417. Multiplicative

417.1. Newsvendor with multiplicative demand

Relax one dimension from basic newsvendor: the demand model is no longer additive. Instead of with additive noise of fixed scale , the demand multiplies a deterministic baseline:

where is a positive random factor with mean 1 and known distribution . In additive form, (independent of mean). In multiplicative form, — uncertainty scales with the baseline.

When does multiplicative make sense? Whenever demand uncertainty is proportional — sales of a product 10× as popular have 10× the absolute swing, not the same. Common in retail, fashion, and seasonal goods.

417.1.1. Setup

Same costs as basic newsvendor:

Demand: , where with .

Decision: , but it’s natural to write (factor relative to baseline).

417.1.2. Critical-ratio derivation (unchanged)

The marginal-analysis logic from basic newsvendor is unchanged. Order one more unit while expected marginal profit is positive. The condition still holds, just rewritten in terms of :

Set :

417.1.3. Form of the answer

Write . Then:

Compare to additive form:

Same critical ratio, same logic, but multiplicative scales as a fraction of instead of adding to it.

417.1.4. Lognormal demand specialization

Common choice: with . Note in general — if you want exactly, use .

In log-space, . Then:

Example

Given (same newspaper baseline but multiplicative noise):

  • , , → , ,
  • Baseline demand:
  • Multiplicative noise: has , coefficient of variation (i.e., 20% relative noise)
  • Specifically: (so )

Step 1 — critical ratio

(Same as basic newsvendor — the critical ratio depends on costs only.)

Step 2 — quantile in -space

Step 3 — order quantity

Step 4 — compare to basic (additive) newsvendor

In the basic example, (absolute), so .

In multiplicative form with , (same absolute uncertainty at ). .

Slight difference (109 vs 107) comes from the lognormal having a different shape than normal — same coefficient of variation, but right-skewed (long right tail). The lognormal puts more probability on the right, so the upper quantile is actually less far above the mean than for a normal of the same .

Why multiplicative matters at scale

If baseline demand grows to , additive-form predicts still (a 2% relative swing — implausibly small). Multiplicative-form predicts (a 20% swing, structurally consistent).

At :

  • Additive: (essentially deterministic).
  • Multiplicative: (still 6.8% above baseline).

Choose multiplicative when uncertainty is proportional to the baseline; additive when it’s a fixed scale (e.g., shipping noise, fixed measurement error).