424. METRIC

Multi-Echelon Technique for Recoverable Item Control (Sherbrooke 1968). A closed-form model for spare-parts networks with Poisson demand and one-for-one (S - 1, S) base-stock policy.

The original was developed for US Air Force repairable items — engines, avionics, hydraulics. The model is now standard for any service-parts network: defense, aerospace, semiconductors, medical devices.

424.1. Setup

A two-echelon network:

• One depot (central repair facility)
• bases (forward sites consuming and returning parts)

At each base :

• Poisson demand at rate
• One-for-one policy: maintain inventory level ; every demand triggers a replenishment order to the depot

Failed parts go to the depot for repair (mean repair time ), then redistributed.

424.2. Palm’s theorem (the key fact)

If demand is Poisson and lead times are independent, the number of orders in the replenishment pipeline at any moment is Poisson-distributed with mean = (demand rate) × (mean lead time).

For base :

where is the expected lead time the base sees — which depends on whether the depot has stock or the base must wait.

424.3. Expected backorders

For Poisson pipeline with mean and base stock level :

— the expected shortfall above in a Poisson distribution. Tables exist; computer-tractable to evaluate.

424.4. Effective lead time

If the depot is stocked, base gets resupplied in standard shipping time . If the depot is stocked-out, the base waits for the depot’s repair pipeline.

where is the expected wait at the depot, which depends on depot’s stock and the aggregate demand hitting the depot.

424.5. Optimization

Decision variables: (depot) and for each base.

Objective: minimize total backorders (or expected number of units down) subject to a budget constraint:

Solved by marginal analysis: at each step, add one unit of inventory at whichever stocking location gives the biggest reduction in expected backorders per dollar.

424.6. VARI-METRIC

Sherbrooke’s original METRIC approximates depot resupply time as deterministic. VARI-METRIC (Graves 1985) fixes this by modeling the depot pipeline as Negative Binomial — preserving both the mean and the variance of pipeline orders. Sometimes large efficiency gains. See VARI-METRIC.

424.7. Where it shows up

• Military / aerospace logistics — original application
• Semiconductor fab spare parts — single fab + multiple tools
• Service parts networks — automotive, industrial equipment
• Computer / IT spares — server / network gear

424.8. Limitations

• Poisson demand only — assumes one-by-one ordering; doesn’t handle bulk demand
• Stationary — no demand trends or seasonality
• Two-echelon — extensions exist for echelons (Multi-Indenture / Multi-Echelon)

424.9. See also

• VARI-METRIC — variance-correction extension
• Stochastic-service — generalization
• Poisson Distribution
• Base Stock Policy