The aim of this paper is to introduce a formulation of linear programming problems involving intuitionistic fuzzy variables. Here, we will focus on duality and a simplex-based algorithm for these problems. We classify these problems into two main different categories: linear programming with intuitionistic fuzzy numbers problems and linear programming with intuitionistic fuzzy variables problems. The linear programming with intuitionistic fuzzy numbers problem had been solved in the previous literature, based on this fact we offer a procedure for solving the linear programming with intuitionistic fuzzy variables problems. In methods based on the simplex algorithm, it is not easy to obtain a primal basic feasible solution to the minimization linear programming with intuitionistic fuzzy variables problem with equality constraints and nonnegative variables. Therefore, we propose a dual simplex algorithm to solve these problems. Some fundamental concepts and theoretical results such as basic solution, optimality condition and etc., for linear programming with intuitionistic fuzzy variables problems, are established so far. Moreover, the weak and strong duality theorems for linear programming with intuitionistic fuzzy variables problems are proved. In the end, the computational procedure of the suggested approach is shown by numerical examples.

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