Shortage Penalty

420. Shortage Penalty

420.1. Newsvendor with explicit shortage penalty

Relax one dimension from basic newsvendor: excess demand costs more than just lost margin. In the basic model, $𝐶_{𝑢} = 𝑃 - 𝐶$ is the foregone profit on a missed sale. But stockouts often have additional costs — customer goodwill, expedited replacement, reputation damage, contractual penalties.

Add an explicit penalty $𝜋$ per unit of unmet demand, on top of lost margin.

420.1.1. Setup

Same as basic newsvendor, plus:

$𝜋$ = stockout penalty per unmet unit (dollars per unsold-due-to-stockout)

Underage cost now has two components:

𝐶_{𝑢} = (𝑃 - 𝐶) + 𝜋

The form of the answer is unchanged — only the value of $𝐶_{𝑢}$ is larger.

420.1.2. Critical ratio

CR = \frac{𝐶_{𝑢}}{𝐶_{𝑢} + 𝐶_{𝑜}} = \frac{(𝑃 - 𝐶) + 𝜋}{(𝑃 - 𝐶) + 𝜋 + (𝐶 - 𝑆)}

A larger $𝐶_{𝑢}$ pushes CR upward → larger $𝑧^{*}$ → larger $𝑄^{*}$ . More penalty for stocking out → order more.

420.1.3. Bounds on CR

$𝜋 = 0$ : collapses to basic newsvendor.
$𝜋 \to \infty$ : $CR \to 1$ (almost certain to satisfy demand). $𝑄^{*}$ grows without bound.
$𝜋$ very large but finite: $𝑄^{*}$ saturates at the upper tail of demand.

420.1.4. Decision rule (continuous demand)

𝑄^{*} = 𝐹^{- 1} (\frac{𝑃 - 𝐶 + 𝜋}{𝑃 - 𝐶 + 𝜋 + 𝐶 - 𝑆})

For normal demand $𝐷 \sim 𝒩︀ (𝜇, 𝜎^{2})$ :

𝑄^{*} = 𝜇 + Φ^{- 1} (CR) 𝜎

Example

Given (same newspaper baseline + a $$ 1$ goodwill penalty per missed sale):

Sale price: $𝑃$ = $3 / unit
Purchase cost: $𝐶$ = $1 / unit
Salvage: $𝑆$ = $0
Stockout penalty: $𝜋$ = $1 / unit (lost goodwill)
Demand: $𝐷 \sim 𝒩︀ (𝜇 = 100, 𝜎 = 20)$

Step 1 — underage and overage

𝐶_{𝑢} = (𝑃 - 𝐶) + 𝜋 = 2 + 1 = 3

𝐶_{𝑜} = 𝐶 - 𝑆 = 1

Step 2 — critical ratio

CR = \frac{3}{3 + 1} = 0.75

Compared to basic newsvendor ( $CR = 2 / 3 \approx 0.667$ ), the penalty pushes CR from 67% up to 75%.

Step 3 — quantile and order quantity

𝑧^{*} = Φ^{- 1} (0.75) \approx 0.674

𝑄^{*} = 100 + 0.674 \cdot 20 \approx 113 newspapers

Step 4 — compare to basic newsvendor

Without penalty: $𝑄_{basic}^{*} \approx 109$ . With $1 stockout penalty: $𝑄_{𝜋}^{*} \approx 113$ .

Order 4 more units to buy down the goodwill cost. Doubling the penalty ( $𝜋 =$ $2) would push $𝑧^{*} \approx 0.84$ and $𝑄^{*} \approx 117$ .

The penalty acts like a thumb on the scale in favor of overstocking. In settings where stockouts are existential (medical supplies, contractual deadlines), $𝜋$ can dwarf $𝑃 - 𝐶$ and push the optimal $𝑄^{*}$ deep into the upper tail of demand.