Sensitivity analysis
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Sensitivity analysis is a method for studying mathematical or simulation models. It assesses how changes in a model's input data, parameters, or assumptions affect its output results (e.g., the optimal solution, the value of the objective function, or other key indicators).
Sensitivity analysis is an important tool in operations research, optimization, decision theory, risk management, economic analysis, and systems analysis.
Essence and Purpose
The main goal of sensitivity analysis is to understand how stable (robust) the results of modeling or optimization are to uncertainty or variations in the initial data. It helps answer questions such as:
- "What will happen to the optimal solution if a resource's cost changes by 10%?"
- "How much will profit change with fluctuations in market demand?"
- "Which model parameters have the greatest impact on the final result?"
- "How reliable is the forecast obtained using the model?"
Key tasks of sensitivity analysis include:
- Assessing robustness:* To determine if the optimality or acceptability of a solution is maintained when parameters change.
- Identifying critical parameters: To identify input data or model parameters whose small changes lead to significant changes in the output results.
- Increasing confidence in the model: To demonstrate that the model behaves predictably and logically in response to changes in input data.
- Supporting decision-making: To provide the decision-maker with information about the range of possible outcomes and the risks associated with the uncertainty of the initial data.
- Guiding further research: To indicate which data collection or refinement of which model parameters is most important.
Methods of Sensitivity Analysis
Various methods for conducting sensitivity analysis exist, ranging from simple to complex:
- Local analysis (One-at-a-Time, OAT/OFAT): One input parameter is changed at a time, while all other parameters remain fixed. This is the simplest method, but it does not allow for assessing the interaction effects between parameters.
- Derivative-based analysis (local sensitivity): Assesses the impact of small parameter changes through the partial derivatives of output variables with respect to input parameters.
- Scenario analysis: Several discrete scenarios (e.g., optimistic, pessimistic, most likely) corresponding to different sets of input parameter values are considered.
- Global sensitivity analysis: Studies the effect of simultaneously changing all (or many) parameters within their ranges of uncertainty. Statistical methods are often used:
- Monte Carlo methods: A large number of random sets of input parameters are generated to estimate the distribution of output results.
- Regression analysis: A regression model is built to link output results with input parameters.
- Analysis of variance (ANOVA) and variance-based methods: Allow for quantitatively assessing the contribution of each parameter (and their interactions) to the total variance (uncertainty) of the output result.
Role in Operations Research and Optimization
In operations research, sensitivity analysis is a standard step after finding an optimal solution. It allows for determining:
- Stability limits of the optimal solution: The range in which the parameters of the objective function or constraints can vary while the found optimal solution remains optimal.
- Shadow prices (dual values): How much the value of the objective function will change with a small change (relaxation) of a constraint (e.g., by adding one unit of a scarce resource). Shadow prices indicate the value of resources.
- Allowable ranges for parameter changes: The ranges of change for the coefficients of the objective function or the right-hand sides of the constraints within which the current structure of the optimal solution (the set of basic variables in linear programming) is preserved.
These results help the decision-maker understand how critical the initial data is and which resources are the most valuable ("bottlenecks").
Role in Modeling and Decision-Making
In the broader context of modeling and decision-making, sensitivity analysis helps to:
- Assess risks: To identify the factors that contribute the most uncertainty to the result.
- Validate the model: To check the adequacy of the model's behavior under changing conditions.
- Compare alternatives: To evaluate which of the alternatives is more robust to changes in external conditions.
- Improve understanding of the system: To identify key driving forces and relationships in the modeled system.
Interpretation of Results
- High sensitivity to a parameter means that even small errors in its estimation or its variability can significantly affect the result. Such parameters require special attention.
- Low sensitivity indicates that the result of the model or solution depends little on changes in a given parameter within the considered range, i.e., the solution is robust with respect to that parameter.
Advantages
- Increases the reliability and validity of models and solutions.
- Helps identify risks and uncertainties.
- Improves understanding of the system and the model.
- Guides efforts in data collection and model refinement.
Limitations
- Can be computationally expensive, especially with a large number of parameters (global analysis).
- Simple methods (OAT) may fail to identify interaction effects between parameters.
- The results depend on the chosen ranges of parameter variation and the model's assumptions.
Literature
- Saltelli, A., et al. Global Sensitivity Analysis: The Primer. — Wiley, 2008.
- Hillier, Frederick S.; Lieberman, Gerald J. Introduction to Operations Research. — McGraw-Hill Education. (11th ed., 2021) (Contains sections on sensitivity analysis in LP)
- Taha, Hamdy A. Operations Research: An Introduction. — Pearson. (10th ed., 2017) (Contains sections on sensitivity analysis in LP)