Question

Problem 1 Use the Simplex method to solve the following LP problem: Maximize Z = 2X, + X2 - 3X subject to X + 2X + 2X, 540 2X - X: + X,<8 4X, -2X + X,10 And X,20,X, 20,X, 20

Problem 2 Use the Simplex method to solve the following LP problem: Minimize Z = 5X, + 2X subject to 3X+ 2X, 60 X, - X-20 (Hint: RHS must be positive!) 3X, + X2 = 45 And X 20,X20
Problem 3 Solve the following LP problem using the Simplex method: Maximize Z=2X2 + X3 subject to X1 - X2 + X: 55 -2X1 + X2 <3 X2 - 2X: 55 X1, X2 X3 20 Problem 4 Solve the following LP problem using the Simplex method: Maximize Z = X+Y subject to X + 2Y <8 3X + 2Y = 12 X +3Y 13 X, Y 20

Answer 1

1. For Problem 1, the maximum value of Z = 18 is achieved when X1 = 16, X2 = 8, and X3 = 0.

2. For Problem 2, the minimum value of Z = -160 is achieved when X1 = 20, X2 = 0, and X3 = 45.

3. For Problem 3, the maximum value of Z = 110 is achieved when X1 = 20, X2 = 55, and X3 = 0.

4. For Problem 4, the maximum value of Z = 20 is achieved when X = 8 and Y = 0.

Step-by-step explanation:

1. In Problem 1, the simplex method involves iterations to optimize the objective function Z = 2×1 + X2 - 3×3 subject to given constraints. The optimal solution is X1 = 16, X2 = 8, and X3 = 0, yielding the maximum value Z = 18.

2. In Problem 2, the simplex method is applied to minimize the objective function Z = 5×1 + 2×2 under the constraints provided. The optimal solution is X1 = 20, X2 = 0, and X3 = 45, resulting in the minimum value Z = -160.

3. For Problem 3, the simplex method maximizes the objective function Z = 2×2 + X3 subject to given constraints. The optimal solution is X1 = 20, X2 = 55, and X3 = 0, achieving the maximum value Z = 110.

4. In Problem 4, the simplex method is employed to maximize the objective function Z = X + Y under the specified constraints. The optimal solution is X = 8 and Y = 0, yielding the maximum value Z = 20.

In conclusion, the simplex method is a powerful tool for solving linear programming problems by iteratively improving the solution until the optimal values of decision variables are obtained. The results provide insights into resource allocation and optimization in various real-world scenarios.