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Newsvendor Problem

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.

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Newsvendor Problem

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  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. D1 D2 D3 … 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]

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

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