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## Single Period Inventory Model The Newsvendor Model

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**Single Period Inventory ModelThe Newsvendor Model**Yang Jinglei**Single Period Ordering**• Seasonal items • Perishable goods • News print • Fashion items • Some high tech products • Risky investments**Selling Magazines**• Weekly demand: Total: 4023 magazines Average: 77.4 Mag/week Min: 51; max: 113 Mag/week**Detailed Histogram**Frequency (Wks/Yr) Cumm Freq. (Wks/Yr) Demand (Mag/Wk) Average=77.4 Mag/wk**Histogram**Cummulative Frequency Cumm Freq. (Wks/Yr) Frequency (Wks/Yr) More Demand (Mag/Wk)**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”**“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.**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**As more units are ordered, the expected benefit from**ordering one unit decreases while the expected loss of ordering one more unit increases.**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**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；**Optimal Order**The “critical ratio”: Cummulative Frequency Frequency (Wks/Yr More Demand (Mag/week)**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.**The Profit Function – Simple Case**Optimal Order: and:**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**Level of Service**REV=$15 COST=$7 Service Level Cycle Service Fill Rate Order**Normal Distribution of Demand**Expected Profit Order Size**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。**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？**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