An Inventory Game. A Game

Transcription

1 An Inventory Game A Game

2 Betting on Uncertain Demand: Newsvendor Model Reference readings: 4 th /3 rd : Ch (up to Example 12-2 (exclusive) )

3 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 $4.0 and then sells at $8.0. At noon goes home for a nap and the unsold is returned for $1. If average daily demand is 50 and he buys just 50 copies daily, then is the average daily profit =50*4 =$200?

4 The basic story of the newsvendor problem is that you have to choose a stocking quantity before knowing what demand will be. To make matters worse, you only get to order inventory once. Among other things, this simple setting drives home that variability cuts into profits: A product that could be profitable if you knew exactly what to produce/buy/sell To meet demand can be unprofitable if demand is sufficiently uncertain.

5 Betting on Uncertain Demand You must take a firm bet (how much stock to order) before some random event occurs (demand) and then you learn that you either bet too much or too little More examples: Products for the Christmas season; winter coats, New- Year Flowers,

6 Bossini -- Winter Clothes Season: Dec. Jan./Feb. Purchase of key materials (fabrics, dyeing/printing, ) takes long times (upto 90 days) Into the selling season, it is too late!

7 Health secretary lays out H1N1 flu vaccine program By Joy Lu (HK Edition) :18 HONG KONG: Hong Kong has ordered 3 million doses of vaccines against the human swine flu virus, with the first 500,000 doses expected by mid-december, Secretary for Food and Health York Chow said yesterday. Unveiling Hong Kong's swine flu vaccination program at a press briefing, he said the government placed the HK$237 million order with the French pharmaceutical company Sanofi-Aventis. The unit price of the order is HK$79 per dose. The remaining 2.5 million doses of the vaccine will be delivered in January, Chow said June: 政府花掉二億三千萬元購買幾乎無人問津的人類豬型流感疫苗, 即將報廢 // 政府即將把購入價超過二億元的二百七十多萬劑人類豬型流感疫苗棄置堆填區

8 Japan has a surplus of 99 million shots of imported H1N1 Flu Vaccines, bought for billion, health ministry officials said Saturday. Canada: Millions could be wasted as H1N1 vaccine tossed

9 乏人問津美 H1N1 疫苗變垃圾 金額高達十六億美元的 H1N1 疫苗計畫, 是美國史上斥資最大的防疫計畫 ; 華府當時領軍趕工製造疫苗, 但製造過程卻發生問題導致延誤, 一直拖到第二波 H1N1 疫情高峰才出貨 ; 而在這段期間苦無疫苗可用的數百萬民眾也飽受焦慮沮喪之苦 豈料如今竟然傳出大批疫苗乏人問津並恐將因過期而被丟棄

10 Paying for H1N1 cancellations What happens when a newsvendor decision goes bad? It depends on what the contract says. That is what Novartis is now finding out (Novartis Seeking Vaccine Payments, Wall Street Journal, May 6, 2010). Novartis was one of the major pharmaceutical firms scrambling to produce flu vaccine. When the flu proved relatively mild, many countries canceled orders, leaving some ambiguity over who was owed what.

11 Paying for H1N1 cancellations About half of the 15 governments that ordered H1N1 vaccine from Novartis AG ended up canceling part of their orders, and some are still negotiating with the company over payment, The French, apparently, offered to pay only 16% of the contract price, terms that the Novartis ultimately accepted. The US meanwhile ordered nearly 230 million doses (not all from Novartis) and has used something like 91 million. The Department of Health and Human Services right now is in discussions with suppliers about the excess.

12 Paying for H1N1 cancellations This is partly a problem of Novartis own making. They are saddled with unused doses and cancellations because they were late in delivering the goods. There was a point in time in which everyone in America was willing to stand in line for hours for a flu shot. Unfortunately, that time came and went before Novartis and the other vaccine makers could produce the vaccine in volume.

13 Imagine you were selling newspapers/magazines/seasonal products How would you decide on How many copies to order for Today? Matching Supply with Demand: An Introduction to Operations Management, 2 nd Edition, by Gerard Cachon and Christian Terwiesch. McGraw-Hill, 2009.

14 Alternative Method of Estimating Distribution An Example Demand for a Weekly Magazine in the past 52 weeks (Computer Today)

15 Frequency Histogram 7 6 Weeks Actual Demand

16 Continuous Distribution 7 6 Frequenc Demand

17 How much should he order? Marginal analysis Suppose you decide to order 1 unit If the demand turns out to be 1 or more, then the (realized) profit =4*1 = $4 (more than 1 unit, opportunity profit is ignored); If there is no demand, you lose $3 Weighted average = $4*Prob(d 1) -3* Prob (d=0)=$4 Suppose you decide to add +1 (=total 2 units) If the demand turns out to be 2 or more, then you make a hold 1, earning $1 profit. If demand =0, loss = $6 Weighted average = $8*Prob (d 2) +1*Prob (d=1) - 6* Prob (d=0) = $8*99/100+1*1/100-6*0 ~ $8 profit of 4*2 = $8; but if it is 1, then you sell 1 and Suppose demand follows a uniform distribution between 1 and 100. What is a uniform distribution? Additional gain from 1 to 2: 8 or 8-4=4? Weighted aver. = expected (actual) value, why use it?

18 Newsvendor : Marginal Analysis Order one unit if Exp. Profit (1) > Exp. Profit (0) Stock 2 units (instead of 1 unit) if... If demand to be Stock 1 Stock 2 Stock 3 d = d = 1 1 1,+1 1, +2 d = 2 1, , +1 d = 3 1, -2 2, -1 3 d = 4 1, -3 2, -2 3, -1

19 Should we order three units, and so on? As we continue this process: Prob. that we sell the (additional) unit decreases Prob that we do not sell that unit increases Imagine we order the 99 th unit. The prob. of selling that unit is quite low (=2/100) -> the expected gain is nearly zero; the prob. of not selling that unit is very high (=98/100) -> the expected loss is nearly $3. Does it make sense to order the 99 th unit?

20 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

21 How to determine the Quantity? 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

22 Too much and too little costs C o = overage cost/ overstocking cost The cost of ordering one more unit than what you would have ordered had you known demand. In other words, suppose you had left over inventory (i.e., you over ordered). C o is the increase in profit you would have enjoyed had you ordered one fewer unit. C u = underage cost/understocking cost The cost of ordering one fewer unit than what you would have ordered had you known demand. In other words, suppose you had lost sales (i.e., you under ordered). C u is the increase in profit you would have enjoyed had you ordered one more unit. In the newspapers example: C 0 = 3, C u =

23 Let P(d<=k) = Prob (d <= k) keep order size at k order k+1 instead of k 1 more unsold 1 fewer lost sale Additional contribution 0 C o C u Order k+1 instead of k if C u *P (d >k) C o * Pr(d<k) > 0 or [1-P(d<k)]*(C u ) - P(d< k )*( C o ) > 0 Increase order from k to k+1 if Prob(Demand < k) < C u = 4/(3+4) = 0.57 C o + C u

24 How much should you order? Ask yourself subjective prob. /confidence, what is the chance that the demand is below/at 1, 2,.., k? Or from historical observation, P (d <= k) = That k+1 is the optimal Q* = k+1 =58! Weighted Profit For continuous demand, just k. k

25 Newsvendor Model- Demand Distribution Continuous Order Q such that Q Prob(Demand < Q) = C u Critical ratio r(q) C o + C u Continuous variable, r(q) = Prob (d <= Q), Q Interpretation of r(q)

26 If r* = 0.65, z = Q*= *20 If r* =0.45, z= Q* = *20 Normal Dist. F(z) 0 z If r* = 0.65, z = Q*=aver.+z*stdev If r* =0.45, z= Q* = aver. + z*stdev z

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

28 Normal Distribution Parameters (mu, sigma^2): µ D =11.73 σ s = 4.74

29 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? r(q) = c o c u = = F( Q * = = ) = = 0.77

30 f(x) Area=0.77 σ 4.74 µ * Q demand, d From the previous normal table,z=0.74 Thus optimal weekly order quantity Q * = =

31 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 r( Q ) = C u Cu + C 0

32 Interpretation of r(q) r( Q) = C u C u + C 0 r (Q): critical ratio The probability that the demand is less than or equal to y = p(0)+p(1)+ +p(q) For a newsvendor, if y=850, r(850) = 0.999, then the chance is extremely high to have some newspapers unsold at the day end

33 Critical Ratio Formula (Marginal Analysis) Item Gross Margin x (1 In-Stock Rate) = The Cost of an Unsold Item x In-Stock Rate r( Q) = Prob (Demand Q) = In - Stock Rate = Cu /[C0 + C u ] In-stock can be understood as the case when all realized demands are met. Imagine you have it 90%; about 10%: stockout,

34 Example: Selling parkas at LL Cost per parka = c = $45 Bean (Sect ) Sale price per parka = p = $100 Inventory holding (until season end) and transportation cost (to outlet store) per parka = $10 Discount price per parka (season end sales) = $50 Salvage value per parka = $50 -$10 = $40 = s Cost of overstocking = C o = $45 + $10 - $50 = c - s = $ 5 Marginal profit from selling parka = cost of understocking = C u = $100 - $45 = p - c = $55

35 LLBean: Expected Profit Expected demand Expected profit if order = i= 4 p i [ i( p c) (10 i)( c s)] + [1 P10 Expected profit if order k k = i= 4 p i C u [ i( p c) ( k 17 = j= 4 i)( c j C 0 p j 10 s)] + [1 P k ](10)( p c) C u ]( k)( p c) Why expected values?

36 Summary of Insights For each product, it is insufficient to have just a forecast of expected demand Important to keep track of the past forecasts Left-over or sold-out is not a good indicator for order decision makers 3 defining characteristics: about uncertain demand, costs/price, single order quantity Constructing a Demand Distribution, Source: Matching Supply and Demand, Cachon and Terwiesch, 2005, McGraw-Hill. Chapter 9.

37 LLBean : Demand distribution Demand Probability Sum(d(i)xp(i)) Cumulative Prob. d(i) p(i) P(i) = Pr( D < d(i) ) Price p Cost c 45.00

38 LLBean: Expected profit Demand Probability Sum(d(i)xp(i)) Cumulative Prob. Prob. demand greater Expected profit d(i) p(i) P(i) = Pr( D < d(i) ) 1 - P(i) = Pr( D > d(i) ) Price p Cost c Salvage s 40.00