Answer to Question #186884 in Operations Research for hami

Question

Solve the following problem using the simplex method:

Maximise: z = −x1 + 2x2 + x3

subject to

3x2 + x3 ≤ 120,

x1 −x2 −4x3 ≤80,

−3x1 +x2 +2x3 ≤100

(no non-negativity constraints). You should follow the following steps.

(a) First reformulate the problem so that all variables have non-negativity constraints.

(b) Then work through the simplex method step by step to solve the problem.

(c) State the values of the decision variables x1, x2, x3 as well as the objective function z in an optimal solution. 

Answer 1

Max $Z=-x_1+2x_2+x_3$

sub. to

$3x_2+x_3\le 120\\ x_1-x_2-4x_3\le 30 \\ -3x_1+x_2+2x_3\le 100$

(a) Max $Z=-x_1+2x_2+x_3+0.S_1+0.S_2+0.S_3$

 

    sub. to

$- 3x_2+x_3+S_1=120\\ x_1-x_2-4x_3+S_2=80 \\ -3x_1+x_2+3x_3+S_3=100,\\\ x_1,x_2,x_3\ge 0$

(b) The table is -

(c) since all $Z_j-_j\ge 0$ , Hence The optimal solution is $x_1=0, x_2=28,x_3=36 \text{ and } Max Z=92$