# Leveraging Industry 4.0 for Sustainable Manufacturing: A Quantitative Analysis Using FI-RST
## 2 Methods
### FI-RST Model  
#### 2.2.1 FI-RST Model: Application and Conditions  
The Fuzzy Integration-Rough Set Theory (FI-RST) model was employed to quantitatively assess the impact of Industry 4.0 technologies on the sustainability performance of the 36 manufacturing firms included in this study. The FI-RST model integrates fuzzy logic to account for uncertainty in sustainability metrics and rough set theory to classify and evaluate the contribution of each technology to sustainability outcomes. The application of the FI-RST model followed a three-phase approach that is detailed below (Figure 4) [12; 17]. These phases were aligned with the specific conditions of data availability, technology adoption levels, and the firms' sustainability metrics. The steps were as follows:  
Phase 1: Data Preparation and Fuzzification  
In this phase, raw data from the questionnaire responses and field studies were collected and pre-processed. The questionnaire included data points related to economic, environmental, and social dimensions of sustainability. The collected data were then fuzzified, meaning the crisp values (e.g., waste reduction percentages and energy savings) were converted into fuzzy numbers using fuzzy membership functions. These membership functions were constructed based on ranges derived from the sustainability performance of the firms. For example, energy savings were categorized into three fuzzy sets: \"low,\" \"medium,\" and \"high,\" with corresponding membership values assigned to each company's reported energy savings. Phase 2: Application of the Fuzzy Integral  
The Choquet fuzzy integral was applied to aggregate the fuzzified sustainability scores for each firm across the three dimensions of sustainability. The fuzzy integral  
Figure 4: FI-RST model application process. allowed the model to account for the interdependencies between different sustainability indicators (e.g., energy savings and waste reduction). The weighting factors for the Choquet integral were derived from the relative importance of the sustainability indicators, which was determined based on expert input from industry professionals, as well as the literature on sustainable manufacturing. Formula Used [18; 19; 20]:  
\[S(T)=\int Th(x)d\mu(x)\]  
where:  
\(S(T)\) is the sustainability score for a specific technology TTT;  
\(h(x)\) is the fuzzified score for each sustainability criterion (e.g., energy savings and waste reduction);  
\(\mu(x)\) is the fuzzy measure, reflecting the relative importance of each criterion. Phase 3: Rough Set Theory for Classification  
After calculating the sustainability scores using the fuzzy integral, rough set theory was applied to classify the technologies based on their impact on sustainability. The firms were divided into decision classes (high, medium, and low) depending on their overall sustainability scores. The rough set approach involved computing upper and lower approximations for the decision classes, identifying the key technologies that most strongly influenced sustainability outcomes. The rough set analysis allowed the model to identify core and redundant attributes (i.e., which technologies had the greatest or least impact on sustainability), by examining the dependency between technology adoption levels and sustainability performance. Example of Rough Set Formulas Used:  
For a set of manufacturing firms U and sustainability criteria A, the lower approximation R(X) is defined as  
\[\text{R(X)}_{\text{lower}}=\{\text{x}\in\text{U}\mid[\text{x}]R\subseteq X\}\]  
The upper approximation R(X) is defined as  
\[\text{R(X)}_{\text{upper}}=\{\text{x}\in\text{U}\mid[\text{x}]R\cap\text{X}\
eq \varnothing\}\]  
These approximations helped us determine which Industry 4.0 technologies (e.g., MES, IIoT, or AM) had the highest impact on firms achieving their sustainability goals.