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This solver evaluates a single decision variable under uncertainty. The page compares nominal and protected solutions using box or ellipsoidal style buffering.
Example data table
Illustrative scenarios based on the default sample parameters. Higher protection narrows feasibility and changes the protected objective.
| Scenario | Protection level | Robust x* | Protected objective | Utilization % | Resource slack |
|---|---|---|---|---|---|
| Protection 0 | 0.00 | 95.0000 | 4,240.0000 | 100.00 | 0.0000 |
| Protection 0.5 | 0.50 | 88.6667 | 3,752.3333 | 100.00 | 0.0000 |
| Protection 1 | 1.00 | 82.8205 | 3,314.3590 | 100.00 | 0.0000 |
| Protection 1.5 | 1.50 | 77.4074 | 2,920.5556 | 100.00 | 0.0000 |
| Protection 2 | 2.00 | 72.3810 | 2,566.1905 | 100.00 | 0.0000 |
Formula used
Protected objective
Maximization: ZR = (c − mΔc)x + F
Minimization: ZR = (c + mΔc)x + F
Protected constraints
(a + mΔa)x ≤ (b − mΔb)(1 − r)
x ≥ q + mΔq
l ≤ x ≤ u
Protection multiplier
Box mode: m = ρ
Ellipsoidal mode: m = √ρ
Decision rule
The calculator builds a protected feasible interval [max(l, q + mΔq), min(u, ((b − mΔb)(1 − r)) / (a + mΔa))]. It then selects the endpoint that best fits the chosen objective direction.
How to use this calculator
- Choose whether you want to maximize or minimize the protected objective.
- Select an uncertainty model. Box is direct and conservative. Ellipsoidal is smoother.
- Enter the nominal objective coefficient, its uncertainty, and any fixed term.
- Enter the nominal constraint coefficient, capacity, and their uncertainty levels.
- Set the minimum target, lower bound, upper bound, and optional safety reserve.
- Choose a protection level. Larger values increase robustness and may reduce performance.
- Press Solve Robust Model to show the protected result above the form.
- Review the chart, summary table, and export the results in CSV or PDF format.
FAQs
1) What does robust optimization mean here?
It means the solver adjusts coefficients, targets, and capacity against uncertainty before choosing a decision. The result is designed to remain safer when inputs shift unfavorably.
2) What is the difference between box and ellipsoidal uncertainty?
Box uncertainty applies the protection level directly to each uncertain term. Ellipsoidal uncertainty softens the adjustment by using a square-root radius, which often produces less conservative results.
3) Why is the protected objective often lower than the nominal one?
Robust solutions reserve room for adverse changes. That protection usually reduces optimistic performance, especially in maximization problems with tight resource limits.
4) What is the price of robustness?
It measures how much performance you give up for protection. The page compares the nominal optimum against the protected optimum to quantify that trade-off.
5) Why can the solver show an infeasible result?
Infeasibility appears when the protected lower requirement exceeds the protected upper limit. That usually means the target is too high, capacity is too low, or uncertainty is too severe.
6) Does this tool solve large multi-variable models?
No. This page is a focused single-variable solver for quick analysis, education, and planning. It explains robust logic clearly without requiring a full optimization package.
7) How should I choose the protection level?
Start small, then increase it until the solution matches your risk tolerance. If performance falls sharply or feasibility disappears, the model may be overprotected.
8) Why compare nominal and protected solutions together?
The comparison shows the benefit and cost of robustness. It helps you understand whether extra protection meaningfully improves decision stability for your case.