triangular type-2 fuzzy variable

Seikh et al. (Iranian Journal of Fuzzy Systems, 18(5) (2021), 155-172) pointed out that there does not exist any approach to solve type-2 fuzzy variable matrix games (matrix games in which each element of the payoff matrix is represented by a type-2 fuzzy variable). To fill this gap, Seikh et al. proposed an approach to solve triangular type-2 fuzzy variable (TT2FV) matrix games (matrix games in which each element of the payoff matrix is represented by a TT2FV). Seikh et al. also solved a real-life TT2FV matrix game (plastic ban problem) to show the applicability of their proposed approach. In future, other researchers may use Seikh et al.’s approach to solve other real-life TT2FV matrix games. However, it is inappropriate to use Seikh et al.’s approach as it may fail to find a correct optimal solution of TT2FV matrix games. To validate this claim, a TT2FV matrix game is solved by Seikh et al.’s approach and it is shown that the obtained optimal solution is not correct. Also, it is pointed out that this inappropriateness occurs as a mathematically incorrect result is considered in Seikh et al.’s approach. Furthermore, Seikh et al.’s approach is modified to resolve its inappropriateness.

The novelty of the type-2 fuzzy variable (T2FV) is its fuzzy secondary possibility distribution. Here, we aim to formulate a new defuzzification model of triangular type-2 fuzzy variables (TT2FVs) based on type reduction. We also show its superiority compared to other existing models by providing some numerical examples. Later, we develop a matrix game problem in the type-2 fuzzy environment to show the applicability of the proposed defuzzification method in a real-world situation. Based on this defuzzification method, two crisp linear programming models are derived, which are subsequently solved by the simplex method using LINGO 17.0 software. Also, the expected pay-off for the maximizing player is calculated as TT2FV, which is desirable. Finally, to show the validity and applicability, the proposed methodology is illustrated with the plastic ban problem.