A Cost Model of Partial Postponement Strategy of the Single-Period Product under Stochastic Demand

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1 Research Journal of Appled cences, Engneerng and Technology 4(): , 0 IN: Mawell centfc Organzaton, 0 ubmtted: December 0, 0 Accepted: January, 0 Publshed: June 0, 0 A ost Model of Partal Postponement trategy of the ngle-perod Product under tochastc Demand Yanhong Qn chool of Management, hongqng Jaotong Unversty, hongqng , hna Abstract: The target of our study s to set a new cost model to solve the partal postponement problem by addng penalty cost parameter of shortage under stochastc demand. By dong ths, we hope the new model can be better appled to real condtons. Postponement s an mportant strategy to acheve mass customzaton and t has been adopted by many companes to mprove producton operaton, nventory and logstcs management and supply chan operaton, but the postponed actvty wll cause addtonal costs at the same tme. There have been many lteratures tradng off between the relatve cost and the postponement benefts of product manufacture, and our paper s to solve the smlar problem. Key words: Inventory, mass customzaton, partal postponement INTRODUTION Postponement strategy s an effectve strategy to acheve mass customzaton and t has been perceved as one of the major supply chan management practces, because t can better deal wth the product prolferaton wthout ncurrng large operatng costs caused by postponement actvtes (L et al., 007). By holdng some or all common nventory n the early stage of product wth uncertanty demands and delayng the customzng fnal product untl the demand nformaton s certan, the postponement strategy can save nventory levels, reduce delvery lead tmes, and varous product portfolo to satsfy varous customer demand, by dong ths, the postponement strategy can lessen the msmatch between the forecast-drven producton and the actual demand, reduce the effect of bullwhp n the supply chan and mprove customer satsfacton obvously (Wong et al., 00, 009). But how many nventores can be hold to reduce cost and response to customer quckly. If the nventory s all hold as fnal products, t wll derve the scale economy of mass producton wth hgh producton effcency, but t wll not meet the varous customer requrements or t wll cause very hgh nventory level caused by the varous product portfolos (hod et al., 006). In the contrary, f the nventory s all hold as common nventory of modules or basc products, maybe t wll reduce the nventory level fully for the rsk poolng effect and acheve the scope economy, but the lost sale wll occur when the delvery tme eceeds the epect tme of customer watng for the fnal customzaton (Ronald and Bassok, 004). o there s a dlemma to the common nventory level and fnal product nventory (Anupnd and Jang, 008). There have been many researches on the postponement problem, typcally, as n Graman and Magazne (00), denoted G&M, the authors studed the mpact of postponement capacty on the benefts of nventory savngs, gven a defned customer servce level, denoted as fll rate and whch s decded by managers n the paper, and then a model of sngle-perod and two products capactated-postponement nventory s analyzed, where the non-postponed fnshed fnal product nventory and the generc product nventory (.e., postponed nventory), whch wll be customzed untl the real customer demand s known are both held by the manufacture. The two fnal dfferent products can be made from the generc products (.e., postponed products) by packagng, addtonal parts or other customzed servce tem etc. to meet dfferent demand. The fnshed product nventory wll be used to meet the real demand frst, but once the demand are ecess of fnshed product nventory, some or all of the generc product wll be completed to meet the ecess demand as much as possble wthn the specfed delvery tme. The author obtaned some mportant conclusons, such as when the fll rate, the coeffcent of varaton, the number of products beng postponed ncrease, the demands are more negatvely correlated and the demand dstrbuton of dfferent products are approachng each other, and then nventory savng wll ncrease and most mportantly, the author observed an mportant phenomenon that a relatvely small amount of postponement capacty (about 40% of total epected demand) can be acheve all of the benefts of completely postponng all demands,.e., keepng all the nventory as generc product and dfferent customzed parts rather than any fnal fnshed products, and the customzed parts can be assembled to the generc product to form the fnal customzed products quckly untl the 494

2 Res. J. Appl. c. Eng. Technol., 4(): , 0 Demand and nventory level for product + {4} {} {} {8} {6} {7} {3} {5} {9} + Demand and nventory level for product Fg. : Graphcal depcton of nventory levels and regons for a partal postponement scenaro as same n Graman and Magazne (00) real demands s certan. Ths mportant phenomenon wll nspre many frms to adopt the partal postponement strategy, and especally, postponement strategy s relatve to some addtonal postponement cost, such as the nvestment cost, processng cost, handlng cost of common nventory. But there s a potental prelmnary behnd the numercal analyss and observaton,.e. the fll rates of dfferent products are the same. o n the later research of Graman (00), denoted G, the author further set a smlar model to the model n Graman and Magazne (00), but there are some dstnct dfferences. The frst dfference s Graman (00) focused on solvng the mnmum-cost objectve functon by consderng some addtonal costs caused by postponement, where the postponed manufacturng or assembly wll cause the more frequent setups of producton lne to process smaller lot sze, addtonal handlng, packagng to facltated the handlng and mantanng ntegraton of generc products, denoted as assemble labor and materal cost. econdly, Graman (00) showed some addtonal dfferent conclusons based on the research of Graman and Magazne (00), such that when the value of generc products and relatve postponement cost decrease (ncludng packagng postponed nventory to mantan product ntegrty, addtonal operaton, watng caused by nablty to delvery wthn the specfed lead tme), the holdng cost ncrease, etc., the total nventory and epected total cost wll decrease. The thrd dfference between the two contnuous researches s: n Graman and Magazne (00), the assumpton that each fll rate or the epected stock-out number for each customzed fnal product equal are the bass for comparson among dfferent level of partal or capactated postponement strategy and other senstvty analyss of dfferent parameters change on the nventory level of generc products and fnal products. But n the verson of Graman (00), the fll rate or the epected stock-out number s set as an constrant n solvng the mnmal cost non-lnear programmng problem, and n the process of reasonng and analyss, Graman (00) ddn t dscrmnate the regons Anupnd and Jang (008), Ronald and Bassok (004) and L et al. (007) n Fg..That s to say, Graman (00) treat the condton happened n regon Anupnd and Jang (008) and Ronald and Bassok (004) as the same n regon L et al. (007) n computng the epected stockout number, as shown n the Eq. (.4) and (.5) n the Append n Graman (00), whch wll cause error reasonng n solvng the non-programmng problem. Importantly, the fll rates for dfferent customer are often dfferent accordng to the profts obtaned from the sales for dfferent customer, the mportant degree of the orders, or the penalty cost caused by the demand unmet. The more mportant of the order, the more proft obtaned from the order or the bgger penalty cost, the respondng order should be met n the more anteror sequence, whch s a unversal phenomenon n many enterprses, so there should be some dfference between the fll rates for dfferent customers or products, and we wll demonstrate that the problem formulaton and computaton wll be smplfed and drect by ntroducng the penalty cost parameter substtutng for the fll rate. o the man targets of our paper nclude two aspects: Why the epresson of epected stock-out product should consder the regon L et al. (007), Anupnd and Jang (008) and Ronald and Bassok (004) ndvdually and dscrmnate the ntegraton epresson of epected stock-out number happened n regon L et al. (007), Anupnd and Jang (008) and Ronald and Bassok (004) When the ntroduce of the dfferent penalty cost parameter rather than the equal fll rate parameter for dfferent customzed product, and when the shortagecost tems set as an tem n the objectve total cost functon, but not as a constrant ncluded n Graman (00), the reasonng and computaton can be smplfed, and more mportantly, t s more ft the decson condton as many companes meet For convenence, the followng notaton and defntons used are same n Graman (00) ecept the new parameter of penalty cost t and nventory number of left ove. Ths study wll compute the optmal nventory level of both common product and fnal product by tradng off between the relatve cost and the postponement benefts of product manufacture nvolvng penalty cost of shortage under stochastc demand. Assumpton and notaton: Of course, there may be some potental assumptons n the Graman and Magazne (00) and Graman (00) and n ths study as followng: The two products completed from the same generc nventory can be substtuted for the other one, so the nventory of product can be used to meet demand of product, vce versa. 495

3 Res. J. Appl. c. Eng. Technol., 4(): , 0 Ether of the fnshed products can be further revsed or customzed to meet the demand of the other product. ustomzaton lead tmes for both products are zero. Each product has a lnear customzaton cost,.e. the addtonal epense of usng postponed manufacturng mode over the cost of non-postponement mode. The unt customzaton cost for both products s same. Besdes, the customzed producton capacty s assumed to be unlmted, so the fed cost assocated wth postponement s assumed to be mnmal and can be set equal to zero. When the postponed nventory s avalable, once there s a shortage of ether or both of products, customzng the products takes place. Each fnal product contans one unt of the generc product and the dfference between fnal products s cosmetc, so all quanttes are n terms of the generc product. If we further assume that the order wth more unt penalty cost s met frst by the postponed capacty, the problem wll be smplfed when the demand eceed the fnshed product, and there s no need to dscrmnate the demand happened n regon L et al. (007) and Ronald and Bassok (004),.e., we can adopt the Fg. drectly to solve the capactated postponement problem. :, products X : Random varable for demand for product,x observe normal dstrbuton N(u, F ) : The realzed demand durng the perod for product n terms of the common tem m : The cost of labor and materal to assemble one unt of the common tem w : The cost of postponement materals and actvtes for one unt of the common tem d : The cost to package one unt of the common tem for delvery r : The holdng cost rate per perod epressed as a percent of product value v F : The value of one unt of fnshed goods nventor v P : The value of one unt of the common tem n the postponed nventory h F = r v F : The cost to hold one unt n fnshed-goods nventory for one perod h P = r v P : The cost to hold one unt n postponed nventory for one perod P : The amount of postponed nventory allocated to meet demand for fnal product : The number of unts of the common tem packaged as fnshed-goods nventory for product j at the begnnng of the perod Demand and nventory level for product + {4} {} {} {8} {6} {7} {3} {5} {9} + Demand and nventory level for product Fg. : Graphcal depcton of nventory levels defnng regons where realzatons of demand can occur for a partalpostponement scenaro, as same n Graman (00) LO T PP T NON : The number of unts of the fnal product-left over : The amount of postponement capacty n terms of the common tem; also the number of postponed tems held n bulk-storage : The epected total cost of a partal postponement strategy, > 0 : The epected total cost of a nonpostponement strategy, = 0 Epressons for the epected stock-out nventory: In model of Graman (00), the objectve functon was to mnmze the cost made up of assembly cost, postponement cost, packagng cost and holdng cost of combned fnshed goods wth postponed nventory. The constrants nclude fll rate constrants, boundary condton constrants, postponement capacty allocaton constrants and non-negatvty constrants. The epresson of the epected stock-out products E[O ] (I =, ) n the regon hod, et al., (006), Graman and Magazne (00), Graman (00) Wong et al. (009) and Wong et al. (00) s same to Graman and Magazne (00) and Graman (00), when the real demand of both products can t be met by all fnshed product nventory and generc or postponed nventory whch s happened n regon L et al. (007) and Ronald and Bassok (004) of Fg. or n regon L et al. (007) of Fg., then the computaton of epected stock-out number for each product s based on the decson rule: equalze the fll rates of each product Graman (00),.e. P P and P + P #, whch s same to E[O ] = E[O ], but ths constrant was not ncluded or reflected n the model of Graman (00). Besdes, Graman (00) ddn t 496

4 Res. J. Appl. c. Eng. Technol., 4(): , 0 dscrmnate the regons Anupnd and Jang (008), Ronald and Bassok (004) and L et al. (007), n Fg.. That s to say, Graman (00) treat the condton happened n regon Anupnd and Jang (008) and Ronald and Bassok (004) as the same n regon L et al. (007), n the computaton the epected stock-out number, as shown n the Eq. (.4) and (.5) n the append, whch wll cause error reasonng n solvng the non-programmng problem. The reason for ths s as followng: When the demand happened n the regon Anupnd and Jang (008) and Ronald and Bassok (004) all fnshed and postponed nventory wll all be eerted where P + P =, and ths s bass to compute E[O ]. At the same tme, the equaton wll decde how the postponed capacty (or generc products) s allocated to each type of fnal products, but as shown n Graman and Magazne (00), when the demand happened n regon Anupnd and Jang (008) or Ronald and Bassok (004) one of the product demand s much more larger than the other one, so even all the postponement capacty s allocated to complete the product of large demand, the fll rate can t be rased to equal that of lower product demand. For eample, n regon Anupnd and Jang (008) demand for product s much more lager than product, so all the postponed capacty s allocated to product to attempt to equalze the fll rate, as a result, the boundary functon between L et al. (007) and Anupnd and Jang (008), a s: smlarly, the boundary functon between L et al. (007) and Ronald and Bassok (004) s so we can get the epected stock-out number for product n regon L et al. (007), and Ronald and Bassok (004): / EO [ { 7}] P f, dd / E[ O{ 8}] f, dd 9 / EO f, dd The equaton for product E[O {7}] E[O {8}]and E[O {9}] can be denoted by the same reason. But not the general epresson (.4) and (.5) n the Append n Graman (00) whch wll gnore the condton n regon Anupnd and Jang (008) and Ronald and Bassok (004) n the process of computaton whch wll nfluence the result of program solvng. The new model ncludng penalty cost parameter: In ths secton, we wll ntroduce the unt penalty cost parameter t denotng the penalty cost of unt shortage of product/ (whch s also often used to reflect the customer satsfacton level from the vewpont of manufacture),then there s no need to dscrmnate the condton n L et al. (007), and Ronald and Bassok (004), as shown n Fg.. For the demand for the products of larger unt penalty cost must be always satsfed before the demand of smaller one, or the total cost can t be mnmzed at all. After the penalty cost parameter s ntroduced to substtute for the fll rate, the fll rate constrants can be relaed, the epresson and computaton of E[O ] can be smplfed. When none of products stock out, the generc nventory wll not be used, and some fnshed nventory and generc nventory wll be left over n the end of perod, as the demands n regon hod et al. (006) n Fg.. When only one of fnshed products stock outs,.e. the demands n regon Graman and Magazne (00), Graman (00) Wong et al. (009), Wong et al. (00), some or all generc nventory wll be allocated to the product of shortage. When both demands can t be met from the fnshed nventory drectly n regon Lee et al. (997) but + # + +, so the postponed nventory can be allocated as: for product and for product. Obvously, there s none shortage of product n regon hod et al. (006), Graman and Magazne (00), Graman (00) Wong et al. (009), Wong et al. (00) and Lee et al. (997), so there s no penalty cost when demands happened n these regons. When all the postponed nventory are ehausted, both of the demands can t be met at all, as the condton n regon L et al. (007), and Ronald and Bassok (004) n Fg. or the regon L et al. (007), n Fg.. The allocaton prncple s that the demand of larger penalty cost wll be all satsfed n the frst place attemptng to mnmze the total cost, so there s no need to dscrmnate the epresson of E[O j ] whch s dfferent from Graman and Magazne (00) and Graman (00). The reason behnd ths s smple, as long as the product can reduce the cost to more etent, the postponed nventory wll be allocated to t, even all the generc nventory wll be allocated to the product of ma {t, t }: 497

5 Res. J. Appl. c. Eng. Technol., 4(): , 0 33 EO [ { 7}] [ ma{ 0, sgt ( t3)}] f (, 3) dd3 [ 3 3 ma{ 0, sg( t t )}] f (, ) d d EO [ { 7}] [ ma{ 0, sgt ( 3 t3)}] f (, 3) dd3 [ 3 3 ma{ 0, sg( t t )}] f (, ) dd The nventory level of fnal product, and common product, can be llustrated n Table (obvously, E[LO ]+E[P +P ] = ), based on the depcton n Fg.. The total cost ncludes assembly labor and materal cost, postponement cost, packagng cost, holdng cost of fnshed product and generc product, and shortage cost, so the objectve functon s: ET [ PP] m ( ) m w d ( ) d EP [ P] hf ELO [ LO] hp LO t E[ O] t E[ O] ( m d) ( ) ( m w hp) ( d hp) EP [ P] hf ELO [ LO] t E[ O ] t E[ O ] subject to., 0,. when = 0, t means that the non-postponement strategy s adopted,.e. n the ntal of perod, the nventory only ncludes the fnshed fnal product but none generc product, and there s none customzaton actvty. o the best choce would be the one wth the lowest total cost. In the equaton: 0 E[ P P ] ( ) f (, ) d d 0 ( ) f (, ) d d ( ) f (, )] d d ( ma{ 0, sg( t t3)}) f (, ) d d ELO [ ] f(, ) dd 0 0 ( ) f (, ) d d 0 0( ) f (, )] dd ( ) f (, ) d d Table : Inventory level of fnal product, and common product Regon (t >t )7(t <t ) LO LO P P O O LO E[ LO LO ] ( ) f (, ) d d 0 0 ( ) f (, ) d d 0 EO [ ] ( ) f(, ) dd [ ma{ 0, sg( t t )}] f (, ) dd [ ma{ 0, sg( t t )}] f (, ) d d 0 EO [ ] ( ) f(, ) dd [ ma{ 0, sg( t t )}] f (, ) dd [ ma{ 0, sg( t t )}] f (, ) d d There are three decson varables,, and only some non-negatvty constrants compared wth the fve decson varables and more addtonal constrants n Graman (00), so the computaton process wll be smplfed and t wll be easy to observe the effects of varable on the value of and the total cost, when the postponed capacty s set to dfferent value There are three decson varables,, and only some non-negatvty constrants compared wth the fve decson varables and more addtonal constrants n Graman (00), so the computaton process wll be smplfed and t wll be easy to observe the effects of varable on the value of and the total cost, when the postponed capacty s set to dfferent value. ase study: The correspondng parameters are denoted as m = 0.5, d = 0.5, r, h P = 0.5, h F = 0.5, h F = 0.5, h F 0.5, w = 0.5u = 000, u = 000, F = 00, F = 00. The Genetc Algorthm s adopted to solve the problem, the soluton s: = , = = 70.30, T = 806, T = 806 The mpact of common nventory on the number of each fnal product nventory and total cost s computed by 498

6 Res. J. Appl. c. Eng. Technol., 4(): , 0 Table : ommon nventory level, fnal product nventory and total cost Total nventory Total cost MatLab7.0, as llustrated n Table. From the Table, we can see that the partal postponement strategy can save nventory level and reduce the total manufacture cost. Besdes, the nventory level of product s hgher than product, the reason behnd ths s the penalty cost of product s hgher, so the nventory level of fnal product and the number of common nventory allocated to product s more than product to reduce the cost to mamal etent. ONLUION It s dffcult to fgure out the condton where more than two products customzed from the generc nventory, but computaton process wll be smplfed for the allocaton rule of the postponement capacty proposed n ths paper by ntroducng the dfferent penalty cost parameter. In future research, we can stll consder two fnal products whch are stll assembled or customzed from a common nventory or product platform, but the two products can be partal substtuted for each other, such that one of the product wth better characterstcs can be used to substtute for the other one to meet demand, but the reverse substtuton can t be accepted or prce of the product wth better characterstcs s general hgher than the other one and the substtuton s relatve to tradeoff between product prce and fll rate of customer demand. Besdes, some mportant factors should be consdered such as product characterstcs (electroncs, automotve, clothng, and so on), specal delvery tme wndow, customzaton tme from generc nventory, the optmal numbers of parts n the generc product, or the rato of the volume of generc product to fnal customzed product, the dffculty degree of customzed assembly and packagng, etc. whch wll nfluence the postponement cost, delvery tme and customer satsfacton. REFERENE Anupnd, R. and L. Jang, 008. Apacty nvestment under postponement strateges, market competton, and demand uncertanty. Manag. c., 54(): hod, J., D. Pyke and N. Rud, 006. He Value of Postponement: Market Drvers and Input ommonalty Workng Paper, arroll chool of Management, Boston ollege hstnut Hll. Graman, G.A., 00. Partal-postponement decson cost model. Euro. J. Operat. Res., 0(): Graman, G.A. and M.J. Magazne, 00. Numercal analyss of capactatedpostponement Prod. Operat. Manag., (3): Lee, H.L. and Tang.., 997. Modelng the costs amd benfts of delayed product dfferentaton. Manage. c., 43(3): L,., B. Ragu-Nathan and T.. Ragu-Nathan, 007. He mpact of supply chan management practces on compettve advantage and organzatonal performance. Omega, 34(): Ronald,. and Y. Bassok, 004. An nventory model for delayed customzaton: A hybrd approach. Europ. J. Operat. Res., 65(3): Wong, H., J. Wkner and M. Nam, 009. Nalyss of form postponement based on optmal postonng of the dfferentaton pont and stockng decsons. Int. J. Prod. Res., 47(5): 0-4. Wong, H., A. Potter and M. Nam, 00. Valuaton of postponement n a soluble coffee supply chan: A case study Int. J. Product. Econ., 3():