4.3 Special cases: degeneracy, unboundedness, and infeasibility

The simplex method, a powerful tool for solving linear programming problems, can encounter special cases that complicate the optimization process. Degeneracy, unboundedness, and infeasibility are three such scenarios that require careful handling to ensure accurate and meaningful results.

Understanding these special cases is crucial for effectively applying the simplex method in real-world situations. By learning to identify and address these challenges, you'll be better equipped to tackle complex optimization problems and interpret their solutions accurately.

Degenerate Solutions in the Simplex Method

Understanding Degeneracy in Linear Programming

• Degeneracy occurs when basic feasible solution has one or more basic variables equal to zero
• Degenerate solutions can lead to cycling in simplex method
  • Algorithm repeatedly visits same set of basic feasible solutions
  • Objective function fails to improve
• Detect presence of degeneracy by examining values of basic variables in each iteration
• Degeneracy does not affect optimality of final solution
• Can significantly increase computational time required to solve problem

Strategies to Handle Degeneracy

• Bland's rule prevents cycling in degenerate problems
  • Selects entering and leaving variables based on their indices
  • Example: Choose variable with lowest index as entering variable
• Lexicographic ordering resolves degeneracy
  • Introduces systematic tie-breaking mechanism in selection of pivot elements
  • Example: Compare entire rows lexicographically to break ties
• Perturbation techniques eliminate degeneracy
  • Slightly modify right-hand side values of constraints
  • Example: Add small positive values ($\epsilon$) to right-hand side of constraints

Unbounded Linear Programming Problems

Identifying Unboundedness

• Unbounded problem occurs when feasible region extends infinitely in direction of improvement for objective function
• Detect unboundedness in simplex method
  • Non-basic variable can enter basis
  • No basic variable needs to leave
• Graphically represented by open feasible region extending infinitely in direction of optimization
• Indicates problem formulation may be incomplete or incorrect
  • Most real-world optimization problems have natural bounds
• Unbounded problems lack optimal solution
  • May have improving direction or ray along which objective function value improves indefinitely

Analyzing and Resolving Unboundedness

• Analyze constraints and objective function coefficients to identify source of unboundedness
• Add appropriate constraints to resolve unboundedness
  • Example: Include upper bounds on production quantities in manufacturing problem
• Adjust objective function to lead to meaningful optimal solution
  • Example: Incorporate diminishing returns in profit function
• Reviewing problem context can reveal missing real-world limitations
  • Example: Consider storage capacity constraints in inventory management problem

Infeasible Linear Programming Problems

Detecting Infeasibility

• Infeasible problem occurs when no solution satisfies all constraints simultaneously
• Detect infeasibility in simplex method
  • Artificial variable remains in basis at positive level after Phase I
• Graphically represented by empty feasible region
  • Constraint lines do not intersect to form feasible area
• Indicates constraints are inconsistent or contradictory
• Use sensitivity analysis to identify constraints causing infeasibility
  • Determine extent constraints need relaxation

Methods for Handling Infeasibility

• Big M method detects infeasibility
  • Introduces artificial variables with high penalties in objective function
  • Example: Add artificial variable with coefficient M (large positive number) to objective
• Two-phase simplex method identifies infeasible problems
  • Phase I minimizes sum of artificial variables
  • If minimum value greater than zero, problem infeasible
• Resolve infeasibility by relaxing certain constraints
  • Example: Increase available budget in resource allocation problem
• Remove conflicting constraints to achieve feasibility
  • Example: Eliminate mutually exclusive production requirements
• Reformulate problem to allow for feasible solution
  • Example: Introduce slack variables to convert strict inequalities to non-strict ones

Special Cases in Optimization

Impact on Optimization Process

• Special cases (degeneracy, unboundedness, infeasibility) significantly impact efficiency and effectiveness of optimization algorithms
• Require additional computational effort and specialized techniques for proper handling
• Crucial for developing robust optimization algorithms
• Essential for interpreting results accurately

Practical Implications

• Degeneracy increases solution time due to potential cycling
  • Example: Manufacturing problem with multiple optimal production schedules
• Unboundedness indicates no finite optimal solution
  • Requires reevaluation of problem formulation
  • Example: Unconstrained profit maximization without considering market saturation
• Infeasibility suggests no solution satisfies all constraints
  • Necessitates review and possible relaxation of problem constraints
  • Example: Over-constrained resource allocation problem in project management
• Understanding special cases helps formulate realistic optimization problems
  • Example: Incorporating realistic capacity constraints in transportation problems
• Improves model accuracy in practical applications
  • Example: Refining financial portfolio optimization models to account for market limitations