A multi-expert Fermatean fuzzy approach to improve the critical path method under uncertainty

Uncertainty in project scheduling remains a central challenge, as real-world activity durations frequently involve ambiguity that traditional methods cannot fully capture. Project scheduling often struggles to manage uncertainty effectively, as traditional methods such as the Critical Path Method (CPM) and the Program Evaluation and Review Technique (PERT) rely on either rigid deterministic estimates or predefined probability distributions. These approaches frequently fall short in capturing the inherent ambiguity and variability present in real-world projects. Existing studies generally suffer from (i) restrictive uncertainty representations, (ii) limited ability to incorporate diverse expert judgments, and (iii) insufficient flexibility when activity information is imprecise. To address these limitations, this study introduces an innovative interval-valued Fermatean fuzzy CPM framework, offering a more expressive and resilient approach to modeling uncertainty in project timelines. The proposed framework integrates multi-expert assessments through interval-valued Fermatean fuzzy sets, capturing both the range of possible activity durations and the associated degrees of confidence. An aggregation approach is employed to combine expert opinions, allowing uncertainty information to be reasonably reflected in the critical path analysis and improving the interpretability of the results. These components collectively overcome the limitations of existing CPM- and PERT-based approaches by enabling richer uncertainty modeling and more transparent integration of expert assessments. An illustrative example is presented to demonstrate the applicability and comparative strengths of the proposed method, highlighting its ability to generate more realistic and risk-sensitive scheduling insights compared to traditional approaches. The results show that the proposed Fermatean CPM method provides an approximately 16% improvement in capturing project risk and uncertainty (calculated as the difference between the PERT probability of completing the project within the baseline duration and the fully covered interval provided by Fermatean CPM) compared to baseline CPM and PERT models, offering a more reliable and realistic project schedule. This research offers a new perspective on uncertainty modeling in project scheduling by applying interval-valued Fermatean fuzzy sets for the first time in this context. It contributes a mathematically grounded and practically useful aggregation approach and provides empirical evidence that conventional techniques may significantly underestimate project risks. Ultimately, the proposed framework empowers project managers to make more informed and resilient scheduling decisions under uncertainty.