1. An Alternative Approach If you have a sufficient history & the demand is relatively stable over time, then use an empirical distribution In the case Sport Obermeyer (Simchi- Levi’s textbook)

2. An Example Demand for a Weekly Magazine in the past 52 weeks (Computer Today) 151991292247811 14116 9181001412 89544171814158 671215 19910916 811 1815171914 17 1312

3. Frequency Histogram

4. Continuous Distribution

5. Normal Distribution Parameters (mu, sigma^2):

6. The Newsboy Model: an Example Mr. Tan, a retiree, sells the local newspaper at a Bus terminal. At 6:00 am, he meets the news truck and buys # of the paper at $3.0 and then sells at $7.0. At noon he throws the unsold and goes home for a nap. If average daily demand is 50 and he buys just 50 copies daily, then is the average daily profit =50*4 =$200? NO!

7. The Intuition There is a tradeoff between ordering too much and ordering too little To balance these forces, it’s useful to think in terms of a cost for ordering too much and a cost for ordering too little –A cost here can be a loss of profit

8. Overstocking cost C 0  = loss incurred when a unit unsold at end of selling season Understocking cost C u  = profit margin lost due to lost sale (because no inventory on hand) In the example: C 0 = -3, C u = 4 Deriving the Formula: Critical Ratio

9. Increase order from k to k+1 if Prob(Demand < k) < 4 3 + 4 Order k+1 instead of k if 4*P(d  k+1)  P(d 0 or 4* [1-P(d 0 order k+1 keep order size at k instead of k 1 more unsold 1 fewer lost sale -3 = C o P(d  k+1) P(d<k)  C u Additional contribution =0.57 Let P(d<=k) = Prob (d <= k)

10. Increase order from k to k+1 if Prob(Demand < k) < C u C o + C u Order k+1 instead of k if C u *P (d >k)  C o *  Pr(d 0 or [1-P(d 0 order k+1 keep order size at k instead of k 1 more unsold 1 fewer lost sale 0 CuCu P(d<=k) P(d>k) CoCo Additional contribution

11. Newsvendor Model- Demand Distribution Continuous Order Q such that Prob(Demand < Q) = C u C o + C u Q Critical ratio r(Q)

12. F(z) 0 z0 z Normal Dist. If F* = 0.65, z = 0.385 Q*=aver.+z*stdev If F* =0.45, z= -0.125 Q* = aver. + z*stdev If F* = 0.65, z = 0.385 Q*=100 +0.385*20 If F* =0.45, z= -0.125 Q* = 100 – 0.125*20

13. Purchasing cost $0.25/per copy ， selling price =$ 0.75 。 If unsold after the week, each copy can be salvaged at $0.1 (to be returned to the publisher) 。 What’s the optimal order quantity Q? F(Q) =

14. From the previous normal table ， z=0.74 。 Thus optimal weekly order quantity Area=0.77 demand ， d f(x)

15. Who is the better manager? Consider two managers using the newsvendor model Manager A never has inventory left over Manager B usually has inventory left over Cu = 0.5 Co = 2 r (y ) = 0.5/2.5 = 0.2

16. Who is the better manager? Consider two managers using the newsvendor model Manager A never has inventory left over Manager B usually has inventory left over What if they both sell Valentine’s cards: C u =$2.00 C o =$0.20 r (Q) = 2/2.2 = 0.91 Cu = 0.5 Co = 2 r (y ) = 0.5/2.5 = 0.2

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