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)

An Example Demand for a Weekly Magazine in the past 52 weeks (Computer Today)

Frequency Histogram

Continuous Distribution

Normal Distribution Parameters (mu, sigma^2):

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!

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

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

Increase order from k to k+1 if Prob(Demand < k) < 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)

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

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

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

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) =

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

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

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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