Consider the following problem.

Maximize Z 2x1  2x2  3x3, subject to 2x1  x2  2x3 4 x1  x2  x3 3 and x1  0, x2  0, x3  0.

Let x4 and x5 be the slack variables for the respective functional constraints. Starting with these two variables as the basic variables for the initial BF solution, you now are given the information that the simplex method proceeds as follows to obtain the optimal solution in two iterations: (1) In iteration 1, the entering basic variable is x3 and the leaving basic variable is x4; (2) in iteration 2, the entering basic variable is x2 and the leaving basic variable is x5.

(a) Develop a three-dimensional drawing of the feasible region for this problem, and show the path followed by the simplex method.

(b) Give a geometric interpretation of why the simplex method followed this path.

(c) For each of the two edges of the feasible region traversed by the simplex method, give the equation of each of the two constraint boundaries on which it lies, and then give the equation of the additional constraint boundary at each endpoint.

(d) Identify the set of defining equations for each of the three CPF solutions (including the initial one) obtained by the simplex method. Use the defining equations to solve for these solutions.

(e) For each CPF solution obtained in part (d), give the corresponding BF solution and its set of nonbasic variables. Explain how these nonbasic variables identify the defining equations obtained in part (d).