Recipe

Compute the price of robustness

Use when: you have a nominal design and either an uncertainty radius or a cost budget, and need what protection actually costs

Goal: the robust cost at a given radius, or the radius a given cost budget affords

Taught in How much does robustness cost?

Quick recipe

Identify which model you're pricing
Compute the sensitivity $\|c\|$
Compute the cost at a given radius
Or work backward from a cost budget to the radius it buys
For the chance-constrained model, price cost as capacity minus nominal

1
Check whether the problem gives a linear objective with an uncertainty ball (radius $r$ ) or a chance-constrained model (risk level $\alpha$ ).
- linear ball model, objective coefficient vector $c$ and radius $r$ — go to Step 2.
- chance-constrained model (Sections 8-9) — skip to Step 4; cost is capacity minus nominal, with no $\|c\|$ involved.
Why: the two models this chapter builds price robustness through different formulas, even though both measure the same idea, a gap from the nominal value.
2
For the linear ball model, compute the norm of the objective's coefficient vector, $\|c\|=\sqrt{c_1^2+c_2^2+\cdots}$ .

Example

$c=(3,4)$ : $\|c\|=\sqrt{3^2+4^2}=\sqrt{25}=5$ .

Why: $\|c\|$ is exactly the constant of proportionality in $\text{Cost}(r)=r\|c\|$ — how steeply this objective reacts to the direction uncertainty pushes it.
3
Multiply: $\text{Cost}(r)=r\|c\|$ .

Example

With $\|c\|=5$ : $\text{Cost}(1)=5$ , $\text{Cost}(2)=10$ .

Why: this is the chapter's own derived formula for the price of robustness under the linear ball model.
4
Given a maximum affordable extra cost, set $r\|c\|\le\text{budget}$ and solve for $r$ .

Example

Budget $=15$ , $\|c\|=5$ : $r\le15/5=3$ , so $r=3$ is the largest affordable radius.

Why: since cost rises linearly in $r$ , dividing the budget by the fixed rate $\|c\|$ gives the largest radius the budget still covers.
5
Compute the capacity at a chosen $\alpha$ (Section 8's formula), then subtract the nominal baseline value.

Example

Demand uniform on $[100,200]$ , nominal $150$ : at $\alpha=0.10$ , capacity $190$ , cost $=190-150=40$ ; at $\alpha=0$ , capacity $200$ , cost $=50$ .
- the distribution is genuinely uniform — the rate stays constant, as in this chapter's models.
- the distribution concentrates mass away from its tail (bell-shaped, for instance) — the rate is not guaranteed constant; treat a straight-line result as a property of this specific model, not a universal law.
Why: this measures the same "extra cost for extra protection" idea as Steps 2-3, just through capacity instead of $r\|c\|$ .

Common mistakes

Assuming every uncertainty model prices robustness on a straight line — that's a property of the ball and uniform models this chapter builds, not a universal law.
Forgetting that a larger $\|c\|$ makes the exact same radius $r$ more expensive — price depends on the objective's own sensitivity, not just the uncertainty size.
Mixing up the two models' formulas — dividing by $\|c\|$ when working with chance-constrained capacity, or vice versa.
Treating $\text{Cost}(r)\le\text{budget}$ as an equality only, missing that any smaller $r$ is also affordable.