5.4 Sensitivity analysis and parametric programming
3 min read•july 30, 2024
and are powerful tools for understanding how changes in input parameters affect optimal solutions in linear programming. These techniques help decision-makers assess the 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 . 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
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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 () indicate the rate of change in the objective function value per unit change in the right-hand side of a constraint
measure the penalty in the objective function for forcing a non-basic variable into the solution
The 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
and define the range within which a parameter can change without affecting the optimal solution's structure
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
for objective function coefficients represents the interval within which the coefficient can vary without changing the optimal solution
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
have a zero slack or surplus and directly impact the optimal solution (production capacity limits)
do not affect the solution within their feasibility range (minimum production requirements)
Factors Affecting Solution Stability
Concept of in linear programming can affect the uniqueness and of sensitivity analysis results
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
in parametric programming are values of the parameter where the optimal solution structure changes
represents how the objective function value changes with respect to the varying parameter
Parametric programming solutions often involve that describe the behavior of decision variables and the objective value
Advanced techniques such as the 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
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 using sensitivity analysis results, aiding in trade-off evaluations
Enhance 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 for various scenarios (economic downturns, supply chain disruptions)