Location Pooling

433. Location Pooling

Holding inventory at one central location (or fewer central locations) instead of at every demand point. The simplest and most direct application of risk pooling.

433.1. Setup

You serve $𝑁$ markets / regions / stores. Demand at each is random with mean $𝜇_{𝑖}$ and standard deviation $𝜎_{𝑖}$ . Two extremes:

Decentralized: each region holds its own safety stock — independent inventories totalling $\sum 𝑧 𝜎_{𝑖}$
Centralized: one warehouse holds aggregate safety stock $𝑧 \sqrt{\sum 𝜎_{𝑖}^{2}}$ (for independent demands), then ships on demand

Centralized cuts safety stock by $\frac{\sqrt{𝑁}}{𝑁} = \frac{1}{\sqrt{𝑁}}$ (the square-root law).

433.2. Numerical illustration

$𝑁 = 25$ regions, each $𝜇 = 1000, 𝜎 = 200$ per week, $𝑧 = 1.65$ for 95% service.

	Decentralized	Centralized
Mean demand	$25 {,} 000$	$25 {,} 000$
Std dev	$25 \cdot 200 = 5 {,} 000$ (sum)	$200 \sqrt{25} = 1 {,} 000$ (pool)
Safety stock at 95%	$25 \cdot 1.65 \cdot 200 = 8 {,} 250$	$1.65 \cdot 1 {,} 000 = 1 {,} 650$
Savings	—	$80 %$

433.3. The trade-offs

Centralization isn’t free:

Transport cost — central → customer distance increases (more outbound shipping)
Response time — last-mile lead time grows
Demand visibility — aggregated demand loses regional / time-of-day texture
Customer experience — same-day delivery may be impossible from central locations
Single point of failure — central warehouse outage hits all customers

Some industries (Amazon, parts distribution) trade off these costs explicitly via partial pooling — multiple regional DCs rather than one central or $𝑁$ stores.

433.4. Optimal degree of pooling

The trade-off is a facility-location problem:

More facilities → less safety stock pooling savings, but lower transport costs
Fewer facilities → more safety stock pooling savings, but higher transport costs

The classical square-root facility-location model (Daganzo continuous approximation; covered in facility location) computes the optimal number of facilities for a given demand density and cost structure.

433.5. Other forms

Inventory pooling: physical centralization (one warehouse)
Virtual pooling: separate physical inventories that share via inter-location transfers (transshipments)
Substitution-based pooling: customer accepts a different but similar product
Lead-time pooling: covered separately at lead-time pooling

433.6. Real-world examples

Walmart regional DCs — centralized 1980s; revolutionized big-box retail
Amazon fulfillment centers — now mostly partially pooled with last-mile depots
Spare parts depots — defense, automotive
Hospital networks — central pharmacy for slow-moving / expensive drugs