The Mathematics of Newsvendor Optimization: When Demand Is Uncertain
The foundational model for inventory under uncertainty. Critical fractile, service level, and the cost of mismatch.
The Core Problem
You stock a product once. Demand is random. Order too much: leftover inventory, salvage value loss. Order too little: stockout, lost margin, customer defection.
This is the newsvendor problem. Named for the newsstand operator who must decide how many papers to stock before knowing demand.
The mathematics applies everywhere: fashion, electronics, perishables, seasonal goods.
The Model
Decision variable: Q (order quantity)
Random variable: D (demand), with cumulative distribution function F(D)
Cost parameters:
- c = unit cost
- p = selling price
- s = salvage value (leftover)
- g = shortage cost (stockout)
Underage cost (Cu): Cost of ordering one unit too few Cu = p - c + g (lost margin + goodwill loss)
Overage cost (Co): Cost of ordering one unit too many Co = c - s (cost minus salvage)
The Critical Fractile
Optimal service level:
Critical fractile = Cu / (Cu + Co)
This is the probability that demand will be less than or equal to optimal order quantity.
Interpretation: The optimal stockout probability is Co / (Cu + Co)
Numerical Example
Fashion item:
- Cost c = $20
- Price p = $50
- Salvage s = $10
- Shortage cost g = $5 (goodwill)
Cu = 50 - 20 + 5 = $35 Co = 20 - 10 = $10
Critical fractile = 35 / (35 + 10) = 35/45 = 0