Question B2. Use the TWO-PHASE SIMPLEX METHOD to find the optimal solution for the below LPP.
(8 Marks)
Min Z = 10𝑥1 + 6𝑥2 + 2𝑥3
Subjected to
−𝑥1 + 𝑥2 + 𝑥3 ≥ 1
3𝑥1 + 𝑥2 − 𝑥3 ≥ 2
𝑥1, 𝑥2, 𝑥3 ≥ 0
To start the two-phase simplex method, we first convert the inequalities into equations by introducing slack variables. Let:
S1 = -𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 = 1
S2 = 3𝑥1 + 𝑥2 - 𝑥3 + 𝑥5 = 2
The initial Simplex Tableau will be as follows:
[ 1 1 1 | 1 ]
[ 3 1 -1 | 2 ]
[10 6 2 | 0 ]
The pivot element is 1, so we select the first column for pivoting. The entering variable into the basis is x4, and the leaving variable is x1.
After the first iteration:
[ 0 2 -2 | 1 ]
[ 3 1 -1 | 2 ]
[10 6 2 | 0 ]
Now the pivot element is -2, so we select the second column for pivoting. The entering variable into the basis is x2, and the leaving variable is x3.
After the second iteration:
[ 0 0 0 | 3 ]
[ 3 1 0 | 5 ]
[10 6 0 | 12 ]
The optimal solution is Z = 3 when x1 = 0, x2 = 5, and x3 = 0.
Therefore, the optimal solution for the given linear programming problem using the two-phase simplex method is Z = 3 when x1 = 0, x2 = 5, and x3 = 0.