运筹学与供应链管理3

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1、运筹学与供应链管理3Multi-Echelon Inventory in Supply ChainTwo Stage Echelon InventorySequential stocking points with level demandTwo-stageprocessTwo 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 anyth

2、ing but an integer multiple of .Therefore,we can think of two alternative decision variables and where(4.1)Two Stage Echelon InventoryTwo-stage process:The first stage costThe second stage cost The total costTwo Stage Echelon InventoryTwo-stage process:The warehouse echelon inventory is valued atwhi

3、le the retailer echelon inventory is valued at only Two 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=average value of the retailer echelon inventory,in units Two Stage Echel

4、on InventoryTwo-stage process:Substituting from equation(4.1)and noting that the echelon stocks follow sawtooth patterns,Two Stage Echelon InventorySelect (an integer)and in order to minimizePartial derivation of TRC Two Stage Echelon InventorySubstitute the result into the cost equationWe recognize

5、 that the n that minimizes the simpler expressionTwo Stage Echelon InventoryA convenient way is to first setwhich givesThis solves for 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

6、 function is convex in n).Two Stage Echelon InventoryTwo-stage process:Step 1ComputeStep 2Ascertain the two integer values,and ,that surround .Two Stage Echelon InventoryTwo-stage process:Step 3Two Stage Echelon InventoryTwo-stage process:Step 4Step 5Two Stage Echelon InventoryExample 1:Let us consi

7、der 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 the retailer corresponds to the inventory after the repackaging operation.The demand for this item can be assumed

8、 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 by the transforming(break and package)operation is$4/liter.The fixed component of the purchase charge()is$10,while

9、 the setup cost for the break and repackage operation()is$15.Finally,the estimated carrying charge is 0.24$/$/yr.Two Stage Echelon InventoryExample 1:Step 1:Step 2:Two Stage Echelon InventoryExample 1:Step 3:that is,Thus,use n=2.Two Stage Echelon InventoryExample:Step 4:Step 5:Two Stage Echelon Inve

10、ntoryExample 1:In other words,we purchase 334 liters at a 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

11、 liters of raw material.Inventory Control with Uncertain DemandThe demand can be decomposed into two parts,where =Deterministic component of demandand =Random component of demand.Inventory Control with Uncertain DemandThere are a number of circumstances under which it would be appropriate to treat a

12、s being deterministic even though is not zero.Some of these are:vWhen the variance of the random component,is small relative to the magnitude of .vWhen the predictable variation is more important than the random variation.vWhen the problem structure is too complex to include an explicit representati

13、on of randomness in the model.Inventory 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 uncer

14、tainty is desirable.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:Inventory Control with Uncertain DemandExample 2:Inventory Control with Uncertain DemandExample 2:Estimate

15、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 probability 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 f

16、ewer 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 Demand We generally approximate the demand history using a continuous distribution.By far,the most popular distribution for inventory applications is the normal.A normal distribution

17、is determined by two parameters:the mean and the variance Inventory Control with Uncertain Demand These can be estimated from a history of demand by the sample mean and the sample variance .Inventory Control with Uncertain Demand The normal density function is given by the formulaWe substitute as th

18、e estimator for and as the estimator for .Inventory Control with Uncertain Demand Optimization Criterion In 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

19、 what the optimization 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 devia

20、tion 4.74.Each copy 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 oth

21、er hand,he is 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

22、 be 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.Th

23、e Newsboy Model(Continuous Demands)Determining the optimal policy:The cost functionThe optimal solution equationThe Newsboy Model(Continuous Demands)Determining the optimal policy:The Newsboy Model(Continuous Demands)Example 2(continued):Normally distributed with mean =11.73 and standard deviation =

24、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 cents.The Newsboy Model(Continuous Demands)Example 2(continued):The critical ratio is =0.50/0.65=0.77.Purc

25、hase enough copies to satisfy all of the weekly demand with probability 0.77.The optimal is the 77th percentile of the demand distribution.The Newsboy Model(Continuous Demands)Example 2(continued):The Newsboy Model(Continuous Demands)Example 2(continued):Using the data of the normal distribution we

26、obtain a standardized value of =0.74.The optimal isHence,he should purchase 15 copies every week.The Newsboy Model(Discrete 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 generali

27、zation of the continuous case.The optimal solution procedure is to locate the critical ratio between two values of and choose the corresponding to the higher value.That isThe Newsboy Model(Discrete Demands)Example 2:The Newsboy Model(Discrete Demands)Example 2:The critical ratio for this problem was

28、 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 approximation.The Newsboy Model(Discrete Demands)Extension to Include Starting Inventory:The optimal policy when there

29、 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 target or base stock level.Multiproduct SystemsABC analysis:vThe trade-offs between the cost of controlling the system and the p

30、otential benefits that accrue from that control.vIn multiproduct inventory systems not all products are equally profitable.vA large portion of the total dollar volume of sales is often accounted for by a small number of inventory items.Multiproduct SystemsABC analysis:Multiproduct SystemsABC analysi

31、s: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 cost parame

32、ters 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 analysis:vThe mi

33、nimum 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.vFor expensive C items with very low demand,the best policy is generally not to hold any inventory.On

34、e would simply order these items as they are demanded.Lot Size-Reorder Point SystemsvIn 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 variables.Lot Size-Reorder Point SystemsA

35、ssumptionsvThe system is continuous-reviewvDemand is random and stationaryvThere is a fixed positive lead time for placing an ordervThe following costs are assumedSetup cost at$per order.Holding cost at$per unit held per year.Proportional order cost of$per item.Stock-out cost of$per unit of unsatisf

36、ied demandLot Size-Reorder Point SystemsDescribing demand:The demand during the lead time is a continuous random variable with probability density function(or pdf),and accumulative distribution function(or cdf).Let and be the mean and standard deviation of demand during lead time.Lot Size-Reorder Po

37、int SystemsDecision variables:There are two decision variables for this problem,and ,where =the lot size or order quantity and =the reorder level in units of inventory.Lot Size-Reorder Point SystemsDecision variables:Additional Discussion of Periodic-Review SystemsDefine two numbers,and ,to be used as follows:When the level of on hand inventory is less than or equal to ,an order for the difference between the inventory and is placed.If is the starting inventory in any period,then the policy is:vIf ,order .vIf ,dont order.