Consider the following problem.

Maximize Z 3x1  x2  4x3, subject to 6x1  3x2  5x3 25 3x1  4x2  5x3 20 and x1  0, x2  0, x3  0.

The corresponding final set of equations yielding the optimal solution is

(0) Z  2x2  

1 5

x4  

3 5

x5 17

(1) x1  

1 3

x2  

1 3

x4  

1 3

x5 

5 3



(2) x2  x3  

1 5

x4  

2 5

x5 3.

(a) Identify the optimal solution from this set of equations.

(b) Construct the dual problem.

(c) Identify the optimal solution for the dual problem from the final set of equations. Verify this solution by solving the dual problem graphically.

(d) Suppose that the original problem is changed to Maximize Z 3x1  3x2  4x3, subject to 6x1  2x2  5x3 25 3x1  3x2  5x3 20 and x1  0, x2  0, x3  0.
Use duality theory to determine whether the previous optimal solution is still optimal.

(e) Use the fundamental insight presented in Sec. 5.3 to identify the new coefficients of x2 in the final set of equations after it has been adjusted for the changes in the original problem given in part (d).

(f) Now suppose that the only change in the original problem is that a new variable xnew has been introduced into the model as follows:
Maximize Z 3x1  x2  4x3  2xnew, subject to 6x1  3x2  5x3  3xnew 25 3x1  4x2  5x3  2xnew 20 and x1  0, x2  0, x3  0, xnew  0.
Use duality theory to determine whether the previous optimal solution, along with xnew 0, is still optimal.
(g) Use the fundamental insight presented in Sec. 5.3 to identify the coefficients of xnew as a nonbasic variable in the final set of equations resulting from the introduction of xnew into the original model as shown in part ( f ).
D,I