Inventory Management and Risk Pooling. Xiaohong Pang Automation Department Shanghai Jiaotong University

Transcription

1 Inventory Management and Risk Pooling Xiaohong Pang Automation Department Shanghai Jiaotong University

2 Key Insights from this Model The optimal order quantity is not necessarily equal to average forecast demand. The optimal quantity depends on the relationship between marginal profit and marginal cost. As order quantity increases, average profit first increases, and then decreases. As production quantity increases, risk increases. In other words, the probability of large gains and of large losses increases.

3 Topics Today

4 A Multi-Period Inventory Model Often, there are multiple reorder opportunities. Consider a central distribution facility which orders from a manufacturer and delivers to retailers. The distributor periodically places orders to replenish its inventory.

5 Why the DC holds inventory? The DC holds inventory in order to: Satisfy demand during lead time Protect against demand uncertainty Balance fixed costs and holding costs

6 Normal distribution: 正 态 分 布 Reminder: The Normal Distribution Standard Deviation = 10 Standard Deviation = 5 Average =

7 Understanding Inventory Tasks for managing inventory Forecasting demand Calculating order quantity Determining when and how much to order Continuous review policy Periodic review policy

8 Continuous Review Policy: Assumptions Daily demand is random, follows a normally distribution Fixed order cost plus a cost proportional to amount ordered Inventory holding cost is charged per item per unit time If an order arrives and there is no inventory on hand, the order is lost The distributor has a required service level This is expressed as the likelihood that the distributor will not stock out during lead time. Intuitively, how will this affect our policy?

9 Notation AVG = average daily demand STD = standard deviation of daily demand LT = replenishment lead time in days h = holding cost of one unit for one day K = fixed cost SL = service level (for example, 95%). This implies that the probability of stocking out is 100%-SL (for example, 5%) Also, the Inventory Position at any time is the actual inventory plus items already ordered, but not yet delivered, minus items backordered.

10 Continuous Review Policy It is known as a (Q, R) policy Inventory is reviewed continuously Whenever inventory level falls to a reorder point, R, we place an order for Q units.

11 The ( Q, R ) Policy Inventory Level Q+R R Lead Time Inventory Position 0 Time

12 The ( Q, R ) Policy (Q, R) Policy: Whenever the inventory position drops below a certain level, R, we order for units Q to raise the inventory position. The reorder point, R, is a function of: The Lead Time Average demand Demand variability Service level

13 Analysis of (Q, R) policy The reorder point (R) has two components: The first accounts for average demand during lead time: L AVG The second accounts for deviations from average, so it is called safety stock: z STD where z is chosen from statistical tables to ensure that the probability of stocking out during lead time is less than100%-sl. L

14 Analysis of (Q, R) policy Service level z value is determined by the statistical table, ensuring that the probability of stocking out during lead time is less than100%-sl. For example, If SL is 98%, z should be 2.05 If SL is 90%, z is1.29

15 Analysis of (Q, R) policy What about the order quantity Q? Since it is necessary to make a tradeoff between fixed order cost and inventory holding cost, we use EOQ model to calculate Q: Q K AVG = 2 h The total order-up-to level is: Q+R The average inventory level is: Q + 2 z STD L

16 Example 2-8 Take 2 min. to read example 2-8, consider the following problems. Why is average weekly demand, rather than average monthly demand, used to calculate reorder level?

17 Example 2-8 The distributor has historically observed weekly demand of: AVG = 44.6 STD = 32.1 Replenishment lead time is 2 weeks, and desired service level SL = 97% Average demand during lead time is: = 89.2 Safety Stock is: = 86.2 Reorder point is thus 176, or about 3.9 weeks of supply at warehouse and in the pipeline.

18 Example 2-8, Cont. Weekly inventory holding cost: 0.87 Therefore, Q=679 Order-up-to level thus equals: Reorder Point + Q = = 855 The average inventory level is: 679/2+86.2=426 The distributor needs to keep about 10 (=426/44.58)weeks of inventory at the warehouse and in the pipeline.

19 Variable Lead Time In many cases, the assumption that the lead time to the warehouse is fixed and known in advance does not hold. In many practical situations the lead time must be assumed to be normally distributed with Average lead time, AVGL Standard deviation, STDL

20 Variable Lead Time The reorder point, R, is calculated as follows: s R = AVG AVGL + z AVGL STD + AVG STDL Where AVG AVGL represents average demand during lead time, AVGL STD + AVG STDL is the standard deviation of demand during lead time. Thus, the amount of safety stock that has to be kept is equal to: z AVGL STD + AVG STDL The order-up-to-level is the sum of reorder level and Q S = Q+ AVG AVGL + z AVGL STD + AVG STDL

21 Periodic Review Policy Suppose the distributor places orders every month. What policy should the distributor use? What about the fixed cost?

22 Periodic Review Policy r r Base-stock stock Level L L L Inventory Position Inventory Level 0 Time

23 How to determine base-stock level? Each review echelon, inventory position is raised to the base-stock stock level. The base-stock level includes two components: Average demand during r+l days (the time until the next order arrives): (r+l)*avg Safety stock during that time: z*std* r+l Where z is the safety factor

24 How to calculate average inventory level? r The maximum inventory level is achieved immediately after receiving an order, it is r AVG + z STD r + The minimum inventory level is achieved just before receiving an order, it is z STD r + L The average inventory level is the average of these two values, it equals: AVG 2 + z STD r + L L

25 Example 2-9 Take 2 min. to read example 2-9.

26 What is Risk Pooling? Risk pooling suggests: Demand variability is reduced if one aggregates demand across more locations. It is true, as we aggregate demand across different locations, it becomes more likely that: High demand from one customer will be offset by low demand from another. This reduction in variability allows a decrease in safety stock, and thus reduce average inventory. Aggregate: 总 计, 聚 集

27 Standard Deviation and Coefficient of Variation Standard Deviation Measure how much demand tends to vary around the average Coefficient of Variation( 变 异 系 数 ) Ratio of standard deviation to average demand Standard deviation Coefficien t of variation = Average demand The standard deviation measures the absolute variability of customer demand, while the coefficient of variation measures variability relative to average demand

28 CASE: Risk Pooling Take 2 min. to read case Risk Pooling on page 41, consider the following questions. Comparing centralized distribution systems with decentralized distribution systems, answer the questions: With the same service level, which system will require more inventory? With the same total inventory level, which system will have better service?

29 CASE: Risk Pooling Consider these two systems: Supplier Warehouse One Warehouse Two Market One Market Two Supplier Warehouse Market One Market Two

30 CASE: Risk Pooling Compare the two systems: two products maintain 97% service level $60 order cost $.27 weekly holding cost $1.05 transportation cost per unit in decentralized system, $1.10 in centralized system 1 week lead time

31 CASE: Risk Pooling Historical Data for Product A and B Week Prod A, Market 1 Prod A, Market 2 Prod B, Market 1 Product B, Market

32 CASE: Risk Pooling Summary of Historical Data Warehouse Product AVG STD CV Market 1 A Market 2 A Market 1 B Market 2 B

33 CASE: Risk Pooling Inventory Levels Warehouse Product AVG STD CV s S Avg. Inven. Market 1 A % Dec. Market 2 A Market 1 B Market 2 B Cent. A % Cent B %

34 Think Pair Share Take 2 min. to think about the questions below, then discuss with your neighbors. With the same service level, which system will require more inventory? With the same total inventory level, which system will have better service?

35 Risk Pooling: Important Observations Centralizing inventory control reduces both safety stock and average inventory level for the same service level. This works best for products with High coefficient of variation, which increases required safety stock. Negatively correlated demand. Why? Are there other kinds of risk pooling?

36 Think Pair Share Take 2 min. to think about some examples that take advantage of risk pooling concept, then discuss with your neighbors. Types of Risk Pooling Risk Pooling Across Markets Risk Pooling Across Products Risk Pooling Across Time Give a specific example of risk pooling (a) across locations, (b) across time, and (c) across products.

37 To Centralize? Or not to Centralize? What is the effect on: Safety stock? Service level? Overhead costs? Lead time? Transportation Costs?

38 Centralized vs Decentralized System Safety stock Clearly, safety stock decreases as a firm moves from a decentralized to a centralized system. Service level When the centralized and decentralized systems have the same total safety stock, the service level provided by the centralized system is higher. Lead time Since the warehouses are much closer to the customers in a decentralized system, response time is much lower.

39 Centralized vs Decentralized System Transportation Costs In a decentralized system, outbound transportation costs decrease because the warehouses are much closer to the customers, inbound transportation costs increase. Overhead costs These costs are much greater in a decentralized system because there are fewer economics of scale.

40 Managing Inventory in the SC Supplier Warehouse Retailers The warehouse echelon inventory( 级 库 存 )

41 Centralized Distribution Systems Echelon inventory: In a distribution system, each stage or level is called an echelon. Thus, the echelon inventory at any stage or level is equal to the inventory on hand, plus all downstream inventory. Supplier Warehouse Question: Retailers How much inventory should management keep at each location?

42 Centralized Distribution Systems A good strategy: The retailer uses (Q,R) policy, raises inventory to level S r each period. The supplier raises the sum of inventory in the retailer and supplier warehouses and in transit to S s If there is not enough inventory in the warehouse to meet all demands from retailers, it is allocated so that the service level at each of the retailers will be equal. Supplier Warehouse Retailers

43 Example 2-11 Take 2 min. to read example 2-11 (TV distributor), consider the question below. How does echelon inventory help the distributor better make its decisions?