Uncertainty in Optimization: Why Deterministic Models Fail

Up to this point, every linear program we have solved

$$\max\ c^T x \quad \text{s.t.}\quad Ax \le b,\ x \ge 0$$

has treated the parameters $c$, $A$, and $b$ as known constants. In real applications they almost never are. Future demand, fuel prices, crop yields, machine failures, travel times, exchange rates -- all of these enter the model as numbers we must guess.

The most common shortcut is nominal-value optimization: pick a single value for each uncertain parameter (often its mean or a point forecast), plug it in, and solve the resulting deterministic LP. This is convenient and dangerous. Two failure modes appear:

1. Infeasibility under realization. When the true parameter turns out to be worse than the nominal value, the optimal plan can violate the constraint.
2. Optimistic objective. The deterministic objective value is a biased estimate of what the plan actually delivers on average.

Illustration. Suppose capacity $b$ is random with $b = 80$ with probability $1/2$ and $b = 100$ with probability $1/2$, so $\bar b = 90$. The deterministic problem $\max\ x$ s.t. $x \le 90$, $x \ge 0$ has $x^* = 90$. But $P(x^* > b) = P(b = 80) = 1/2$: the plan exceeds true capacity half the time.

A stochastic optimization model treats the uncertain parameter as a random variable and asks for a decision that performs well across the distribution, not just at one nominal point. The rest of this lesson quantifies why that matters.