Three Solution Approaches for Single-objective and Multi-objective Intuitionistic Fuzzy Matrix Games: a Two-level Optimization Framework and its Applications

Intuitionistic fuzzy sets (IFSs) enhance traditional fuzzy sets by introducing a hesitation margin-the difference between one and the sum of the membership and non-membership degrees-capturing uncertainty more effectively. This added flexibility makes IFSs particularly suitable for addressing vague, imprecise, or conflicting information in practical matrix game problems. The aim of this research is to present three solution approaches-pessimistic, optimistic and mixed-to determine maxmin-minmax solutions in intuitionistic fuzzy matrix games. These games are modeled as two-player zero-sum matrix games involving intuitionistic fuzzy goals (IFGs) and intuitionistic fuzzy payoffs (IFPs). In the proposed framework, the membership and non-membership functions associated with goals and payoffs are assumed to be linear. A new two-level methodology is introduced to derive optimal solutions for both single-objective and multi-objective matrix game problems. For the single-objective case under the pessimistic approach, it has been proved that the optimization problem of each player is equivalent to a fractional programming problem. This establishes a connection between fractional programming and goal programming (GP). The corresponding nonlinear GP can be effectively solved using the proposed two-level method to obtain optimal strategies for both players. Furthermore, due to its flexibility, the methodology is extended to multi-objective matrix games with intuitionistic fuzzy data. The applicability and effectiveness of the proposed approaches are demonstrated through several illustrative examples. In addition, the implementation of the methodology is carried out on two real-world applications: ( i ) neutralizing adverse drug effects and ( ii ) analyzing market share strategies in the electric vehicle (EV) sector.