421. Clark-Scarf

The foundational result for stochastic multi-echelon inventory in serial systems (Clark & Scarf 1960). Shows that the multi-echelon problem decomposes into independent single-echelon problems — one per stage, each solved as a newsvendor-like base-stock problem with an induced penalty cost.

421.1. Setup: serial system

Stages numbered (most upstream) to (most downstream = retailer). Customer demand hits stage only. Each stage:

• Holds inventory
• Receives shipments from upstream stage with lead time
• Ships to downstream stage
• Stage also faces backorder cost per unit per period

Holding cost rates: at stage (assume — inventory is more expensive downstream where it carries more added value).

421.2. Echelon stock

Define echelon stock at stage :

— what’s at stage plus everything below it. Holding cost on echelon stock is incremental:

(with ). This is the marginal holding cost as inventory moves from stage to stage .

421.3. Optimal policy: echelon base-stock

The optimal policy is echelon base-stock: at each review, stage orders to bring its echelon stock up to a target .

The Clark-Scarf decomposition shows that at each stage solves a single-stage newsvendor problem with:

• Demand: lead-time demand from the most upstream (stage 1) feeding through
• Underage cost: an induced penalty that propagates upstream
• Overage cost: (incremental echelon holding cost)

421.4. Induced penalty

The recursion (downstream → upstream):

Formally: each stage’s penalty depends on the optimal cost-to-go from downstream — a backward DP. The decomposition magic is that, despite this coupling, the form of each stage’s optimal policy is base-stock with parameters depending only on local data + the induced penalty.

421.5. Worked sketch (2-stage)

Stages: warehouse (1) → retailer (2). lead times. . Backorder cost at retailer.

Retailer (stage 2): classical newsvendor with critical ratio . Solve for .

Warehouse (stage 1): newsvendor with critical ratio where is the expected cost the warehouse imposes on the retailer when warehouse stockouts cause retailer delays.

421.6. Why echelon stocks decouple the problem

If stage ‘s order pulls from infinite supply (or upstream is always available), each stage is independent. Real life: upstream isn’t always available. The induced penalty charges upstream for the downstream cost of its stockouts — capturing the coupling exactly.

421.7. Limitations

• Serial systems only: pure chains, no branching
• Stationary demand: i.i.d. demand each period
• Costly to compute for many stages and long horizons

For general networks, see Graves-Willems guaranteed-service.

421.8. See also

• Multi-Echelon — overview
• Graves-Willems — alternative for general networks
• Base Stock Policy
• Newsvendor