Two-Stage Three-Dimensional Transportation Optimization Under Elliptic Intuitionistic Fuzzy Quadruples: An Index-Matrix Interpretation

The transportation problem (TP) is a canonical linear programming model for minimizing the cost of distributing goods from multiple sources to multiple destinations. Classical TPs assume deterministic costs, supplies, and demands, whereas real supply chains are affected by volatility in fuel prices, inflation, disruptions, and weather, making such parameters uncertain. Fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs) have been widely used to handle vagueness; however, while Atanassov’s IFSs incorporate hesitation in addition to membership and non-membership, they remain limited to isotropic representations of uncertainty. This paper introduces an index-matrix interpretation for a two-stage three-dimensional transportation problem (2-S 3-D TP) defined under Elliptic Intuitionistic Fuzzy Quadruples (E-IFQs). Within this framework, transportation costs, supplies, and demands are represented as E-IFQs, allowing the modeling of anisotropic and correlated uncertainty along the membership and non-membership axes. The two-stage formulation extends previous intuitionistic fuzzy approaches by adding a temporal dimension and incorporating practical constraints such as cost thresholds and feasibility checks. The objective is to determine optimal producer–hub–buyer allocations that minimize the total E-IFQ cost while preserving consistency across all stages and time periods. A detailed case study on EV battery module distribution demonstrates the effectiveness of the proposed model. Compared with conventional fuzzy and intuitionistic fuzzy formulations, the 2-S 3-D E-IFTP yields more robust and precise decisions under complex, multidimensional uncertainty, offering improved interpretability and policy integration over time.