437. Capacity-Constrained

Relax one dimension from basic EOQ: capacity is no longer unlimited. A shared resource — warehouse space, working-capital budget, or shelf footprint — caps the total inventory you can hold at any moment. Multiple items compete for the same budget.

437.0.1. Setup

items indexed :

• , , as in basic EOQ (item-specific)
• = capacity consumed per unit of item (e.g., $ per unit if budget-constrained, ft³ per unit if warehouse-constrained)
• = total capacity available

Constraint (peak inventory after a fresh order arrives):

If individual EOQs already satisfy the constraint, the constraint is not binding — use them directly. The interesting case is when .

437.0.2. Lagrangian formulation

Minimize total cost subject to the capacity constraint:

Introduce Lagrange multiplier for the constraint:

437.0.3. KKT / FOC in

For each , :

This is basic EOQ with an inflated holding cost . The Lagrange multiplier raises the effective holding cost on every item in proportion to its capacity consumption — items that hog more capacity get penalized more, shrinking their .

437.0.4. Find from the constraint

If the constraint binds, . This is one equation in one unknown — solve numerically (or in closed form for some special cases).

Special case: budget constraint with proportional holding. If (capacity is dollars), (carrying-rate model), and we substitute into :

Then . Sum:

Solve for :

For other constraint forms, comes out of a 1-D root-finding pass over (monotone decreasing in , so bisection works).

437.0.5. Algorithm

1. Compute unconstrained for each .
2. Check . If yes, done — constraint not binding.
3. Else, solve for (closed form above, or numerical).
4. Each .

437.0.6. Final formulas

Sanity check: if (no constraint), and — basic EOQs ✓.