Newsvendor Problem must decide how many newspapers to

Newsvendor Problem • must decide how many newspapers to buy before you know the day’s demand • q = #of newspapers to buy • b = contribution per newspaper sold • c = loss per unsold newspaper • random variable D demand

Analytical Solution • P(D ≤ q*) = b/(b+c) • round up if q* integer

Previously • Optimization • Probability Review – pdf, cdf, E, Var – Poisson, Geometric, Normal, Binomial, …

Agenda • Hwk due date postponed • Projects • Inventory…

Newsvendor Problem • must decide how many newspapers to buy before you know the day’s demand • q = #of newspapers to buy • b = contribution per newspaper sold • c = loss per unsold newspaper • random variable D demand • R(q) = expected profit when ordering q newspapers

Last Time… • spreadsheet approach – calculate profit Y(demand k, q) for all pairs (k, q) – calculate P(Demand = k) – calculate R(q) = E[Y(D, q)] = ∑k P(D=k) Y(k, q)

Benefit of Ordering 1 More R(q+1) - R(q) = P(D≥q+1) b (extra newspaper sold) P(D≤q) c (extra newspaper not sold) = (1 -P(D≤q)) b - P(D≤q) c = b - P(D≤q) (b+c) maximum when R(q+1)-R(q) = 0 or P(D≤q) = b/(b+c)

Analytical Solution • P(D ≤ q*) = b/(b+c) • round up if q* integer

Newsvendor Model • • Single-period model Uncertain demand Lost-sales (no backordering) Perishable supply Q: How much supply to have?

Base Stock Model • • Multi-period model Uncertain Demand Lost-sales model (no backordering) Inventory (nonperishable supply) Q: How much supply to have?

Base Stock Model D 1 D 2 D 3 … • D distribution of demand in each period is iid (independent, same distribution) • Inventory replenished at beginning of period • Decision: level q to which inventory replenished “order-up-to quantity” • safety-stock: q-E[D]

Base-Stock Model • 95% service level • demand distribution • order up to quantity q* P(D ≤ q*) = 95%

Base Stock Model • Ex. D~Poisson( ) – q=E[D] + 2. 35 √E[D] • Rule of thumb: – q= E[D] + constant √E[D] – constant depends on service level and distribution

Choosing a Service Level • Inventory holding cost (h per unit) vs. • contribution (c per unit) -> Newsvendor problem

Order Quantity Model • Continuous review (instead of periodic) • Ordering costs vs. • Inventory costs inventory Q: When to reorder? reorder times time