1. 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

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

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

4. Agenda Hwk due date postponed Projects Inventory…

5. 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

6. 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)

7. 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)

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

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

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

11. Base Stock Model 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] D1D1 D2D2 D3D3 …

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

13. 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

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

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