Newsvendor Problem

1
Newsvendor Problem

• must decide how many newspapers to buy before you
  know the days 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/(bc)
• 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 days 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) EY(D,q) ?k P(Dk) Y(k,q)

7
Benefit of Ordering 1 More

• R(q1) - R(q)
• P(Dq1) b (extra newspaper sold)
• -
• P(Dq) c (extra newspaper not sold)
• (1-P(Dq)) b - P(Dq) c
• b - P(Dq) (bc)
• maximum when R(q1)-R(q) 0 or
• P(Dq) b/(bc)

8
Analytical Solution

• P(D q) b/(bc)
• 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-ED

12
Base-Stock Model

• 95 service level
• demand distribution
• order up to quantity q
• P(D q) 95

13
Base Stock Model

• Ex. DPoisson(?)
• qED 2.35 vED
• Rule of thumb
• q ED constant vED
• constant depends on service level and distribution

14
Choosing a Service Level

• Inventory holding cost (h per unit) vs.
• contribution (c per unit)
• -gt Newsvendor problem

15
Order Quantity Model

• Continuous review (instead of periodic)
• Ordering costs vs.
• Inventory costs
• Q When to reorder?