In optimization problems, a feasible solution is a specific outcome that meets all the required conditions or limitations.
Based on the reference provided, a feasible solution is a set of values for the decision variables that satisfies all of the constraints in an optimization problem. These solutions are the only candidates considered valid when searching for the best possible outcome (the optimal solution) among all possibilities.
Understanding the Components
To grasp what a feasible solution is, it's important to understand its core components within the context of an optimization problem:
- Decision Variables: These are the unknown values that you are trying to determine in the problem. For example, in a manufacturing problem, decision variables might be the quantity of each product to produce. A "set of values" for these variables represents a specific plan or option.
- Constraints: These are the limitations or restrictions that the decision variables must adhere to. Constraints can represent resource limitations (like budget, labor, or raw materials), physical boundaries, demand requirements, or logical conditions. They are typically expressed as mathematical inequalities or equalities.
A potential set of values for the decision variables is tested against every single constraint. If all constraints are satisfied simultaneously, then that set of values constitutes a feasible solution. If even one constraint is violated, the solution is considered infeasible.
The Feasible Region
The reference states that the set of all feasible solutions defines the feasible region of the problem.
Think of the feasible region as the complete collection or space containing every single possible combination of decision variable values that obeys all the rules (constraints) of the problem.
- Any point inside or on the boundary of this region represents a feasible solution.
- Any point outside this region represents an infeasible solution.
Finding the optimal solution involves searching within this feasible region for the point that maximizes or minimizes the objective function (the goal of the optimization problem, like maximizing profit or minimizing cost).
In summary, feasible solutions are the valid options that are mathematically possible and allowed by the problem's rules before you even consider which option is the best one according to your objective.
