In This Article
Introduction
In the real world, parameters in optimization models are rarely known with certainty. What-if analysis and sensitivity analysis help managers understand how changes in parameters affect optimal solutions, enabling better decision-making under uncertainty.
These techniques answer questions like: How would profit change if we had more capacity? How much can costs increase before our solution changes? Which constraints are binding our performance?
What-If Analysis
What-if analysis explores how changes in input parameters affect model outputs.
Types of What-If Analysis
| Type | Description | Excel Tool |
|---|---|---|
| One-variable | Vary one input, observe output | Data Table (1-way) |
| Two-variable | Vary two inputs simultaneously | Data Table (2-way) |
| Scenario | Compare multiple scenarios | Scenario Manager |
| Goal Seek | Find input for desired output | Goal Seek |
When to Use What-If Analysis
- Testing impact of uncertain parameters
- Planning for different scenarios (best case, worst case)
- Understanding model behavior
- Presenting options to stakeholders
The Sensitivity Report
After solving a linear program, the sensitivity report provides detailed information about how the optimal solution responds to parameter changes.
Two Main Sections
- Variable Cells: Information about decision variables
- Constraints: Information about each constraint
Key Information in Sensitivity Report
| Element | What It Tells You |
|---|---|
| Final Value | Optimal value of variable or constraint |
| Reduced Cost | How much objective must improve for zero variables to become positive |
| Shadow Price | Value of one additional unit of resource |
| Allowable Increase/Decrease | Range where shadow price or reduced cost is valid |
Shadow Prices (Dual Values)
The shadow price indicates the improvement in the objective function per unit increase in the right-hand side of a constraint.
Shadow Price Interpretation:
If Machine 1 has shadow price = ₹5, then:
One additional hour of Machine 1 capacity would increase profit by ₹5
Important Notes
- Shadow price is only valid within the allowable range
- Non-binding constraints have shadow price = 0
- Shadow price indicates maximum you should pay for additional resource
Example
If Machine 1 has shadow price ₹5 and you can rent extra machine time for ₹3/hour, you should rent more because value (₹5) > cost (₹3).
Allowable Ranges
Allowable Range for Objective Coefficients
The range within which an objective function coefficient can change without changing the optimal solution (though the objective value will change).
Allowable Range for RHS
The range within which the right-hand side of a constraint can change while the shadow price remains valid.
Business Applications
- Resource acquisition: Should we buy more capacity? At what price?
- Contract negotiation: How much can supplier prices increase?
- Risk assessment: How sensitive is our plan to demand changes?
- Strategic planning: Which constraints are limiting growth?
- Budgeting: Where should we invest for maximum impact?
Conclusion
Key Takeaways
- What-if analysis explores how input changes affect outputs
- Sensitivity reports come from LP solutions automatically
- Shadow prices show value of additional resources
- Use shadow prices to decide whether to acquire more resources
- Allowable ranges indicate where shadow prices are valid
- Non-binding constraints have shadow price = 0
- Sensitivity analysis enables better decisions under uncertainty