METRIC

428. 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.

428.1. Setup

A two-echelon network:

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

At each base $𝑗$ :

Poisson demand at rate $𝜆_{𝑗}$
One-for-one policy: maintain inventory level $𝑆_{𝑗}$ ; every demand triggers a replenishment order to the depot

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

428.2. 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 $𝑗$ :

Pipeline orders at base 𝑗 \sim Poisson (𝜆_{𝑗} 𝐸 [𝐿_{𝑗}])

where $𝐸 [𝐿_{𝑗}]$ is the expected lead time the base sees — which depends on whether the depot has stock or the base must wait.

428.3. Expected backorders

For Poisson pipeline with mean $𝜌_{𝑗}$ and base stock level $𝑆_{𝑗}$ :

𝐸 [𝐵_{𝑗} (𝑆_{𝑗})] = \sum_{𝑥 = 𝑆_{𝑗} + 1}^{\infty} (𝑥 - 𝑆_{𝑗}) 𝑒^{- 𝜌_{𝑗}} \frac{𝜌_{𝑗}^{𝑥}}{𝑥!}

— the expected shortfall above $𝑆_{𝑗}$ in a Poisson distribution. Tables exist; computer-tractable to evaluate.

428.4. Effective lead time

If the depot is stocked, base $𝑗$ gets resupplied in standard shipping time $𝑇_{𝑗}^{ship}$ . If the depot is stocked-out, the base waits for the depot’s repair pipeline.

𝐸 [𝐿_{𝑗}] = 𝑇_{𝑗}^{ship} + 𝐸 [𝑊_{𝑗}^{depot}]

where $𝐸 [𝑊_{𝑗}^{depot}]$ is the expected wait at the depot, which depends on depot’s stock $𝑆_{0}$ and the aggregate demand $\sum_{𝑗} 𝜆_{𝑗}$ hitting the depot.

428.5. Optimization

Decision variables: $𝑆_{0}$ (depot) and $𝑆_{𝑗}$ for each base.

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

min \sum_{𝑗} 𝐸 [𝐵_{𝑗} (𝑆_{𝑗})] s.t. \sum_{𝑗} 𝑐_{𝑗} 𝑆_{𝑗} + 𝑐_{0} 𝑆_{0} \leq budget

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.

428.6. 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.

428.7. 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

428.8. 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 $> 2$ echelons (Multi-Indenture / Multi-Echelon)

428.9. See also

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