# Consider the three-variable linear programming problem shown.

1. Suppose that the three-variable linear programming problem given  has the objective function Maximize Z _ 3×1 _ 4×2 _ 3×3. Without using the algebra of the simplex method, apply just its geometric reasoning (including choosing the edge giving the maximum rate of increase of Z) to determine and explain the path it would follow in Fig. 5.2 from the origin to the optimal solution.
2. Consider the three-variable linear programming problem shown.

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(a) Construct a table like, giving the indicating variable for each constraint boundary equation and original constraint.
(b) For the CPF solution (2, 4, 3) and its three adjacent CPF solutions  (4, 2, 4), (0, 4, 2), and (2, 4, 0), construct a table lik, showing the corresponding defining equations, BF solution, and nonbasic variables.
(c) Use the sets of defining equations from part (b) to demonstrate that (4, 2, 4), (0, 4, 2), and (2, 4, 0) are indeed adjacent to
(2, 4, 3), but that none of these three CPF solutions are adjacent to each other. Then use the sets of nonbasic variables from part (b) to demonstrate the same thing.

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