Solve a linear programming problem with bounded variables.

Question: Solve a linear programming problem with bounded variables.

Answer: To solve a linear programming problem with bounded variables, we need to follow the steps of the simplex method.

Linear programming is a mathematical technique used to optimize a linear objective function subject to linear constraints. The simplex method is an algorithm used to solve linear programming problems. It involves converting the problem into a standard form, where all variables are non-negative and the objective function is to be maximized. The simplex method then iteratively improves the solution by moving from one feasible solution to another until the optimal solution is found.

To illustrate the process, let's consider the following linear programming problem:

Maximize Z = 3x + 2y
Subject to:
x + y ≤ 4
2x + y ≤ 5
x, y ≥ 0

We first convert the problem into standard form by introducing slack variables:

Maximize Z = 3x + 2y
Subject to:
x + y + s1 = 4
2x + y + s2 = 5
x, y, s1, s2 ≥ 0

We then construct the initial simplex tableau:

| 3 2 0 0 | 0 |
| 1 1 1 0 | 4 |
| 2 1 0 1 | 5 |

The first row represents the objective function coefficients, and the last column represents the constants. The remaining rows represent the constraints, with the slack variables added. The pivot element is the smallest positive coefficient in the objective function row, which is 2 in this case. We choose the pivot column as the one corresponding to this element, and the pivot row as the one with the smallest non-negative ratio of the constant to the pivot column coefficient. In this case, the pivot row is the second row, and the pivot element is 1.

We then perform row operations to make the pivot element 1 and all other elements in the pivot column 0. This gives us the new tableau:

| 3 0 2 0 | 2 |
| 1 0 1 0 | 3 |
| 0 1 -2 1 | 1 |

We repeat this process until all coefficients in the objective function row are non-negative. In this case, the optimal solution is Z = 10, x =