SPECIAL CASE GRAPHICAL METHOD/SIMPLEX METHOD

In linear programming, certain special cases can arise when using the graphical method to solve problems. Here are four notable special cases:

  1. Infeasible Solution: This occurs when there are no possible solutions that satisfy all constraints simultaneously. Graphically, this is represented when the feasible region does not exist, often due to conflicting constraints that make it impossible to find a common area where all conditions are met.

  2. Unbounded Solution: An unbounded solution arises when the feasible region extends infinitely in at least one direction. In this case, the objective function can increase (for maximization problems) or decrease (for minimization problems) without bound. Graphically, this is seen when the objective function lines can be moved indefinitely without hitting a boundary of the feasible region.

  3. Multiple Optimal Solutions: This situation occurs when the objective function is parallel to a constraint line and touches the feasible region along that line segment. As a result, there are infinitely many solutions along that segment which yield the same optimal value for the objective function.

  4. Degenerate Solution: A degenerate solution happens when more than one constraint intersects at a single corner point of the feasible region. In this case, while there is an optimal solution, the point may represent multiple validity conditions that can lead to difficulties during the solution evaluation process,


In linear programming, certain special cases can arise when using the simplex method to solve problems. Here are four notable special cases:

  1. Infeasible Solution: Similar to the graphical method, this occurs when there are no possible solutions that satisfy all constraints simultaneously. In the simplex method, this indicates a lack of viable basic feasible solutions, often due to conflicting constraints.

  2. Unbounded Solution: An unbounded solution is identified when there is a feasible region that extends indefinitely in at least one direction. In the simplex method, this might be the case when the objective function can increase indefinitely, indicating that the optimal solution cannot be determined.

  3. Multiple Optimal Solutions: This situation arises when the objective function is such that there are multiple combinations of decision variables that provide the same optimal value. The simplex method may reveal this when multiple basic feasible solutions yield the same objective value at optimality.

  4. Degenerate Solution: A degenerate solution occurs when a basic feasible solution corresponds to multiple constraints intersecting at a single point. In this case, while an optimal solution exists, the presence of degeneracy can complicate the pivoting process within the simplex algorithm, causing cycling issues if not addressed.