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1、http:/ An Assemble-to-Order System with Two End Products 1 Wei Jie Tu Fengsheng College of Information Technical Science College of Information Technical Science Nankai University Nankai University Beijing,P.R.China 300071 Beijing,P.R.China 300071 Abstract A single period assemble-to-order system co

2、nsisting of a manufacturer and three suppliers was considered.The manufacturer produces two different types of correlated end products,each product needs a different mix of a common set of component parts,which belong to two types,one type is specific components and the other is common components.Ho

3、w to manage inventories of components in such a system is of managerial interest.We formulate the inventory management problem for the system as a two-stage stochastic nonlinear integer program.We prove the decomposition of total expected profit function and give two algorithms.We solve the first st

4、age of the problem to determine the optimal stock levels of the components by algorithm 2 and solve the second stage problem by algorithm 1,where the component allocation decisions are made.At last,we give a numerical example.Keywords:Assemble-to-order,inventory(stock)management,optimal stock levels

5、.1 Introduction Over the past decade,the assemble-to-order(ATO)system has become a widely accepted business model in electronic industry.Many high-tech firms,which face heightening customer expectations,shrinking product life cycles,increasing demand for product varieties,and rapid technology breakt

6、hroughs,have successfully used ATO to broaden their customized product offerings,to lower inventory cost,and to reduce time-to-market.In contrast to the traditional Make-to-Stock system,which keeps inventory at the end product level,ATO keeps inventory at the component level.Components are acquired

7、in advance.When a customer order arrives,required components are pulled from inventory and the end product is assembled and delivered to the customer.ATO postpones the point of commitment of components to specific products and increases the probability of meeting customized demand on time.Similar st

8、rategies have been termed delayed customization and postponement(Aviv and Federgruen(2001a,b),Brown,Lee,and Petrakin(2000),Johnson and Anderson(2000),Van Hoek(2001).In this paper,we consider a single period ATO system consisting of a manufacturer and three suppliers.The manufacturer produces two dif

9、ferent types of correlated end products(such as,printer,computer,TV set,ect.),one is the innovation of the other.Using a common set of inputs(e.g.,component parts)from the suppliers.Each end product needs a different mix of these component parts.These component parts belong to two types,one type we

10、shall call specific components and the other we shall call common components.Product demands are integer-valued,possibly correlated,random variables.At the beginning of the period(that is the first stage),the manufacturer makes the order quantities of 1 Support by the National Natural Science Founda

11、tion of China(Grant No.60474062).1http:/ components.In the second stage,the manufacturer observes product demands and makes component allocation decisions based on the inventory on hand and the realized demands.There have been a large number of studies of ATO systems.Some early work in this area inc

12、ludes analysis of the repair-kit problem by Smith et al.(1980)and Graves(1982).Lu and Song(2003)formulate a customer-order level cost-minimization model to determine the joint optimal base-stock levels in the multi-product ATO system and compare it with the single-item cost minimization model.Lu,Son

13、g and Yao(2003)treat the ATO system as a set of queues driven by a common,multiclass batch Poisson input.Bernstein and DeCroix(2004)study the issue of modular assembly in a multi-tier assembly system.We refer to Song and Zipkin(2003)for a recent survey paper in this research areas.Other related work

14、 includes the study of component commonality.Relevant research on component commonality includes Collier(1982),Baker et al.(1986),Gerchak et al.(2004)and Gerchak and Henig(1989).Differ from all existing papers,we model the inventory(stock)management problem for two end-product ATO system as a two-st

15、age stochastic nonlinear integer program.In this paper,we demonstrate the decomposition of total expected profit function,and give two algorithms.We solve the second stage problem by algorithm 1,where the component allocation decisions are made.We solve the first stage of the problem to determine th

16、e optimal stock levels of the components by algorithm 2.The rest of the paper is organized as follows.Section 2 introduces the model,notation,and demonstrates the properties of optimal order quantities,the decomposition of total expected profit function,and gives two algorithms.In section 3,we give

17、a numerical example.Finally,section 4 provides some concluding remarks.2 Model and notation Consider a single period ATO system consisting of a manufacturer and three suppliers.The manufacturer produces two correlated end products(labeled 1 and 2)using three components(labeledA,Band).Product 1 consi

18、sts of one unit each of component A andCB,while product 2 consists of one unit each of componentsBandC.(More general component quantity requirements can be reduced to this case by re-scaling the problem parameters.)Product 1 is the innovation type of product 2.We say component A and Care dedicated t

19、o products 1 and 2 respectively,and call them specific components.We call component B is common component.ComponentA,Band are produced by suppliers A,B andC respectively.Demand for end product during a single selling season,denoted by,is a stochastic integer-valued variable.Assume that lead-times fo

20、r assembly and delivery lead-time from the supplier to the manufacturer are negligible,so that final products can be assembled to order after demand is observed.Any unsatisfied demands are lost.The sequence of events is as follows.First,at the beginning of the period and before demand realization,th

21、e optimal stock level is determined and orders are placed.Second,after observing demands,components are allocated to assembly to maximize profit.Each sale generates revenue,and any remaining components are salvaged at unit salvage.To summarize the notations:C)2,1(=jjjDiqjDjrisiq=order quantity for c

22、omponent (decision variable),iCBAi,=；q=(,);AqBqCqjD=demand for end product during a single selling season(random variable)，；j2,1=jD=(,);1D2D2http:/)(ABY=sales quantity of product 1 with components,given qandBA,)(D;)(BCY=sales quantity of product 2 with component,given andCB,q)(D;iw=unit cost for com

23、ponenti,CBAi,=;),(CBAwwww=；jr=unit revenue for product,j2,1=j;is=salvage value for component,iCBAi,=.We give the following assumptions about the model:(1)assume which follows directly from the definition of product 1 and product 2.(2)A similar assumption,21rr CAssrr21,states that the difference in p

24、rice between the products is greater than the difference in the salvage of the products specific components.(3)To avoid trivialities,we assume that01BAssr,which insure that it is not beneficial to buy components simply to sell them for salvage.(4)We also assume that the vector of wholesale prices is

25、 such that both products earn a positive margin for the manufacturer,and that product 1 has the higher margin,e.g.,02CBssrw,021CBBAwwrwwr This assumption means that manufacturers profit from product 1 is bigger than that from product 2.After making the order quantity for component iqCBAi,=in advance

26、,the manufacturer begins to assemble end products,when learns about the customer orders of products.For a demand realization)(D=()(1D,)(2D);and given=(,),let qAqBqCq)(,(DqWbe the maximal total reward attainable from the demand=()(D)(1D,)(2D).Further,let be the expected value of .The follows is the t

27、wo-stage stochastic optimization problem.)(qW)(,(DqW )(0qWMaxMaxqq=,(1)subject to and integer for 0iq,CBAi=(2)where )(,()()(DqWEqWD=,(3)and )()(,(CCBBAAqwqwqwVDqW=,(4)where +=)()()()(21)(),(ABAABCABYYYqsYpYpMaxVBCAB ),()()(BCCCBCABBBYqsYYqs+(5)3http:/ subject to ),()(1DYAB (6),()(2DYBC (7),)(AABqY (

28、8),)(CBCqY (9),)()(BBCABqYY+(10),0)(),(BCABYYand integer.(11)Note1.Consider that(5)is linear,and use the above three assumptions above the model,it is obvious that there exists one and only one pair of)(),(BCABYY,given a demand realization and =(,),such that)(DqAqBqCqVattains its maximal value.First

29、 we identify some properties of the optimal values for the decision variables.Let superscript*denote the optimal value of the respective decision variables,we have Note2.Consider that the only component shared by the two products is componentB.Since each unit product 1 and product 2 uses the same am

30、ount of the component Band product 1 is more profitable,it is optimal for the manufacturer to give priority to product 1 when choosing production quantities.That is,the manufacturer satisfies as much demand for product 1 as possible given capacities of componentsA,B and then uses any remaining capac

31、ity of component B along with capacity of component C to satisfy as much demand for product 2 as possible.Property 1 ,(12)*BAqq .(13)*BCqqProof Note(10),we have(12)and(13).property 2.(14)*CABqqq+=Proof Note that,together with the descriptions about the model,we have the following:if,without loss of

32、generality,assume that.It is easy to see that whatever the demands of the two products are,there will always be units of component CBAiswii,=*CABqqq+*CABqqq+)(*CABqqq+B are salvaged.This is not beneficial to the manufacturer.Similarly,if,It is also easy to see that whatever the demands of the two pr

33、oducts are,there will always be units of component*CABqqq+*BCAqqq+Aand are salvaged.This is also not beneficial to the manufacturer.C 4http:/ Using the above four assumptions about the model,for given q=(,)and realized demand=(AqBqCq)(D)(1D,)(2D)=,we can solve the value of),(21dd)(V with the followi

34、ng algorithm 1,and it is easy to see that the results)(ABY and )(BCY are optimal.Algorithm 1 Begin Step 1 Set unfilled demands and;1d2d Initialize on-hand inventoryCBAiqi,=.Step 2 Compute effective capacity for product 1:.,min1BAqqf=Step 3 Fill units of product 1.,min)(11dfYAB=Step 4 Compute effecti

35、ve capacity for product 2:.),(min2CABBqYqf=Step 5 Fill units of product 2.,min)(22dfYBC=End Using property 2 and algorithm 1,we can easy have:)(,min)(),(,min)(21DqYDqYCBCAAB=(15)Theorem 1 There exist functions and )(AqW)(CqW such that )()()(CqAqqqWMaxqWMaxqWMaxCA+=,(16)where+=)()()(1)()(1ABAAABYDAYq

36、sYrMaxEqWAB)(ABAAABBABqwqwYsqs,(17)subject to(6),(8),(11).+=)()()(2)()(2BCCBBCYDCYqsYrMaxEqWBC)(CCCBBCCCCqwqwYsqs,(18)subject to(7),(9),(11).Proof Using property 2,(4)and(5)can be wrote as 5http:/)()()()()()()(,(21)(),(CCCBBCCCCBCCBBCABAAABBABABAAABYYqwqwYsqsYqsYrqwqwYsqsYqsYrMaxDqWBCAB+=)19(,)()()(

37、)()()(2)(1)(CCCBBCCCCBCCBBCYABAAABBABABAAABYqwqwYsqsYqsYrMaxqwqwYsqsYqsYrMaxBCAB+=subject to(6),(7),(8),(9),(11).We have by(3),and(18),)(,()()(DqWEqWD=,)()()()()()(2)(1)()(CCCBBCCCCBCCBBCYABAAABBABABAAABYDqwqwYsqsYqsYrMaxqwqwYsqsYqsYrMaxEBCAB+=)20(,)()()()()()(2)()(1)()(21CCCBBCCCCBCCBBCYDABAAABBABA

38、BAAABYDqwqwYsqsYqsYrMaxEqwqwYsqsYqsYrMaxEBCAB+=So,it is easy to see that theorem 1 holds.Without loss of generality,assume that the joint distribution function of integer-valued random variables and is 1D2D.1,1;,2,1,2,1,),(,2,121,22,11=kikikiddnknipdDdDPLL The marginal distribution function of rando

39、m variables and is respectively.1D2D.1,2,1,)(,11,11=iiidnipdDPL1,2,1,)(,22,22=kkkdnkpdDPLBy(17),we have the first-order difference equation,of expected function in.)(AqW)(AqWAq6http:/.),1min(),min(),min(),1min(1),min(),1min()(,1,1,1,11,1,111BAiAiABBiAiAAniiAiAiAwwdqdqssdqdqsdqdqrpqW+=Similarly,by(18

40、),we can also have the first-order difference equation,)(CqW,of expected function)(CqW in.Cq.),min(),1min(),min(),1min(1),min(),1min()(,2,2,2,21,2,222CBkCkCBBkCkCCnkkCkCkCwwdqdqssdqdqsdqdqrpqW+=Lemma 1.The second-order difference equation of expected function in.0)(AqWAqProof.For any if or ,then ,2,

41、1,1,11nidDiL=Aiqd,12,1+Aiqd,0),2min(),1min(2),min(),min(),1min(2),2min(),min(),1min(2),2min(,1,1,1,1,1,1,11,11,11=+iABiABiABiAAiAAiAAiAiAiAdqsdqsdqsdqsdqsdqsdqrdqrdqr if,then,if ,then 1,1+AiAqdq021,1+AiAqdq0.So the second-order difference equation,have)(2AqW.0),2min(),1min(2),min(),min(),1min(2),2mi

42、n(),min(),1min(2),2min()(,1,1,1,1,1,1,11,11,11121+=iABiABiABiAAiAAiAAiAiAiAniiAdqsdqsdqsdqsdqsdqsdqrdqrdqrpqW the lemma 1 holds.Lemma 2.The second-order difference equation of expected function 0)(CqW in.CqProof.The proof is similar to lemma 1.Theorem 2 There exists,if and,then if and,then also exis

43、ts,if and,then,0*AAdq,0*AAdq*AAqq AqW,0*AAdq*AAqq;0)(AqW,0*CCdq,0*CCdq*CCqq CqW if and,then,0*CCdq*CCqq;0)(=BAAwwrqW,*AAdq=7http:/ 0)(=CBCwwrqW,if ,*CCdq=then 0)(let;if hL=,0)(iqW,CAi=Let.hR=Step 4 If.1+LR,go to step 2;else,go to step 6.Step 5 Fill .stop.,*hqi=Step6 Fill .stop.,*Lqi=End 3 Computatio

44、nal result In this section,we give a numerical example as follows by using algorithm 1,algorithm 2,and Matlab 7.0.We let ,15951=r,10302=r,799=Aw290=Bw,307=CW,150,180,400=CBAsss,and without loss of generality,assume that the distributions of random variables as following:21,DDid,1 1 2 3 4 5 6 7 8 9 1

45、0 ip 0.0002 0.0011 0.0046 0.01480.00370.07390.12010.16020.1762 0.1602id,1 11 12 13 14 15 16 17 ip 0.1201 0.0739 0.0370 0.01480.00460.00110.0002 kd,2 1 2 3 4 5 6 7 8 9 10 kp 0.0005 0.0031 0.0123 0.03500.07460.12440.16590.17970.1597 0.11718http:/ kd,2 11 12 13 14 15 16 kp 0.0710 0.0355 0.0146 0.00490.

46、00130.0003 We have .16,8,8*=BCAqqq4 Conclusion In this paper,we consider an assemble-to-order system involving correlated products.We formulate the inventory(stock)management problem for two end-product ATO system as a two-stage stochastic nonlinear integer program,and give two algorithms.We solve t

47、he first stage of the problem to determine the optimal stock levels of the components by algorithm 2.Using algorithm 1,we solve the second stage problem,where the component allocation decisions are made.Future research should continue to analyze this important problem by examining the multi-period,m

48、ulti-product case.It appears to be quite challenging.References 1 Aviv,Y.,and Federgruen,A.,Capacitated multi-item inventory systems with random and seasonally fluctuating demands:implications for postponement strategies.Management Science,47,(4),.512-531,2001(a).2 Aviv,Y.,and Federgruen,A.,2001(b)D

49、esign For Postponement:A Comprehensive Characterization Of Its Benefits Under Unknown Demand Distributions.Operations Research,47,(4),578-598.3 Baker,K.,M.Magazine and H.Nuttle,(1986)The Effect of Commonality on Safety Stock in a Simple Inventory Model,Management Science,32,982-988.4 Bernstein,F.,an

50、d G.DeCroix,(2004)Decentralized Pricing and Capacity Decisions in a Multitier System with Modular Assembly,Management Science,50,1293-1309.5 Brown,A.,Lee,H.L.,and Petrakian,R.(2000)Xilinx Improves Its Semi-conductor Supply Chain Using Product and Process Postponement,INTERFACES,30(4),65-80.6 Collier