Single Period Inventory Model The Newsvendor Model

1. Single Period Inventory ModelThe Newsvendor Model Yang Jinglei

2. Single Period Ordering • Seasonal items • Perishable goods • News print • Fashion items • Some high tech products • Risky investments

3. Selling Magazines • Weekly demand: 􀂄 Total: 4023 magazines 􀂄 Average: 77.4 Mag/week 􀂄 Min: 51; max: 113 Mag/week

4. Detailed Histogram Frequency (Wks/Yr) Cumm Freq. (Wks/Yr) Demand (Mag/Wk) Average=77.4 Mag/wk

5. Histogram Cummulative Frequency Cumm Freq. (Wks/Yr) Frequency (Wks/Yr) More Demand (Mag/Wk)

6. The Ordering Decision(Spreadsheet) • Assume: each magazine sells for: $15 • Cost of each magazine: $8 Order d/wk Prob Exp.Profit: Newsboy Framework – Panel 1: “basic Scenario”

7. “Too much” and “too little” costs • Co = overage cost 边际成本的概念 • The cost of ordering one more unit than what you would have ordered if had you known demand. • In other words, suppose you had left over inventory (i.e., you over ordered). Co is the increase in profit you would have enjoyed had you ordered one fewer unit. Cu = underage cost • The cost of ordering one fewer unit than what you would have ordered if had you known demand. • In other words, suppose you had lost sales (i.e., you under ordered). Cu is the increase in profit you would have enjoyed had you ordered one more unit.

8. Balancing the risk and benefit of ordering a unit • Ordering one more unit increases the chance of overage … • Expected loss on the Qth unit = Co x F(Q) • F(Q) = Distribution function of demand = Prob{Demand <= Q) 订多的部分 • … but the benefit/gain of ordering one more unit is the reduction in the chance of underage: • Expected gain on the Qth unit • CuProb{Demand >= Q) 比需求少的部分

9. As more units are ordered, the expected benefit from ordering one unit decreases while the expected loss of ordering one more unit increases.

10. Optimal Order (Analytical) 􀂆 • The optimal order is Q* • At Q* the probability of selling one more magazine is the probability that demand is greater than Q* • The expected profit from ordering the (Q*+1)st magazine is: • 􀂄 If demand is high and we sell it: • 􀂆 (REV-COST) x Pr( Demand is higher than Q*) • 􀂄 If demand is low and we are stuck: • 􀂆 (-COST) x Pr( Demand is lower or equal to Q*) • The optimum is where the total expected profit from ordering one more magazine is zero: • (REV-COST) x Pr( Demand > Q*) – COST x Pr( Demand • ≤ Q*) >= 0

11. 解释 • 边际分析 • 再多订一个边际成本会增加，而不是减少，即 • E(Q+1)-E(Q)>=0 • 凸函数，即最优解是使得不等式成立的最小的Q值 • 再多订一个单位时 • 如果D<=Q, 则订购Q+1比订购Q多Co； • 如果D>Q+1, 则订购Q+1比订购Q少Cu；

12. Optimal Order The “critical ratio”: Cummulative Frequency Frequency (Wks/Yr More Demand (Mag/week)

13. The Profit Function • Revenue from sold items • Revenue or costs associated with unsold items. These may include revenue from salvage or cost associated with disposal. • Costs associated with not meeting customers’ demand. The lost sales cost can include lost of good will and actual penalties for low service. • The cost of buying the merchandise in the first place.

14. The Profit Function

15. The Profit Function – Simple Case Optimal Order: and:

16. Level of Service Cycle Service – The probability that there will be a stock-out during a cycle Cycle Service = F(Q) Fill Rate - The probability that a specific customer will encounter a stock-out

17. Level of Service REV=$15 COST=$7 Service Level Cycle Service Fill Rate Order

18. Normal Distribution of Demand Expected Profit Order Size

19. 引言 • 张五常 • 买桔者言 • 香港的年宵市场 • 香港年宵市场，一般指由食物环境卫生署（食环署）所举办的年宵市场。年宵市场的举行日期为每年农历十二月廿四日，至正月初一日清晨结束，全部均为户外举行。 • 每年十一月，食环署都会把翌年的年宵市场档位以价高者得的方式公开竞投。

20. What’s your comments？ • 价格歧视 • 价格策略 • 信息不无安全 • 买卖过程 • 怎样取得最大利润 • 关注平均价格？是不是？？ • 单周期模型告诉我们什么？

21. 库存系统 库存策略：定时盘点（每周末），当库存低于再订货点（21）时，则发出订单订货，订货量为（库存水平（25）－盘点量） 输入可控变量(库存成本、订货成本、缺货成本) 输入不可控变量(每周原材料需求量) 计算不同库存水平下的总成本 (库存成本+订货成本+缺货成本) 统计分析 (各种成本的均值、标准差和占总成本的比例) 不同库存水平下的模拟