418. Multi-Product

418.1. Multi-product newsvendor (risk pooling)

Relax one dimension from basic newsvendor: number of items is no longer one. Multiple products with random demands .

Two cases, dramatically different:

1. Independent decisions, independent inventories. Each product is its own newsvendor. Solve separate problems.
2. Pooled inventory. All products share the same physical pool (interchangeable, or shipped from one location to many destinations). One newsvendor problem on the aggregate demand .

Pooling is often much better — risk pooling / demand aggregation reduces relative uncertainty.

418.1.1. Independent newsvendors (case 1)

Just separate basic newsvendors. Each has its own , , , . Total expected profit is the sum.

418.1.2. Pooled inventory (case 2): aggregate demand statistics

If with pairwise correlation :

For independent products ():

For identical, independent products (, ):

418.1.3. Pooling reduces relative uncertainty

Coefficient of variation (CV = ) for the aggregate vs. individual:

Pooling identical products cuts relative uncertainty by . Pool 4 products → half the relative noise. Pool 100 → 10× less.

This translates directly into safety stock savings: the buffer above scales with , not , so doubling adds inventory only as , while demand grows as 2.

418.1.4. Pooled order decision

Same newsvendor structure, but on aggregate:

where uses the pooled cost structure.

418.1.5. When pooling is worse

Risk pooling is not always beneficial:

• Correlated demands with : aggregation doesn’t reduce variance much. Worst case, perfect correlation (): , no benefit.
• Heterogeneous service-level requirements: pooling forces a single CR; if products need different service levels, separate inventories may be better.
• Substitution / customization: physical pooling requires items to be substitutable. A red and blue jersey can’t be pooled if the customer wants a specific color.