Glossary · metric
Shadow Price
In a linear-programming problem, the marginal change in the objective value caused by a unit increase in the right-hand side of a binding constraint; equivalently, the optimal value of the corresponding dual variable.
Dual PriceMarginal ValueLagrange Multiplier
The shadow price (or dual price) of a constraint in a linear-programming (LP) problem is the marginal change in the optimum objective value caused by a unit increase in the constraint's right-hand-side coefficient $b_r$. For the standard LP $\max c^T x$ subject to $Ax \le b$, $x \ge 0$, the optimal dual variable $y_r^* \ge 0$ is the shadow price of resource $r$; the strong duality theorem (von Neumann 1947) says that if both primal and dual are feasible, the optimal objective values coincide. Shadow prices carry four key interpretations: (1) **capacity investment decision** — they give the marginal contribution-margin value of one additional unit (hour/kg/...) of the scarce resource; e.g. if the bottleneck-machine shadow price is 380 TRY/hour and overtime premium is 250 TRY/hour, opening overtime pays. (2) **Slack constraint** — a resource with zero shadow price is non-binding; it has idle slack and adding more does not improve the objective. (3) **Marginal pricing** — used as an internal transfer price or internal-bid price along a supply chain. (4) **Sensitivity analysis** — the shadow price stays constant inside the validity range $[b_r^-, b_r^+]$; outside the range the dual variable changes. In practice, spreadsheet solver add-ins and commercial/open-source LP solvers report shadow prices directly in the sensitivity-analysis output.
Örnek
In a 10-product food manufacturer, the LP solution gives the filling line a shadow price of 47 TRY/hour; if over 12 months the filling line runs 40 hours per week at the binding limit, the annual marginal value is 40 hours × 50 weeks × 47 TRY ≈ 94 thousand TRY. A new filling line's 9-month payback calculation anchors on this number.