The Price of Robustness
At identical spend, which plan best resists the uncertainty around it? Hardness gives that a number — a continuous robustness measure in [0,1], computed two independent ways that agree, running live in your browser.
Understand exactly what you're looking at
The theory, how the data is acquired and cleaned, how to read the result — and what BIS would do for you next.
“Is this solution robust?” is usually answered yes or no — a strange way to talk about a continuous property. Two plans can both be feasible and both survive the worst case, yet behave nothing alike in between.
For a problem whose coefficients live in intervals rather than at points, Hardness aggregates per-constraint worst-case violation into a single scalar η ∈ [0, 1], so feasible solutions can be ranked rather than merely accepted.
No external dataset — the instance is constructed to make the point. Take a fixed budget (think a renewable-generation portfolio with a hard cap) and sweep one decision variable. Cost is constant by construction, so price cannot rank anything; all that is left to separate the plans is how they behave under uncertainty.
The interval bounds you set become the uncertainty the measures reason about — the same intervals the closed form integrates and the simulator draws from.
The measure is computed two ways, and the fact that they agree is the validation. The closed form evaluates a generalized Irwin–Hall CDF exactly — microseconds per call, right for small problems. Monte Carlo estimates the same quantity from a violations matrix and scales where the closed form does not.
When an exact method and a sampling method built from different mathematics land on the same curve, you have evidence the definition is sound rather than an artifact of one implementation. Part of the theory is machine-checked in Agda under --safe with zero postulates.
As you sweep, three different robustness measures disagree about which plan to buy — that disagreement is the whole argument for caring about the definition. One slopes monotonically the wrong way; another is pinned flat at zero across half the range, unable to express an opinion at all.
η picks a plan whose realized shortfall, under twenty thousand simulated futures, is a small fraction of what the naive choice delivers — at identical spend. The histogram shows that shortfall distribution; the closed form predicted its shape without drawing a single scenario.
Most tools tell you whether a plan is feasible. The expensive question is which of the feasible plans will still be standing when the numbers you assumed turn out to be intervals, not facts.
BIS builds that ranking into your decision: a robustness score your team can sort on, validated two independent ways, so you spend the same money on the plan that survives contact with reality — not merely the one that looked best on nominal data.
This preview runs on real public data. Yours would run on your data.
We start with a conversation, prove the value on a focused POC with your real data, then ship it — in our environment or deployed into yours.