Question

Choose the option that represents the original linear programming problem based on the initial simplex tableau given below, where \(S_i\) represents slack/surplus variables and \(A_i\) represents the artificial variables corresponding to the \(i^{th}\)constraint: 

• A. Minimize Z-15x+25y
  subject to 7x+6y≥ 20, 3x-2y ≤18, 8x+5y ≤ 30; x, y ≥0.
• B. Maximize Z-15x+25y
  subject to 7x+6y≥ 20, 3x-2y=18, 8x+5y ≤ 30 x, y 20.
• C. Minimize Z-15x+25y
  subject to 7x+6y≥ 20, 3x-2y=18, 8x+5y ≥30; x, y ≥0.
• D. Maximize Z-15x+25y
  subject to 7x+6y=20, 3x-2y=18, 8x+5y≤ 30; x, y ≥0.

Answer 1

The Correct Option is B

To determine the original linear programming problem from the given initial simplex tableau, we analyze the constraints and the objective function.

Objective Function

The objective row in the tableau shows the coefficients of the variables \(x\), \(y\), along with \(S_i\) (slack/surplus variables) and \(A_i\) (artificial variables). Since the coefficients of the artificial variables are negative, it indicates a maximization problem.

Constraints Analysis

From the tableau, we observe:

• First constraint: \(7x + 6y - S_1 + A_1 = 20\). Since \(A_1\) is an artificial variable, the original constraint was \(7x + 6y \geq 20\).
• Second constraint: \(3x - 2y + A_2 = 18\). This suggests an equality constraint \(3x - 2y = 18\).
• Third constraint: \(8x + 5y + S_3 = 30\). This is a standard \(\leq\) constraint \(8x + 5y \leq 30\) with slack variable \(S_3\).

Conclusion

We conclude that the original linear programming problem, given the observation that the problem is a maximization one, is:

Maximize \(Z = 15x + 25y\)

Subject to:

• \(7x + 6y \geq 20\)
• \(3x - 2y = 18\)
• \(8x + 5y \leq 30\)
• \(x, y \geq 0\)

Thus, the correct option is:

Maximize \(Z = 15x + 25y\) 
subject to \(7x + 6y \geq 20\), \(3x - 2y = 18\), \(8x + 5y \leq 30\); \(x, y \geq 0\).