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

1. Critical ratio: CR = Cu / (Cu + Co) = (p - c) / (p - v)
2. If demand ~ Normal(μ, σ): Q* = μ + z(CR) × σ where z(CR) = inverse normal CDF at CR
3. Expected profit = Cu × E[min(Q,D)] - Co × E[max(Q-D, 0)]
4. 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