Multi-Product

422. Multi-Product

422.1. Multi-product newsvendor (risk pooling)

Relax one dimension from basic newsvendor: number of items is no longer one. Multiple products $𝑖 = 1, \dots, 𝑛$ with random demands $𝐷_{𝑖} \sim 𝒩︀ (𝜇_{𝑖}, 𝜎_{𝑖}^{2})$ .

Two cases, dramatically different:

Independent decisions, independent inventories. Each product is its own newsvendor. Solve $𝑛$ separate problems.
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 $𝐷_{agg} = \sum_{𝑖} 𝐷_{𝑖}$ .

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

422.1.1. Independent newsvendors (case 1)

Just $𝑛$ separate basic newsvendors. Each has its own $𝐶_{𝑢}^{𝑖}$ , $𝐶_{𝑜}^{𝑖}$ , ${CR}_{𝑖}$ , $𝑄_{𝑖}^{*}$ . Total expected profit is the sum.

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

If $𝐷_{𝑖} \sim 𝒩︀ (𝜇_{𝑖}, 𝜎_{𝑖}^{2})$ with pairwise correlation $𝜌_{𝑖 𝑗}$ :

𝜇_{agg} = \sum_{𝑖 = 1}^{𝑛} 𝜇_{𝑖}

𝜎_{agg}^{2} = \sum_{𝑖 = 1}^{𝑛} 𝜎_{𝑖}^{2} + 2 \sum_{𝑖 < 𝑗} 𝜌_{𝑖 𝑗} 𝜎_{𝑖} 𝜎_{𝑗}

For independent products ( $𝜌_{𝑖 𝑗} = 0$ ):

𝜎_{agg} = \sqrt{\sum_{𝑖 = 1}^{𝑛} 𝜎_{𝑖}^{2}}

For identical, independent products ( $𝜇_{𝑖} = 𝜇$ , $𝜎_{𝑖} = 𝜎$ ):

𝜇_{agg} = 𝑛 𝜇, 𝜎_{agg} = \sqrt{𝑛} \cdot 𝜎

422.1.3. Pooling reduces relative uncertainty

Coefficient of variation (CV = $𝜎 / 𝜇$ ) for the aggregate vs. individual:

{CV}_{agg} = \frac{𝜎_{agg}}{𝜇_{agg}} = \frac{\sqrt{𝑛} 𝜎}{𝑛 𝜇} = \frac{{CV}_{individual}}{\sqrt{𝑛}}

Pooling $𝑛$ identical products cuts relative uncertainty by $\sqrt{𝑛}$ . Pool 4 products → half the relative noise. Pool 100 → 10× less.

This translates directly into safety stock savings: the buffer above $𝜇$ scales with $𝜎$ , not $𝜇$ , so doubling $𝑛$ adds inventory only as $\sqrt{2}$ , while demand grows as 2.

422.1.4. Pooled order decision

Same newsvendor structure, but on aggregate:

𝑄_{agg}^{*} = 𝜇_{agg} + 𝑧^{*} \cdot 𝜎_{agg}

where $𝑧^{*} = Φ^{- 1} (CR)$ uses the pooled cost structure.

422.1.5. When pooling is worse

Risk pooling is not always beneficial:

Correlated demands with $𝜌_{𝑖 𝑗} > 0$ : aggregation doesn’t reduce variance much. Worst case, perfect correlation ( $𝜌 = 1$ ): $𝜎_{agg} = \sum_{𝑖} 𝜎_{𝑖}$ , 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.

Example

Given (4 identical-ish newspaper variants, independent demands):

4 products, each: $𝑃_{𝑖} = 3$ , $𝐶_{𝑖} = 1$ , $𝑆_{𝑖} = 0$ , $𝐷_{𝑖} \sim 𝒩︀ (100, 20^{2})$
Demands independent across products

Step 1 — independent newsvendors (no pooling)

Each product is its own newsvendor, identical to the basic example:

$CR = 2 / 3$ , $𝑧^{*} = 0.44$ , $𝑄_{𝑖}^{*} = 100 + 0.44 \cdot 20 \approx 109$
Total inventory: $4 \cdot 109 = 𝟒𝟑𝟔$

Step 2 — pooled (case 2)

Aggregate:

$𝜇_{agg} = 4 \cdot 100 = 400$
$𝜎_{agg} = \sqrt{4} \cdot 20 = 40$ (only $\sqrt{𝑛}$ , not $𝑛$ )
$CR = 2 / 3$ , $𝑧^{*} = 0.44$
$𝑄_{agg}^{*} = 400 + 0.44 \cdot 40 \approx 𝟒𝟏𝟖$

Step 3 — compare

Independent: 436 units total
Pooled: 418 units total
Save 18 units (≈ 4%) with the same service level.

Step 4 — see the $\sqrt{𝑛}$ scaling

At $𝑛 = 100$ identical products:

Independent: $100 \cdot 109 = 10900$ units
Pooled: $𝜇_{agg} + 𝑧^{*} 𝜎_{agg} = 10000 + 0.44 \cdot 200 = 10088$ units
Save 7% and the relative savings grow with $𝑛$ — at large $𝑛$ , pooled inventory approaches just $𝜇_{agg}$ (deterministic limit), while independent inventory still carries each product’s full safety buffer.

Why pooling works: when one product over-sells, another likely under-sells. Net demand is more predictable than any one product’s demand. Pooling captures this — independent stocks can’t.

Real-world consequence: distribution center designs, online retailers vs. physical stores, fungible commodities — all driven by the $\sqrt{𝑛}$ risk-pooling math.