Consider the following problem.

Minimize W 5y1  4y2, subject to 4y1  3y2  4 2y1  y2  3 y1  2y2  1 y1  y2  2 and y1  0, y2  0.

Because this primal problem has more functional constraints than variables, suppose that the simplex method has been applied directly to its dual problem. If we let x5 and x6 denote the slack variables for this dual problem, the resulting final simplex tableau is For each of the following independent changes in the original primal model, you now are to conduct sensitivity analysis by directly investigating the effect on the dual problem and then inferring the complementary effect on the primal problem. For each change, apply the procedure for sensitivity analysis summarized at the end of Sec. 6.6 to the dual problem (do not reoptimize), and then give your conclusions as to whether the current basic solution for the primal problem still is feasible and whether it still is optimal. Then check your conclusions by a direct graphical analysis of the primal problem.

(a) Change the objective function to W 3y1  5y2.

(b) Change the right-hand sides of the functional constraints to 3, 5, 2, and 3, respectively.

(c) Change the first constraint to 2y1  4y2  7.

(d) Change the second constraint to 5y1  2y2  10.

D,I 6.7-1.* Consider the following problem.

Maximize Z 5x1  5x2  13x3, subject to

x1  x2  3x3 20 12x1  4x2  10x3 90 and xj  0 (j 1, 2, 3).

If we let x4 and x5 be the slack variables for the respective constraints, the simplex method yields the following final set of equations:
(0) Z  2x3  5x4 100.
(1) x1  x2  3x3  x4 20.
(2) 16x1  2x3  4x4  x5 = 10.
Now you are to conduct sensitivity analysis by independently investigating each of the following nine changes in the original model. For each change, use the sensitivity analysis procedure to revise this set of equations (in tableau form) and convert it to proper form from Gaussian elimination for identifying and evaluating the current basic solution. Then test this solution for feasibility and for optimality. (Do not reoptimize.)

(a) Change the right-hand side of constraint 1 to b1 30.

(b) Change the right-hand side of constraint 2 to b2 70.

(c) Change the right-hand sides to  .

(d) Change the coefficient of x3 in the objective function to c3 8.

(e) Change the coefficients of x1 to .

(f) Change the coefficients of x2 to .
(g) Introduce a new variable x6 with coefficients .
(h) Introduce a new constraint 2x1  3x2  5x3 50. (Denote its slack variable by x6.)
(i) Change constraint 2 to 10x1  5x2  10x3 100.