運(yùn)籌學(xué)與供應(yīng)鏈管理-第4講VIP

1、第講第講庫(kù)存管理（庫(kù)存管理（II）Multi-Echelon Inventory in Supply ChainOutside supplier(s)Central warehouseBranch warehouseRetail outletsCustomersBranch warehouseTwo Stage Echelon InventorySequential stocking points with level demandTwo-stageprocessQWWarehouse inventory levelTimeTimeQRRetailerinventory levelActual

2、 physical inventory level at the particular locationEchelon inventory of the warehouse itemTwo Stage Echelon InventoryTwo-stage process:A little reflection shows that at least for the case of deterministic demand it never would make sense to have be anything but an integer multiple of . Therefore, w

3、e can think of two alternative decision variables and where(4.1)WQRQnRQ. 3, 2, 1, nnQQRWTwo Stage Echelon InventoryTwo-stage process:The first stage costThe second stage costThe total costrvvQQrvQDArvQQDACCQQWRWRWRRWWWWWRRW222,TRCrvQQDACRRRRR2rvQrvQQDACWRWWWWW22Two Stage Echelon InventoryTwo-stage p

4、rocess:The warehouse echelon inventory is valued atwhile the retailer echelon inventory is valued at only WWvvWRRvvvTwo Stage Echelon InventoryTwo-stage process:The total relevant (setup plus carrying) costs per unit time are given by = average value of the warehouse echelon inventory, in units= ave

5、rage value of the retailer echelon inventory, in unitsrvIQDArvIQDA,QQRRRRWWWWRWTRCWIRITwo Stage Echelon InventoryTwo-stage process:Substituting from equation (4.1) and noting that the echelon stocks follow sawtooth patterns,222TRCRWRWRRRRRRWRRWRWvnvrQnAAQDrvQQDArvQnnQDA,QQTwo Stage Echelon Inventory

6、Select (an integer) and in order to minimizePartial derivation of TRC nRQ2,TRCRWRWRRRvnvrQnAAQDQn02TRC2RWWRRRvnvrnAAQDQ rvnvDnAAnQRWWRR 2*Two Stage Echelon InventorySubstitute the result into the cost equationWe recognize that the n that minimizes the simpler expression rvnvDnAAnRWWR 2TRC* RWWRvnvnA

7、AnFTwo Stage Echelon InventoryA convenient way is to first setwhich givesThis solves for 0nFn02WWRWRWvnAAnAvnv*WRRWvAvAn Two Stage Echelon InventoryAscertain and where and are the two integers surrounding theWhichever gives the lower value of F is the appropriate n to use (because the F function is

8、convex in n). 1nF2nF1n2n*nTwo Stage Echelon InventoryTwo-stage process:Step 1ComputeStep 2Ascertain the two integer values, and , that surround .*WRRWvAvAn 1n2n*nTwo Stage Echelon InventoryTwo-stage process:Step 3 111RWWRvvnnAAnF222RWWRvvnnAAnF 121 use , IfnnnFnF 221 use , IfnnnFnFTwo Stage Echelon

9、InventoryTwo-stage process:Step 4Step 5rvnvDnAAQRWWRR2RWnQQTwo Stage Echelon InventoryExample 1:Let us consider a particular liquid product that a firm buys in bulk, then breaks down and repackages. So in this case, the warehouse corresponds to the inventory prior to the repackaging operation, and t

10、he retailer corresponds to the inventory after the repackaging operation.The demand for this item can be assumed to be essentially deterministic and level at a rate of 1000 liters per year.Two Stage Echelon InventoryExample 1:The unit value of the bulk material or is $1/liter, while the value added

11、by the transforming (break and package) operation is $4/liter.The fixed component of the purchase charge ( ) is $10, while the setup cost for the break and repackage operation ( ) is $15.Finally, the estimated carrying charge is 0.24$/$/yr.WARAWvWvWRRvvvTwo Stage Echelon InventoryExample 1:Step 1:St

12、ep 2:11n63. 1115410*n22nTwo Stage Echelon InventoryExample 1:Step 3:that is, Thus, use n = 2. 12541110151F 120412210152F 21FFTwo Stage Echelon InventoryExample:Step 4:Step 5:liters 16724. 04121000210152RQliters 3341672WQTwo Stage Echelon InventoryExample 1:In other words, we purchase 334 liters at a

13、 time; one-half of these or 167 liters are immediately broken and repackaged.When these 167 (finished) liters are depleted, a second break and repackage run of 167 liters is made.When these are depleted, we start a new cycle by again purchasing 334 liters of raw material.Inventory Control with Uncer

14、tain DemandThe demand can be decomposed into two parts, where = Deterministic component of demandand = Random component of demand.RanDetDDDDetDRanDInventory Control with Uncertain DemandThere are a number of circumstances under which it would be appropriate to treat as being deterministic even thoug

15、h is not zero. Some of these are: When the variance of the random component, is small relative to the magnitude of . When the predictable variation is more important than the random variation. When the problem structure is too complex to include an explicit representation of randomness in the model.

16、DRanDRanDDInventory Control with Uncertain DemandHowever, for many items, the random component of the demand is too significant to ignore. As long as the expected demand per unit time is relatively constant and the problem structure not too complex, explicit treatment of demand uncertainty is desira

17、ble.Inventory Control with Uncertain DemandExample 2:A newsstand purchases a number of copies of The Computer Journal. The observed demands during each of the last 52 weeks were:15 19 9 12 9 22 4 7 8 11 14 11 6 11 9 18 10 0 14 12 8 9 5 4 4 17 18 14 15 8 6 7 12 15 15 19 9 10 9 16 8 11 11 18 15 17 19

18、14 14 17 13 12 Inventory Control with Uncertain DemandExample 2:01234560246810121416182022Inventory Control with Uncertain DemandExample 2:Estimate the probability that the number of copies of the Journal sold in any week.The probability that demand is 10 is estimated to be 2/52 = 0.0385, and the pr

19、obability that the demand is 15 is 5/52 = 0.0962.Cumulative probabilities can also be estimated in a similar way. The probability that there are nine or fewer copies of the Journal sold in any week is (1 + 0 + 0 + 0 + 3 + 1 + 2 + 2 + 4 + 6) / 52 = 19 / 52 = 0.3654.Inventory Control with Uncertain De

20、mandWe generally approximate the demand history using a continuous distribution.By far, the most popular distribution for inventory applications is the normal.A normal distribution is determined by two parameters: the mean and the variance 2Inventory Control with Uncertain DemandThese can be estimat

21、ed from a history of demand by the sample mean and the sample variance .D2sniiDnD11niiDDns12211Inventory Control with Uncertain DemandThe normal density function is given by the formulaWe substitute as the estimator for and as the estimator for . xxxffor 21exp212DsInventory Control with Uncertain De

22、mand00.020246810121416182022-2-424260.040.060.080.100.12Optimization CriterionIn general, optimization in production problems means finding a control rule that achieves minimum cost. However, when demand is random, the cost incurred is itself random, and it is no longer obvious what the optimization

23、 criterion should be.Virtually all of the stochastic optimization techniques applied to inventory control assume that the goal is to minimize expected costs.The Newsboy Model (Continuous Demands)The demand is approximately normally distributed with mean 11.731 and standard deviation 4.74. Each copy

24、is purchased for 25 cents and sold for 75 cents, and he is paid 10 cents for each unsold copy by his supplier. One obvious solution is approximately 12 copies. Suppose Mac purchases a copy that he doesnt sell. His out-of-pocket expense is 25 cents 10 cents = 15 cents. Suppose on the other hand, he i

25、s unable to meet the demand of a customer. In that case, he loses 75 cents 25 cents = 50 cents profit. The Newsboy Model (Continuous Demands)Notation:= Cost per unit of positive inventory remaining at the end of the period (known as the overage cost).= Cost per unit of unsatisfied demand. This can b

26、e thought of as a cost per unit of negative ending inventory (known as the underage cost).The demand is a continuous nonnegative random variable with density function and cumulative distribution function .The decision variable is the number of units to be purchased at the beginning of the period.ocu

27、cD xf xFQThe Newsboy Model (Continuous Demands)Determining the optimal policy:The cost functionThe optimal solution equation uoucccQF* QuQodxxfQxcdxxfxQcQG0The Newsboy Model (Continuous Demands)Determining the optimal policy:024681012-10-6-21000200300400(Thousands)G(Q)OQ*QThe Newsboy Model (Continuo

28、us Demands)Example 2 (continued):Normally distributed with mean = 11.73 and standard deviation = 4.74.Since Mac purchases the magazines for 25 cents and can salvage unsold copies for 10 cents, his overage cost is = 25 10 = 15 cents.His underage cost is the profit on each sale, so that = 75 25 = 50 c

29、ents. ocucThe Newsboy Model (Continuous Demands)Example 2 (continued):The critical ratio is = 0.50/0.65 = 0.77.Purchase enough copies to satisfy all of the weekly demand with probability 0.77. The optimal is the 77th percentile of the demand distribution. *QuoucccThe Newsboy Model (Continuous Demand

30、s)Example 2 (continued):Area = 0.7711.73Q*xf(x)The Newsboy Model (Continuous Demands)Example 2 (continued):Using the data of the normal distribution we obtain a standardized value of = 0.74. The optimal isHence, he should purchase 15 copies every week.zQ1524.1573.1174. 074. 4*zQThe Newsboy Model (Di

31、screte Demands)Optimal policy for discrete demand:The procedure for finding the optimal solution to the newsboy problem when the demand is assumed to be discrete is a natural generalization of the continuous case.The optimal solution procedure is to locate the critical ratio between two values of an

32、d choose the corresponding to the higher value. That is QFQ uouQxcccQFxp*0*The Newsboy Model (Discrete Demands) Example 2: Q Qf QF Q Qf QF 0 1/52 1/52(0.0192) 12 4/52 30/52 (0.5769) 1 0 1/52(0.0192) 13 1/52 31/52 (0.5962) 2 0 1/52(0.0192) 14 5/52 36/52 (0.6923) 3 0 1/52(0.0192) 15 5/52 41/52 (0.7885

33、) 4 3/52 4/52(0.0769) 16 1/52 42/52 (0.8077) 5 1/52 5/52(0.0962) 17 3/52 45/52 (0.8654) 6 2/52 7/52 (0.1346) 18 3/52 48/52 (0.9231) 7 2/52 9/52 (0.1731) 19 3/52 51/52 (0.9808) 8 4/52 13/52(0.2500) 20 0 51/52 (0.9808) 9 6/52 19/52 (0.3654) 21 0 51/52 (0.9808) 10 2/52 21/52 (0.4038) 22 1/52 52/52 (1.0

34、00) 11 5/52 26/52 (0.5000)The Newsboy Model (Discrete Demands)Example 2:The critical ratio for this problem was 0.77, which corresponds to a value of between = 14 and = 15. Since we round up, the optimal solution is = 15. Notice that this is exactly the same order quantity obtained using the normal

35、approximation. QFQQ*QThe Newsboy Model (Discrete Demands)Extension to Include Starting Inventory:The optimal policy when there is a starting inventory of is:Order if .Dont order if .Note that should be interpreted as the order-up-to point rather than the order quantity when . It is also known as a t

36、arget or base stock level.0uuQ *Qu *Qu 0u*QMultiproduct SystemsABC analysis: The trade-offs between the cost of controlling the system and the potential benefits that accrue from that control. In multiproduct inventory systems not all products are equally profitable. A large portion of the total dol

37、lar volume of sales is often accounted for by a small number of inventory items.Multiproduct SystemsABC analysis:000.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90195 %80 %20 %50 %Production of inventory itemsCumulative fraction of value of inventoryABCMultiproduct SystemsABC

38、 analysis:Since A items account for the lions share of the yearly revenue, these items should be watched most closely. Inventory levels for A items should be monitored continuously.More sophisticated forecasting procedures might be used and more care would be taken in the estimation of the various c

39、ost parameters required in calculating operating policies.Multiproduct SystemsABC analysis:For B items inventories could be reviewed periodically, items could be ordered in groups rather than individually, and somewhat less sophisticated forecasting methods could be used. Multiproduct SystemsABC ana

40、lysis: The minimum degree of control would be applied to C items. For very inexpensive C items with moderate levels of demand, large lot sizes are recommended to minimize the frequency that these items are ordered. For expensive C items with very low demand, the best policy is generally not to hold any inventory. One would simply order these items as they are demanded.Lot Size-Reorder Point Systems In what follows, we assume that the operating policy is of the form. However, when generalizing the EOQ analysis to allow for random demand, we treat and as independent decision varia