Applied Mathematics & Information Sciences

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This paper proposes a kind of bi-level linear programming problem, in which there are two decision makers in a hierarchy and they have a common variable. To deal with this bi-level problem, we introduce a virtual decision maker, who controls the common variable to maximize the sum of the objective functions of the upper and lower level decision maker (the leader and follower). To illustrate the partial cooperation, the virtual decision maker chooses his/her decision before the leader because the leader and the follower exchange the information to maximize their total benefits. Then the leader chooses his/her decision before the follower. Consequently, a tri-level programming model is obtained. Then, a fuzzy approach is presented to solve this tri-level programming. Finally, a numerical example is solved to demonstrate the feasibility of the model after presenting a fuzzy programming approach.

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