Answer:
Explanation:
To put the model into equality form, we introduce slack variables and a surplus variable:
max z = 3x1 + x2
subject to:
3x1 + x2 + x3 = 3 (constraint 1)
x1 + x2 = 2 (constraint 2)
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
Now, let's create the first simplex tableau:
Tableau:
Cj x1 x2 x3 RHS
z -3 0 0 0 0
x3 0 3 1 1 3
x2 0 1 1 0 2
To execute a single iteration of the simplex algorithm, we will follow these steps:
Identify the entering variable (EV): Choose the column with the most negative coefficient in the z row. In this case, x1 has the most negative coefficient.
Identify the departing variable (DV): Divide the RHS column by the column corresponding to the entering variable (x1). Choose the row with the minimum positive ratio. In this case, the minimum positive ratio is 2/1 = 2, which occurs in the second row (constraint 2).
Pivot: Perform a pivot operation to make the DV equal to 1 and update the tableau accordingly. To do this, divide the row of the DV by the DV value (1 in this case).
Tableau after pivoting:
Cj x1 x2 x3 RHS
z -3 0 0 0 0
x3 0 3 1 1 3
x1 0 1 1 0 2
Update the tableau: Perform row operations to make the other entries in the EV column equal to zero. To do this, subtract the corresponding multiple of the pivot row from the other rows.
Tableau after updating:
Cj x1 x2 x3 RHS
z 0 0 3 0 3
x3 0 0 -2 1 1
x1 0 1 1 0 2
The iteration is now complete, and the optimal solution is x1 = 2, x2 = 0, and z = 3. The value of z represents the maximum objective function value.