VARI-METRIC

429. VARI-METRIC

An extension of METRIC (Graves 1985) that captures the variance of pipeline orders, not just the mean. The pipeline is modeled as Negative Binomial instead of Poisson — same mean, larger variance, more realistic for low-stock scenarios.

429.1. Why METRIC under-estimates backorders

METRIC assumes depot resupply time is deterministic and that the resulting order pipeline is Poisson. But when the depot occasionally stocks out, downstream waiting times are random, inflating pipeline variance.

If true variance $>$ Poisson variance ( $=$ mean), the real probability of large pipelines is higher than Poisson predicts → real backorders are higher.

429.2. Negative Binomial pipeline

VARI-METRIC fits the pipeline-order distribution as Negative Binomial with mean $𝜌$ and variance $𝜎^{2} > 𝜌$ :

𝑃 (pipeline = 𝑥) = (\binom{𝑥 + 𝑟 - 1}{𝑥}) {(1 - 𝑝)}^{𝑥} 𝑝^{𝑟}

where $𝑟, 𝑝$ are chosen to match the observed (or computed) mean and variance:

𝜌 = (1 - 𝑝) \frac{𝑟}{𝑝}, 𝜎^{2} = (1 - 𝑝) \frac{𝑟}{𝑝^{2}} = \frac{𝜌}{𝑝}

Solve for $(𝑟, 𝑝)$ given target $(𝜌, 𝜎^{2})$ .

429.3. Computing the variance

Variance of orders at base $𝑗$ :

𝜎_{𝑗}^{2} = 𝜆_{𝑗} 𝐸 [𝐿_{𝑗}] + 𝜆_{𝑗}^{2} Var (𝐿_{𝑗})

— two contributions:

Demand variance during a mean-length lead time (Poisson contribution, $= 𝜆 𝐸 [𝐿]$ )
Lead-time variance itself (when waiting at the depot)

For the depot pipeline (single location aggregating multiple bases), use law of total variance:

Var (pipeline) = 𝐸 [pipeline | depot stockout state] + Var (\dots)

(Exact formulas are detailed; the point is that variance is not equal to the mean.)

429.4. Expected backorders with NB pipeline

𝐸 [𝐵_{𝑗} (𝑆_{𝑗})] = \sum_{𝑥 = 𝑆_{𝑗} + 1}^{\infty} (𝑥 - 𝑆_{𝑗}) 𝑃 (pipeline = 𝑥)

— same form as METRIC, just with the NB instead of Poisson distribution for pipeline orders.

429.5. Practical impact

For low-stock items with appreciable depot-side delay, VARI-METRIC predictions are 20–50% more pessimistic than METRIC — and they match empirical data much better.

The optimization (marginal analysis on the budget vs total backorders) is the same algorithm; only the backorder calculation differs.

429.6. When METRIC is “good enough” vs VARI-METRIC

High stock levels (low backorder probability) → METRIC is close
Low stock / tight budget / many bases → VARI-METRIC corrects substantial under-counting

429.7. See also

METRIC — Poisson-pipeline original
Multi-Echelon overview
Stochastic-service models