Multi-Echelon Inventory
ME-INV · CLARK-SCARF 1960
Logistics · Inventory · StrategicThe Multi-Echelon Inventory Problem asks how to coordinate stock levels across a multi-tier supply chain — typically plant → central DC → regional DCs → retailers — when demand is stochastic and leads are positive. The cornerstone result, Clark & Scarf (1960), proves the optimality of echelon base-stock policies for serial systems, reducing the multi-echelon problem to a sequence of independent newsvendor problems via the echelon-stock decomposition. The same decomposition drives the Eppen & Schrage (1981) allocation model and the Rosling (1989) reduction from assembly to serial.
Clark-Scarf decomposition
Serial system · stochastic demand · base-stock optimality
Setting
Consider a serial supply chain: stage $N$ (highest, e.g., plant) receives from an outside supplier; each stage $j$ supplies stage $j-1$ with a deterministic lead time $L_j$; stage $1$ faces stochastic demand $D^t$. Holding cost $h_j$ per unit per period accrues at each stage. Stockouts at stage $1$ cost $b$ per unit per period (backorder). Each stage follows a base-stock policy: order-up-to level $S_j$ after each demand realisation.
Notation
| Symbol | Meaning |
|---|---|
| $N$ | number of serial stages; stage 1 is the customer-facing tier |
| $L_j$ | lead time from stage $j+1$ to stage $j$ (periods) |
| $h_j$ | holding cost rate at stage $j$ ($h_j \ge h_{j+1}$ for physical consistency) |
| $b$ | backorder cost per unit per period (applied at stage 1) |
| $D^t$ | demand at stage 1 in period $t$ (i.i.d. with known distribution) |
| $S_j$ | echelon base-stock level at stage $j$ (decision variable) |
| $IP_j^t$ | echelon inventory position at stage $j$ at start of period $t$ |
The Clark-Scarf result
Define echelon inventory at stage $j$ as the on-hand inventory at stages $1, \ldots, j$ plus inventory in transit between them (i.e., all stock "downstream of stage $j$"). Let $h_j'$ be the incremental holding rate at echelon $j$, i.e., $h_j' = h_j - h_{j+1}$ with $h_{N+1} = 0$. Then:
where $D_{[L]}$ denotes lead-time demand over $L$ periods. Each term involves only a single decision variable $S_j$ — so the multi-stage problem decomposes into $N$ independent one-stage newsvendor problems. Optimal $S_j^*$ satisfies the critical-fractile condition specific to that stage's holding/penalty trade-off.
Extensions
- Assembly systems (several inputs combining into one output) — Rosling (1989) shows equivalence to serial systems under balanced-echelon conditions.
- Distribution (one-warehouse multi-retailer) — Eppen & Schrage (1981) analyse allocation policies; Axsäter (1990, 2015) gives tighter bounds.
- General tree and network topologies — exact analysis is hard; SIP (stochastic integer programming) and METRIC-family approximations (Sherbrooke 1968) are the standard tools.
- Finite horizon — dynamic-programming extension due to Scarf (1960).