Multiplicative

433. Multiplicative

433.1. Newsvendor with multiplicative demand

Relax one dimension from basic newsvendor: the demand model is no longer additive. Instead of $𝐷 = 𝜇 + 𝜀$ with additive noise of fixed scale $𝜎$ , the demand multiplies a deterministic baseline:

𝐷 = 𝜇 \cdot 𝑍

where $𝑍$ is a positive random factor with mean 1 and known distribution $𝐺$ . In additive form, $𝜎_{𝐷} = 𝜎_{𝑍}$ (independent of mean). In multiplicative form, $𝜎_{𝐷} = 𝜇 𝜎_{𝑍}$ — 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.

433.1.1. Setup

Same costs as basic newsvendor:

$𝐶_{𝑢} = 𝑃 - 𝐶$ , $𝐶_{𝑜} = 𝐶 - 𝑆$
$CR = 𝐶_{𝑢} / (𝐶_{𝑢} + 𝐶_{𝑜})$

Demand: $𝐷 = 𝜇 \cdot 𝑍$ , where $𝑍 \sim 𝐺$ with $𝐸 [𝑍] = 1$ .

Decision: $𝑄$ , but it’s natural to write $𝑄 = 𝜇 \cdot 𝑧$ (factor relative to baseline).

433.1.2. 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 $𝐹_{𝐷 (𝑄^{*})} = CR$ still holds, just rewritten in terms of $𝑍$ :

𝐹_{𝐷 (𝑄)} = 𝑃 (𝐷 \leq 𝑄) = 𝑃 (𝜇 𝑍 \leq 𝑄) = 𝑃 (𝑍 \leq 𝑄 / 𝜇) = 𝐺 (𝑄 / 𝜇)

Set $𝐺 (𝑄^{*} / 𝜇) = CR$ :

𝑄^{*} = 𝜇 \cdot 𝐺^{- 1} (CR)

433.1.3. Form of the answer

Write $𝑧^{*} = 𝐺^{- 1} (CR)$ . Then:

𝑄^{*} = 𝜇 \cdot 𝑧^{*} (multiplicative)

Compare to additive form:

𝑄^{*} = 𝜇 + 𝜎_{𝑍} 𝑧^{*} (additive)

Same critical ratio, same $𝑧^{*}$ logic, but multiplicative scales $𝑄^{*}$ as a fraction of $𝜇$ instead of adding to it.

433.1.4. Lognormal demand specialization

Common choice: $𝑍 \sim Lognormal (𝜇_{𝑍} = 0, 𝜎_{𝑍}^{2})$ with $𝐸 [𝑍] = 𝑒^{𝜎_{𝑍}^{2} / 2}$ . Note $𝐸 [𝑍] \neq 1$ in general — if you want $𝐸 [𝑍] = 1$ exactly, use $𝜇_{𝑍} = - 𝜎_{𝑍}^{2} / 2$ .

In log-space, $log 𝑍 \sim 𝒩︀ (𝜇_{𝑍}, 𝜎_{𝑍}^{2})$ . Then:

𝑧^{*} = exp (𝜇_{𝑍} + 𝜎_{𝑍} \cdot Φ^{- 1} (CR))

Example

Given (same newspaper baseline but multiplicative noise):

$𝑃 = 3$ , $𝐶 = 1$ , $𝑆 = 0$ → $𝐶_{𝑢} = 2$ , $𝐶_{𝑜} = 1$ , $CR = 2 / 3$
Baseline demand: $𝜇 = 100$
Multiplicative noise: $𝑍$ has $𝐸 [𝑍] = 1$ , coefficient of variation $CV = 𝜎_{𝑍} = 0.2$ (i.e., 20% relative noise)
Specifically: $log 𝑍 \sim 𝒩︀ (- 0.02, {0.2}^{2})$ (so $𝐸 [𝑍] \approx 1$ )

Step 1 — critical ratio

CR = \frac{2}{3} \approx 0.667

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

Step 2 — quantile in $𝑍$ -space

𝑧^{*} = exp (- 0.02 + 0.2 \cdot Φ^{- 1} (0.667)) = exp (- 0.02 + 0.2 \cdot 0.430)

𝑧^{*} = exp (- 0.02 + 0.086) \approx exp (0.066) \approx 1.068

Step 3 — order quantity

𝑄^{*} = 𝜇 \cdot 𝑧^{*} = 100 \cdot 1.068 \approx 107 newspapers

Step 4 — compare to basic (additive) newsvendor

In the basic example, $𝜎_{𝐷} = 20$ (absolute), so $𝑄^{*} \approx 100 + 0.44 \cdot 20 \approx 109$ .

In multiplicative form with $CV = 0.2$ , $𝜎_{𝐷} = 𝜇 \cdot 0.2 = 20$ (same absolute uncertainty at $𝜇 = 100$ ). $𝑄^{*} \approx 107$ .

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 $𝜎$ .

Why multiplicative matters at scale

If baseline demand grows to $𝜇 = 1000$ , additive-form predicts $𝜎_{𝐷} = 20$ still (a 2% relative swing — implausibly small). Multiplicative-form predicts $𝜎_{𝐷} = 200$ (a 20% swing, structurally consistent).

At $𝜇 = 1000$ :

Additive: $𝑄^{*} \approx 1000 + 0.44 \cdot 20 = 1009$ (essentially deterministic).
Multiplicative: $𝑄^{*} \approx 1000 \cdot 1.068 = 1068$ (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).