# Definition / feasibility domain

A solution to a linear problem is said to be achievable if all the constraints are satisfied. The feasibility domain contains all the feasible solutions to the problem. The optimal solution is the « best » of the feasible solutions.

## Feasible solution

To see if a solution is feasible, just test if all constraints are satisfied, it can be done manually or in matrix form.

Manually :

Let us check if the solution (3, 1) is feasible.

The first equation gives 3 * 1/3 + 1 = 2, the constraint is satisfied.
The second inequality gives -2 * 3 + 5 * 1 = -1 ≤ 7, the constraint is satisfied.
The third inequality gives 3 + 1 = 4 ≤ 4, the constraint is satisfied, we say that it is saturated.
Both type constraints are satisfied.

The solution is in the feasibility domain. The value of the objective function is z=3-1=2.

With the matrix: multiply the matrix of the linear program by the solution vector and compare the result to the members to the right of the linear program

## Feasibility domain

For example the equationPar exemple l’équation ai*x1 + bi*x2 = ci divides the plane into two half-planes P1 and P2 of equation:

The lower or equal constraint will determine a half plane, the greater or equal constraint will determine the other half plane. To know in which half-plane lies the feasible solutions for the constraints, it suffices to test a simple example and to determine if it is feasible or not.

For example for the constraint: x1 + x2 ≤ 4, the solution (0,0) is reachable, so the origin is in the faisible half-plane.

The intersection of all the achievable half-planes constitutes the feasibility domain. The latter can be bounded or unbounded.