419. Capacity-Constrained

419.1. Capacity-constrained multi-product newsvendor

Relax two dimensions from basic newsvendor: number of items > 1 and capacity is constrained. Each product has its own demand and cost structure, but they share a budget, warehouse space, or other resource.

Without the constraint, each product is its own newsvendor — at its own critical ratio . With a shared constraint, those individual optima may not all be achievable — the constraint forces trade-offs.

419.1.1. Setup

products. For each :

• Demand
• Costs: → ,
• Resource consumption: per unit of product (e.g., $ if budget-constrained, ft³ if space-constrained)

Constraint:

If individual optima already satisfy the constraint, use them. The interesting case is when — must shrink.

419.1.2. Lagrangian

Maximize total expected profit subject to the capacity constraint:

where is the basic newsvendor expected profit for product .

Lagrangian (with multiplier ):

419.1.3. Modified critical ratio

Take for each :

For continuous demand, (from the marginal-analysis derivation in basic newsvendor). Setting equal to :

Compare to basic: . The constraint reduces each product’s effective critical ratio by — products that consume more capacity per unit get bigger CR cuts, ordering less.

Effective per-product CR:

When , the effective CR goes negative — that product is not worth ordering at all given the constraint.

419.1.4. Find

is chosen so that . One equation, one unknown — bisection works (the LHS is monotone decreasing in ).

419.1.5. Algorithm

1. Compute unconstrained .
2. Check . If yes, done.
3. Else, find such that (bisection).
4. Each .