for effective implementation.

The integrated AHP and FIS strategy offers a decision-support tool that assists SMEs in navigating digital transformation with a lower risk of failure and greater benefits. Our hybrid approach also contributes methodologically by automating the creation of FIS rule bases using weights obtained from AHP evaluations, thereby reducing the reliance on extensive expert consultations.

In essence, our work presents a replicable and flexible model suitable for organizations embarking on transformative initiatives. This methodological proposal offers a generalizable framework that companies can readily employ for self-assessment across diverse contexts, with significant implications for professionals across various industries.

Our findings also hold important implications for policymakers, who play a crucial role in providing necessary support through policies, financial assistance, and regulatory frameworks. Furthermore, fostering technological innovation hubs and providing access to affordable, tailored Industry 4.0 solutions can empower organizations to overcome implementation challenges. Collaborative initiatives involving industry associations, academic institutions, and government bodies can further facilitate knowledge-sharing, skill development, and the exchange of best practices. Overall, by leveraging political support, technological innovation, and collaborative networks, SMEs can navigate the complexities of Industry 4.0 more effectively, positioning themselves to thrive in the era of digital transformation.

Finally, it is important to acknowledge a significant limitation of this research related to its findings on the relevance of Industry 4.0 barriers. While the hybrid methodology employed here holds potential for application across organizations of varying sizes and industries, as well as for analyzing other transformation projects, caution is warranted regarding the assessment of barrier significance. This caution stems from the reliance on judgments from a panel of just five experts confined to a relatively small geographic area; the significance of barriers may vary for organizations in different geographical contexts, sizes, or industrial sectors.

In this regard, professionals using this methodological approach to support decision-making in Industry 4.0 implementation or other improvement projects should consider assembling a broad and diverse panel of experts that aligns with the organization’s characteristics and project requirements. Understanding the variability in barrier relevance across company sizes, industries, and geographies, as well as exploring specific barriers associated with different Industry 4.0 technologies, would be valuable research directions for academics.

Moreover, decision-makers may benefit from complementing expert judgments with data from additional sources such as surveys or case studies to strengthen the robustness of findings in specific applications. Additionally, companies could enhance the accessibility, usability, and understanding of the data analysis process and results by developing user-friendly interfaces and utilizing advanced visualization tools.

Another limitation of our study is its exclusive focus on Industry 4.0 barriers, which are crucial for understanding digital transformation. For future academic research or practical application of this methodology, considering enablers alongside barriers could provide a more holistic view of the Industry 4.0 implementation landscape.

Additionally, exploring similar studies on Industry 5.0 implementation would be particularly intriguing, as the methodology used here could offer significant value in analyzing this emerging paradigm. Industry 5.0 aims to transition towards a more sustainable, resilient, and human-centric industrial landscape, making it an area of growing interest among academics and practitioners alike.