Linear Programming: Finding All Optimal Solutions for max z = x1 + x2

To find all optimal solutions for this LP, we need to solve the given linear programming problem.

The given LP can be written in standard form as follows:

Maximize: z = x1 + x2

Subject to: x1 + x2 + x3 ≤ 1 x1 + 2x3 ≤ 1 xi ≥ 0

To solve this LP, we can use the Simplex Method.

First, we convert the LP into canonical form by introducing slack variables:

Maximize: z = x1 + x2

Subject to: x1 + x2 + x3 + x4 = 1 x1 + 2x3 + x5 = 1 xi ≥ 0

The initial tableau is:

| Cb | x1 | x2 | x3 | x4 | x5 | RHS | |----|----|----|----|----|----|-----| | z | -1 | -1 | 0 | 0 | 0 | 0 | | x4 | 1 | 1 | 1 | 1 | 0 | 1 | | x5 | 1 | 0 | 2 | 0 | 1 | 1 |

We can see that the initial basic feasible solution is x4 = 1, x5 = 1, and all other variables are 0.

Next, we perform the Simplex Method to iteratively improve the solution.

Iteration 1:

| Cb | x1 | x2 | x3 | x4 | x5 | RHS | |----|----|----|----|----|----|-----| | z | 0 | -1 | 1 | 0 | 0 | 1 | | x1 | 1 | 1 | 1 | 1 | 0 | 1 | | x5 | 0 | -1 | 1 | 0 | 1 | 0 |

We can see that the optimal solution is: x1 = 1, x2 = 0, x3 = 0, x4 = 0, x5 = 0, with z = 1.

Since the objective function coefficient of x1 is 0, this means that x1 can take any non-negative value without affecting the objective function value. Therefore, there are infinitely many optimal solutions for this LP.

The optimal solutions are:

x1 = 1, x2 = 0, x3 = 0
x1 = 2, x2 = -1, x3 = 0
x1 = 3, x2 = -2, x3 = 0
... and so on, where x1 ≥ 0 and x2, x3 can take any values.