# Using the facts given in, show that the following statements must be true for any linear programming problem that has a bounded feasible region and multiple optimal solutions

1. Using the facts given in, show that the following statements must be true for any linear programming problem that has a bounded feasible region and multiple optimal solutions:
(a) Every convex combination of the optimal BF solutions must be optimal.

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(b) No other feasible solution can be optimal.
2. Consider a two-variable linear programming problem whose CPF solutions are (0, 0), (6, 0), (6, 3), (3, 3), and (0, 2). (Seefor a graph of the feasible region.)
(a) Use the graph of the feasible region to identify all the constraints for the model.
(b) For each pair of adjacent CPF solutions, give an example of an objective function such that all the points on the line segment between these two corner points are multiple optimal solutions.
(c) Now suppose that the objective function is Z__x1 _ 2×2. Use the graphical method to find all the optimal solutions.
(d) For the objective function in part (c), work through the simplex method step by step to find all the optimal BF solutions.
Then write an algebraic expression that identifies all the optimal solutions.

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