In this paper, we propose a new fuzzy group multi-criteria decision making method and apply it to determine the critical path in a project network. The criteria used here are time (expected duration), cost, risk, and quality of the project activities that are considered critical in project management. As each criterion has its independent level of importance in the critical path selection, the weights of the project criteria are also considered in the analysis. Considering that the information in terms of various project activities and criteria weights are often incomplete and/or uncertain in real-world situations, the essential information in terms of the criteria and project activities are obtained using triangular fuzzy numbers and/or linguistic variables that are mapped to triangular fuzzy numbers, wherever appropriate. The proposed method involves fuzzy evaluation based on the fuzzy information of the possible project paths on each criterion leading to the strength and weakness index scores of the project paths. We define a measure of criticality termed as the total performance score of each project path obtained using its strength and weakness index scores. The path that has the highest measure of criticality is selected as the critical path. A numerical illustration is provided to demonstrate working of the proposed methodology. Further, a case study from manufacturing engineering industry is also presented to better justify the applicability and potentials of the proposed methodology. A comparison with closely related fuzzy multi-criteria decision methods for the critical path selection is done to analyze the performance of the proposed methodology.