Newsvendor Model
Overview
The newsvendor model determines optimal order quantity for a single selling period with uncertain demand. Balances overage cost (Co = cost - salvage) against underage cost (Cu = price - cost). Optimal Q* satisfies: P(D ≤ Q*) = Cu / (Cu + Co). Known as the critical ratio solution.
When to Use
Trigger conditions:
- One-time or seasonal purchasing decisions (fashion, holiday goods, event tickets)
- Perishable products with no restocking opportunity
- Setting initial stocking levels before demand is observed
When NOT to use:
- For continuous replenishment with stable demand (use EOQ)
- When backorders are acceptable and demand carries over (multi-period models)
Algorithm
IRON LAW: The Critical Ratio Determines Optimal Service Level
Q* = F⁻¹(Cu / (Cu + Co)) where F⁻¹ is the inverse demand CDF.
If margin is high relative to cost (Cu >> Co), order MORE (high service level).
If margin is low relative to excess cost (Co >> Cu), order LESS (low service level).
The optimal solution almost NEVER equals expected demand.Phase 1: Input Validation
Define: unit cost (c), selling price (p), salvage value (v), demand distribution (mean μ, std σ). Compute: Cu = p - c, Co = c - v. Gate: p > c > v (profitable with positive overage cost), demand distribution estimated.
Phase 2: Core Algorithm
- Critical ratio: CR = Cu / (Cu + Co) = (p - c) / (p - v)
- If demand ~ Normal(μ, σ): Q* = μ + z(CR) × σ where z(CR) = inverse normal CDF at CR
- Expected profit = Cu × E[min(Q,D)] - Co × E[max(Q-D, 0)]
- Expected units sold = μ - σ × L(z) where L(z) is the standard loss function
Phase 3: Verification
Check: Q* > 0, CR between 0 and 1, Q* is above or below μ depending on whether CR > or < 0.5. Gate: Q* directionally correct relative to mean demand.
Phase 4: Output
Return optimal order quantity with profit analysis.
Output Format
{
"optimal_quantity": 130,
"critical_ratio": 0.71,
"expected_profit": 2800,
"expected_leftover": 15,
"expected_stockout_probability": 0.29,
"metadata": {"price": 50, "cost": 20, "salvage": 5, "demand_mean": 100, "demand_std": 30}
}Examples
Sample I/O
Input: p=$50, c=$20, v=$5, D~Normal(100, 30) Expected: Cu=30, Co=15, CR=30/45=0.667, z=0.43, Q*=100+0.43×30=113 units.
Edge Cases
| Input | Expected | Why |
|---|---|---|
| v = 0 (total loss) | Lower Q*, conservative | High overage cost pushes order down |
| p >> c (high margin) | Q* well above mean | Worth risking excess to avoid lost sales |
| σ = 0 (certain demand) | Q* = μ exactly | No uncertainty, order exactly demand |
Gotchas
- Distribution choice matters: Normal allows negative demand. For low-mean items, use Poisson or truncated normal. For high CV, use lognormal.
- Demand estimation: The hardest part is estimating μ and σ. Use historical data, expert judgment, or Bayesian updating from early sales signals.
- Risk aversion: The newsvendor model is risk-neutral. Risk-averse decision makers systematically under-order relative to Q*. Adjust for behavioral bias.
- Multi-product constraints: With a shared budget constraint across products, solve the constrained newsvendor (Lagrangian relaxation).
- Salvage value assumption: Assumes all excess can be salvaged at v. If disposal has a cost (v < 0), the model still works but Q* drops further.
Scripts
| Script | Description | Usage |
|---|---|---|
scripts/newsvendor.py | Compute newsvendor optimal quantity, expected profit, and fill rate | python scripts/newsvendor.py --help |
Run python scripts/newsvendor.py --verify to execute built-in sanity tests.
References
- For multi-product constrained newsvendor, see
references/constrained-newsvendor.md - For demand distribution fitting, see
references/demand-fitting.md