416. Shortage Penalty
416.1. Newsvendor with explicit shortage penalty
Relax one dimension from basic newsvendor: excess demand costs more than just lost margin. In the basic model, 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 per unit of unmet demand, on top of lost margin.
416.1.1. Setup
Same as basic newsvendor, plus:
- = stockout penalty per unmet unit (dollars per unsold-due-to-stockout)
Underage cost now has two components:
The form of the answer is unchanged — only the value of is larger.
416.1.2. Critical ratio
A larger pushes CR upward → larger → larger . More penalty for stocking out → order more.
416.1.3. Bounds on CR
- : collapses to basic newsvendor.
- : (almost certain to satisfy demand). grows without bound.
- very large but finite: saturates at the upper tail of demand.
416.1.4. Decision rule (continuous demand)
For normal demand :
Example
Given (same newspaper baseline + a goodwill penalty per missed sale):
- Sale price: = $3 / unit
- Purchase cost: = $1 / unit
- Salvage: = $0
- Stockout penalty: = $1 / unit (lost goodwill)
- Demand:
Step 1 — underage and overage
Step 2 — critical ratio
Compared to basic newsvendor (), the penalty pushes CR from 67% up to 75%.
Step 3 — quantile and order quantity
Step 4 — compare to basic newsvendor
Without penalty: . With $1 stockout penalty: .
Order 4 more units to buy down the goodwill cost. Doubling the penalty ( $2) would push and .
The penalty acts like a thumb on the scale in favor of overstocking. In settings where stockouts are existential (medical supplies, contractual deadlines), can dwarf and push the optimal deep into the upper tail of demand.