Ch. 5 存贮论 I ：确定性模型

1. Ch. 5 存贮论 I：确定性模型 5.1存贮系统 5.2 确定性时齐库存模型：基本模型(EOQ) 5.3确定性时齐库存模型：EOQ推广 5.4确定性非时齐库存模型

2. 5.1 Inventory Systems5.1.1 Introduction Inventory is the stock of any items used in organizations Items: Manufacturing: raw materials, work-in-process, component parts, finished products, .. .. .. Service: tangible goods, supplies, .. .. .. Various examples ……

3. 5.1.1 Introduction Common characteristics in the examples ? Demand: from customers Supply: from suppliers Match: the two sides Gerard Cachon, Christian Terwiesch

4. Purpose/Reasons of Inventory ? • Economic scale • Take advantage of economic purchase order size • 2) Meet uncertain in demand • Demand is not completely known, a safety stock must be maintained; Provide a safeguard for variation in raw material delivery time

5. Purpose/Reasons of Inventory (Cond') 3) Smooth production Maintain independence of operations, e.g., reduce the number of setups, smooth output of a workstation 4) Investment

6. What do we want to know in IS? Inventory system provides the organizational structure and operating policies for controlling stocks, and is responsible for ordering & receipt of goods: • Inventory policy: a set of commands that monitors levels of inventory, and then determine: • a) Timing the order placement • b) How much and from whom • Keeping track of what has been ordered • a) Has the supplier received the order • b) Has it been shipped • Are the dates correct • How many are in shipping

7. Long View of U.S. Inventory Perform. From 1977 to 2000, typical high-tech company doubled its inventory perform. from 2.5 to 5.0 turns typical company requires 50% less inventory Reasons • Companies steadily improved their inventory turns by • improving their supply chain business processes • improving the information systems to manage SC

8. What Actually Happened to the Inventories of American Companies Between 1981 and 2000? Hong Chen, Murray Z. Frank and Owen Q. Wu Abstract: This paper examines the inventories of publicly traded American: manufacturing companies between 1981 and 2000. The median of inventory holding periods were reduced from 96 days to 81 days. The average rate of inventory reduction is about 2% per year. The greatest reduction was found for work-in-process inventory which declined by about 6% per year. Finished goods inventories did not decline. Firms with abnormally high inventories have abnormally poor long term stock returns. Firms with slightly lower than average inventories have good stock returns, but firms with the lowest inventories have only ordinary returns.

9. 中国经济或已进入新一轮库存周期

10. 库存在人类历史中的作用 • 人类播下的第一粒种子，恐怕就是在与自己过不去。人们不去采集三个橡栗和两个黑刺李，而是收割自己种的小麦。这样，他就能储备粮食，养活更多的人，壮大自己的家庭，...，但，他这是作茧自缚，因为同时他需要管理田野、照料牲畜、开垦森林、磨制小斧挖掉树桩、栽种、种耕、收割、... • 总之是劳动。 • 人类推广了劳动，积蓄和创造了财富、剩余资源。但他同时给自己造就了一个金字塔社会，增加了强制性。新石器时代农耕和城居革命所代表的希望有一部分转而损害了人类自己。人类变成了他所创造的东西的奴隶。他既是胜者，同时也是受害者。人类走出了伊甸园，开始衰落、错误；或者说开始进步、文明，向着越来越舒适和幸福的生活迈进。

11. Purpose of this chapter • To present standard inventory models to help manager • keep the cost down while still • meet production / service requirements

12. Supply Inventory Demand …… items Inform. Fig 1.1 An inventory system 5.1.2 Components of IS

13. I. Demand • Demand: quantities needed for the specific items • Description: • constant demand rate • varying demand rate • Poisson process • periodic & stochastic demand • (complete information, partial information)

14. Multi-item Case • Independent demand • Dependent demand Examples Demands for cars: is independent Demands for wheels: depend on those for cars In a supermarket Demands for beers & diaper: are dependent Substituted, Complementary goods

15. II. Supply Supply: quantities to satisfy demands • Supply Ways: • Order: come together • Produce: according to a production rate • Other Factors: • Economics scale • Finite capacity • Lead-time

16. III. Match • Two Basic Ways: • Backorder • Lost-sale

17. IV. Inventory Costs 1) Setup (or production change) costs:including setup time. Reducing setup costs (time) is the goal of JIT 2) Ordering costs:counting items and calculating order quantities, maintaining the system to track orders 3) Holding (or carrying) costs:costs for storage facilities, handling, insurance, pilferage, breakage, obsolescence, depreciation, taxes, opportunity cost of capital 4) Shortage costs Trade-off between carrying stock and the costs from stock Order more: high H, low SS; Order less: low H, high SS

18. V. Products' Types ? 1) Perishable: foods 2) Durable: TV sets 3) Newsvendor: news paper, air ticket, seats 4) Re-Order: TV sets, foods 5) Non Re-Order: House 6) Multiple Items: generations, substitution, complementary

19. VI. Inspection Ways Continuously inspection Periodically inspection

20. VII. Inventory Policies Inventory Policies: To whom, when, and how many to order • Typical Inventory Policies: • (T, Q) cycling policy: supplying Q items per time T • (s, S) policy: supplying items to lift the inventory to S whenever the inventory is down below s

21. Idle state Waiting for demand Demand occurs Units withdrawn from inventory or backordered Compute I-position=on-hand + on-order - backorder No Is I-position ≤ Reorder point? Yes Issue an order for exactly Q units Q-Model Fixed-order quantity models (economic order quantity, EOQ, Q-model): order Q units when the inventory position drops to some R

22. Idle state Waiting for demand Demand occurs Units withdrawn from inventory or backordered No Has review time arrived? Yes Compute I-position =on-hand + on-order -- backorder Compute order quantity to bring inventory up to required level Issue an order for the number of units needed P-Model Fixed-time period models (periodic system, periodic review system, fixed-order interval system, P-model)

23. Comparison

24. VIII. Objective Minimizing the total costs / average cost per time, or Maximizing the total revenue / discouted total revenue in a given time horizon (finite or infinite) To trade-off the holding costs between the setup costs and the shortage costs

25. 5.1.3 Classification of IS Inspection way Continuously, Periodically Including stochastic factors? Deterministic IS, Stochastic IS Parameters varies? Stationary IS, Nonstationary IS Influence among different periods? Single-period IS, Multi-period IS

26. 5.2 EOQ5.2.1 Fixed-Order Quantity Models • Model / Assumptions: • Demand per unit time is constant • Supply is instantaneously and equal inter-epochs • Shortage is not allowed • All costs are constant • Comments: • The model is the simplest one (all are certainty) • unrealistic, but approximating Benchmark model: basic & starting point

27. Notations D=demand rate c=wholesale price K=setup cost ch=holding cost per unit per time

28. Inventory Q T 2T time Policy (Q-Model): Order Q units EVERY T times Figure.Inventory dynamics in Q-model

29. Average cost per unit time is The problem is to solve Formulae

30. The optimal solution The optimal inventory strategy: order quantity Solution Interpretation

31. Example Annual demand rate (D)=1,000 cost per unit (c)=$12.50 setup cost (K) = $5 per order holding cost per unit (cn)=$1.25 per unit per year

32. D+ΔD How does Q* changes? D Sensitivity Analysis What would happen when some parameter changes? • Does the optimal order quantity change when some parameter changes? • In which area some parameter changes, the optimal order quantity remains constant?

33. Proposition 2.1 : • EOQ is sensitive in smaller D (or K, ch) • EOQ is robust in larger D (or K, ch)

34. 5.2.2 General EOQ Model • Model / Assumptions here • Demand per unit time is constant • Supply continuously, uniformly, & equal periodically • Shortage is allowed & backordered • All costs are constant New notations P=supply rate cs=shortage cost per unit per time Assumption: P>D

35. T 0 st2uTT+ t22T t Inventory dynamics / Policy M m Policy (s, T ): length of a period & when to begin production or (t2 , T ): length of a period & length of shortage time

36. Average cost per unit time for (t2 , T ) is The problem is to solve Formulae

37. Solution The optimal solution

38. The optimal inventory strategy • begins production after shortage duration • with length of s* • length of a production cycle is t* The optimal inventory strategy: 其它表述？

39. The optimal demand quantity Q*, the maximal shortage quantity B*, the minimal costs C* in one period

40. Equilibrium Relationship 1 : length of a period : length of shortage time in one period : length of inventory time in one period 在最优策略下，一个周期中存储时间与缺货时间之比等于单位缺货费用与单位存储费用之比。 或：单位货物的缺货费=单位货物的总存贮费

41. : total cost in one period : total holding cost in one period : total shortage cost in one period 在最优策略下，一个周期中的总存储费用与总缺货费用之间达到一种平衡；存储与缺货费用的总和与固定订购费相等(另一种平衡)。 Equilibrium Relationship 2

42. Assumptions的合理性? • Model / Assumptions here • Demand is constant and uniform • Supply continuously, uniformly, & equal periodically • Shortage is allowed & backordered • All costs are constant Not necessary as an Assumption

43. A B C 原补充点 新补充点 Assumptions的合理性? 引理2.1在库存量为正时补充库存的策略一定不是最优的。

44. 定理2.1最优策略下，各补充周期完全相同，即周期长度相同，缺货时间也相同。定理2.1最优策略下，各补充周期完全相同，即周期长度相同，缺货时间也相同。

45. 5.2.3 Special CasesSpecial Case I 瞬时补充/不允许缺货(经典EOQ):

46. Special Case II 逐渐补充/不允许缺货(经典EOQ):

47. Special Case III 瞬时补充/允许缺货(经典EOQ):

48. 5.2.4 Sensitive Analysis 结论. 最优订购批量Q*相对于需求率D具有鲁棒性: Q*与D同向变化(只要D<P满足)，即需求率上升，最优订购批量也会上升，反之亦然. 通常我们有P<2(P-D)，因此， Q*的变化比率小于D的变化比率

49. Q*对于各种费用参数K, cs, ch来说也具有鲁棒性。 • 其他变量t*,B*,Q*, T*相对于各参数都具有鲁棒性。 结论：应用EOQ公式时，可以比较放心

50. 5.3 确定性时齐库存模型II5.3.1 数量折扣 数量折扣：在订购量超过某一特定值时，在价格上可以享受批发价优待 • 常见形式： • 全体折扣（All-Units Discounts），即一旦订购量达到批发量，所有商品都享受同一折扣价 • 增量折扣（Incremental Discounts），只有超过批发量的商品才享受价格折扣