Economic Interpretation of Dual Variables

Consider a linear program of the form

$$\max\ \mathbf{c}^T \mathbf{x} \quad \text{subject to} \quad A\mathbf{x} \le \mathbf{b},\ \mathbf{x} \ge \mathbf{0}.$$

Each primal constraint $\mathbf{a}_i^T \mathbf{x} \le b_i$ has an associated dual variable $y_i$. Economically, the optimal dual value $y_i^*$ is the shadow price of resource $i$: it measures how much the optimal objective value changes per unit increase in $b_i$.

For small enough changes in the right-hand sides (changes that do not alter the optimal basis),

$$\Delta z^* \;=\; \sum_{i=1}^m y_i^* \, \Delta b_i.$$

For example, suppose a workshop's optimal profit is \500$, with$y_1^* = $3$per unit of raw material. If the supplier delivers$4$ additional units of raw material, the new optimal profit is

$$500 + 3 \cdot 4 \;=\; \$512.$$