Sensitivity analysis and parametric programming are powerful tools for understanding how changes in input parameters affect optimal solutions in linear programming. These techniques help decision-makers assess the robustness of their solutions and make informed choices in uncertain environments.

By examining ranges of optimality and feasibility, we can determine how much parameters can change before altering the optimal solution. This knowledge is crucial for strategic planning and risk management in various fields, from production planning to resource allocation.

Sensitivity Analysis for Optimal Solutions

Examining Parameter Changes and Their Effects

• Sensitivity analysis investigates how changes in input parameters impact the optimal solution of a linear programming problem
• Primary parameters analyzed include objective function coefficients and right-hand side values of constraints
• Shadow prices (dual variables) indicate the rate of change in the objective function value per unit change in the right-hand side of a constraint
• Reduced costs measure the penalty in the objective function for forcing a non-basic variable into the solution
• The 100% rule determines the simultaneous change allowed in objective function coefficients while maintaining the current optimal solution

Methods and Tools for Sensitivity Analysis

• Perform sensitivity analysis using algebraic methods or specialized software tools that provide sensitivity reports
• Allowable increase and allowable decrease define the range within which a parameter can change without affecting the optimal solution's structure
• Graphical methods visualize the ranges of optimality and feasibility for two-dimensional problems (supply and demand curves)
• Sensitivity reports typically include allowable increases and decreases for each parameter

Parameter Range for Solution Validity

Ranges of Optimality and Feasibility

• Range of optimality for objective function coefficients represents the interval within which the coefficient can vary without changing the optimal solution
• Range of feasibility for right-hand side values indicates the interval within which the constraint's right-hand side can vary without changing the basis of the optimal solution
• Binding constraints have a zero slack or surplus and directly impact the optimal solution (production capacity limits)
• Non-binding constraints do not affect the solution within their feasibility range (minimum production requirements)

Factors Affecting Solution Stability

• Concept of degeneracy in linear programming can affect the uniqueness and stability of sensitivity analysis results
• Crossover points occur when parameter changes cause a switch in the optimal solution, marking the boundaries of validity ranges
• Sensitivity analysis results presented in a sensitivity report include allowable increases and decreases for each parameter

Parametric Programming for Optimal Solutions

Types of Parametric Programming

• Parametric programming extends sensitivity analysis by examining how the optimal solution changes as a parameter varies continuously over a specified range
• Objective function parametric programming analyzes changes in the objective function coefficients as a function of a single parameter (raw material costs)
• Right-hand side parametric programming focuses on variations in the constraints' right-hand side values as a function of a parameter (available production hours)

Analyzing Solution Behavior

• Critical points in parametric programming are values of the parameter where the optimal solution structure changes
• Optimal value function represents how the objective function value changes with respect to the varying parameter
• Parametric programming solutions often involve piecewise linear functions that describe the behavior of decision variables and the objective value
• Advanced techniques such as the parametric simplex method efficiently solve parametric programming problems

Sensitivity Analysis in Decision Making

Interpreting Analysis Results

• Sensitivity analysis results provide insights into the robustness and stability of optimal solutions under parameter uncertainty
• Economic interpretation of shadow prices helps assess the marginal value of resources and make resource allocation decisions
• Reduced costs guide decision-makers in evaluating the potential impact of introducing non-basic variables into the solution (new product lines)
• Parametric programming results assist in strategic planning by showing how optimal decisions should change as key parameters evolve over time (market demand fluctuations)

Applying Results to Business Decisions

• Quantify opportunity cost using sensitivity analysis results, aiding in trade-off evaluations
• Enhance risk assessment in decision-making by understanding the ranges within which optimal solutions remain valid
• Identify critical parameters that have the most significant impact on the optimal solution, focusing attention on key areas for data refinement or risk management (raw material prices, labor costs)
• Use sensitivity and parametric analysis results to develop contingency plans for various scenarios (economic downturns, supply chain disruptions)