Linear programming helps optimize systems by finding the best solution within a set of constraints. The feasible region, formed by these constraints, is where all possible solutions lie. Understanding its shape and boundaries is crucial for identifying optimal outcomes.
Optimal solutions typically occur at the corners or edges of the feasible region. By graphing constraints and using iso-profit lines, we can visually locate these solutions. Changing constraints can significantly impact the optimal solution, making sensitivity analysis a valuable tool for decision-making.
Understanding Feasible Regions in Linear Programming
Feasible region in linear programming
- Set of all possible solutions satisfying constraints in linear programming problem bounded by constraint lines or planes
- Geometric representation formed by intersection of constraint inequalities creates convex set (empty, bounded, or unbounded)
- Components include decision variables, constraints, non-negativity conditions (production quantities, resource limits)
Characteristics of optimal solutions
- Optimal solution occurs at corner point (vertex) of feasible region or along edge for multiple optimal solutions
- Maximizes or minimizes objective function value determined by optimization direction
- Single optimal solution at one vertex or multiple optimal solutions when edge parallel to objective function contour
Graphical representation of solutions
- Graph feasible region by plotting constraint lines, identifying area satisfying all constraints, shading feasible region
- Locate optimal solution using iso-profit lines parallel to objective function, moving in optimization direction
- Special cases include unbounded solutions, infeasible problems (empty feasible region), degenerate solutions (optimal point at intersection of >2 constraint lines)
Effects of changing constraints
- Constraint changes include tightening (reducing feasible region), relaxing (expanding feasible region), adding, or removing
- Impact optimal solution by shifting optimal point, changing optimal objective value, potential infeasibility or unboundedness
- Sensitivity analysis determines range of constraint values maintaining current optimal solution, shadow prices show change in objective value per unit change in constraint
- Practical implications involve resource allocation adjustments (labor hours), production capacity modifications (machine availability), market demand fluctuations (product mix)