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I need to know if such a Linear Programming model possible where each basic feasible solution shows degeneracy. If it is not possible, then why.

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To make an example, take any linear programming problem with nontrivial feasible region and make an extra copy of each of the constraints.

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Another way to do it is to set out to optimize any linear function over the unit square in two dimensions. (The objective, and whether you maximize or minimize, is irrelevant.) The four vertices of the square correspond to the BFSes. Now add one constraint at each corner that is tangent to the square at that corner. For instance, at (0, 1) you might add the constraint $-x + y \le 1$. (Make sure the feasible region remains feasible by choosing the direction of the inequality correctly.) You can generalize this to any finite dimension.

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