The Inter-dependent Reductions of Lead Time and Ordering Cost in. Periodic Review Inventory Model with Backorder Price Discount.
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1 e Inter-dependent Reductons of ead me and Orderng Cost n Perodc Revew Inventory Model wt Bacorder Prce Dscount ang-yu Ouyang * Bor-Ren Cuang amang Unversty a Hwa Insttute of ecnology R.O.C. R.O.C. Yu-Jen n S. Jon s Unversty R.O.C. Abstract s paper nvestgates te lead tme and orderng cost reductons are nter-dependent n te perodc revew nventory model wt bacorder prce dscount. e objectve s to mnmze te total related cost by smultaneously optmzng te revew perod lead tme and bacorder prce dscount. e protecton nterval demand s assumed to be normally dstrbuted. A procedure of fndng te optmal soluton s developed and two numercal eamples are gven to llustrate te results. Keywords: nventory; perodc revew; protecton nterval; bacorder prce dscount * ts researc was supported n part by te Natonal Scence Councl awan ROC under NSC 9--E--5.
2 . Introducton In tradtonal economc order quantty EOQ lterature dealng wt nventory problems eter usng determnstc or probablstc models lead tme s vewed as a prescrbed constant or a stocastc varable. erefore lead tme s not subject to control see e.g. Naddor [9] Jonson and Montgomery [5] Slver and Peterson []. However ts may not be realstc. ead tme usually conssts of te followng components: order preparaton order transt manufacture & assembly transt and uncratng nspecton and transport see ersne [5 p.5]. In some practcal cases lead tme can be sortened at an added crasng cost; n oter words t s controllable. By sortenng lead tme we can lower te safety stoc reduce te stocout loss and mprove te servce level to te customer so as to ncrease te compettve edge n busness. Also troug te Japanese eperence of usng Just-In-me JI producton te advantages and benefts assocated wt efforts to control te lead tme can be clearly perceved. Recently tere as been some nventory model lterature consderng lead tme as a decson varable. Intally ao and Syu [6] presented an nventory model n wc lead tme s a unque decson varable and te order quantty s predetermned. Ben-Daya and Raouf [] etended ao and Syu s [6] model to permt bot te lead tme and te order quantty as decson varables. In 996 Ouyang et al. []
3 generalzed Ben-Daya and Raouf s [] model by allowng sortages wt partal bacorders. ater Moon and Cos [8] and Harga and Ben-Daya [] modfed Ouyang et al. s [] model to consder te reorder pont as anoter decson varable. Also based on te Ouyang et al. s [] model Pan and Hsao [] furter dscussed te nventory problem of bacorder prce dscount. It s notced tat tese papers ao and Syu [6] Ben-Daya and Raouf s [] Ouyang et al. s [] Moon and Cos [8] Harga and Ben-Daya [] and Pan and Hsao [] are focusng on te contnuous revew nventory model to derve te benefts from lead tme reducton and te orderng cost s treated as a fed constant. However for te perodc revew nventory model lead tme as a decson varable as rarely been dscussed. e applcatons of te perodc revew nventory model can often be found n managng nventory cases suc as smaller retal stores drugstores and grocery stores see for eample aylor III [ p.779]. In a recent artcle Ouyang and Cuang [] proposed a perodc revew nventory model to study te effects of lead tme and orderng cost reductons. We note tat reducng lead tme and orderng cost n Ouyang and Cuang [] are assumed to act ndependently; owever ts s only one of te possble stuatons. e lead tme and orderng cost reductons may be related closely; te reducton of lead tme may accompany te reducton of orderng cost and vce versa. For eample te mplementaton of
4 electronc data ntercange EDI can reduce bot te lead tme and orderng cost smultaneously see Slver and Peterson [ p.5]. erefore t s more reasonable to assume tat lead tme and orderng cost reductons are dependent and ter functonal relatonsp may be as lnear logartmc eponental and te le. In te real maret as unsatsfed demands occur we can often observe tat some customers may prefer ter demands to be bacordered and some may refuse te bacorder case. Wen a sortage occurs many factors may affect te customers wllngness of acceptng bacorders. For eample for well-famed products or fasonable goods suc as certan brand gum soes -f equpment cosmetcs and clotes customers may prefer to wat for bacorders. Besdes tere s a potental factor tat may motvate te customers desre for bacorders. e factor s an offerng of a bacorder prce dscount from te suppler see Pan and Hsao []. In general provded tat a suppler could offer a bacorder prce dscount on te stocout tem by negotaton to secure more bacorders t may mae te customers more wllng to wat for te desred tems. In oter words te bgger te bacorder prce dscount te bgger te advantage to te customers and ence a larger number of bacorder rate may result. s penomenon reveals tat as unsatsfed demands occur durng te stocout perod ow to fnd an optmal bacorder rate troug controllng a bacorder prce dscount from a suppler to mnmze te relevant
5 nventory total cost s a decson-mang problem wort dscussng. e purpose of ts paper s to study te effect of lead tme reducton on te perodc revew nventory system wt partal bacorders. Specfcally we modfy Ouyang and Cuang s [] model to nclude te controllable bacorder prce dscount and te reducton of lead tme accompanes a decrease of orderng cost. e objectve s to mnmze te total related cost by smultaneously optmzng te revew perod bacorder prce dscount and lead tme. s paper s organzed as follows. In te net secton te notaton and assumptons are presented. e model n wc te protecton nterval demand follows a normal dstrbuton s formulated n Secton. wo numercal eamples are provded to llustrate te proposed models n Secton and Secton 5 s a summary of te wor done n ts artcle.. Notaton and assumptons Frst of all te followng notaton and assumptons are employed trougout ts paper so as to develop te proposed models. Notaton : D average demand per year nventory oldng cost per unt per year 5
6 R target level A orgnal orderng cost before any nvestment s made A orderng cost per order < A A upper bound of te bacorder rate < fracton of te sortage tat wll be bacordered.e. bacorder rate a decson varable margnal proft.e. cost of lost demand per unt bacorder prce dscount offered by te suppler per unt a decson varable lengt of a revew perod a decson varable lengt of lead tme a decson varable X te protecton nterval demand wc as a normal probablty densty functon p.d.f. f X wt fnte mean D and standard devaton were denotes te standard devaton of te demand per unt tme E matematcal epectaton mamum value of and.e. Ma{ }. Assumptons :. e nventory level s revewed every unts of tme. A suffcent quantty s ordered up to te target level R and te orderng quantty s arrved after 6
7 unts of tme.. e lengt of te lead tme does not eceed an nventory cycle tme so tat tere s never more tan a sngle order outstandng n any cycle. at s.. e margnal proft per unt s greater tan te nventory oldng cost per unt per revew cycle.e. >.. e target level R epected demand durng te protecton nterval safety stoc SS and SS standard devaton of protecton nterval demand.e. R D were s te safety factor and satsfes P X > R q q represents te allowable stocout probablty durng te protecton nterval and s gven. 5. e lead tme conssts of n mutually ndependent components. e -t component as a mnmum duraton a and normal duraton b and a crasng cost per unt tme c. Furter for convenence we rearrange c suc tat c c... cn. en t s clear tat te reducton of lead tme sould be frst on component because t as te mnmum unt crasng cost and ten component and so on. n 6. If we let b j and be te lengt of lead tme wt components j crased to ter mnmum duraton ten can be epressed as 7
8 b b a n ; and te lead tme crasng cost n j j j j j C per cycle for a gven [ ] s gven by C c c b a. j j j j 7. e reducton of lead tme accompanes a decrease of orderng cost A and A s a strctly concave functon of.e. A > and A <. 8. Durng te stocout perod te bacorder rate s varable and s n proporton to te bacorder prce dscount offered by te suppler per unt. us were < and see Pan and Hsao [].. Model formulaton As mentoned earler we ave assumed tat te protecton nterval.e. revew perod plus lead tme demand X as a p.d.f. f X wt fnte mean D and standard devaton and te target level R D were s defned as assumpton. Usng te same approac as n Montgomery et al. [7] for te perodc revew case te epected net nventory at te begnnng of te perod s R D E X R and te epected net nventory at te end of te perod s R D D E X R. erefore te epected oldng D cost per year s appromately R D E X R epected stocout cost per year s [ ] E X R and te were E X R 8
9 s te epected demand sort at te end of cycle. By assumptons -7 te total epected annual cost wc s composed of orderng cost nventory oldng cost stocout cost and lead tme crasng cost s epressed by A D R D [ ] E X R E X R C. Note tat te smlar model was proposed by Ouyang and Cuang [] wt te followng features:. e bacorder rate s gven.. e orderng cost A s treated as a fed constant and ndependent of lead tme reducton. Moreover snce R D te epected demand sortage at te end of te cycle E X R R f d were R φ [ Φ ] > φ and Φ denote te standard normal p.d.f. X and dstrbuton functon d.f. respectvely. Furtermore by assumpton 8 durng te stocout perod te bacorder rate s varable and s proportonal to te bacorder prce dscount offered by te suppler per unt tat s. us te bacorder prce dscount offered by te suppler per unt can be treated as a decson varable nstead of 9
10 te bacorder rate. us te total epected annual cost n Equaton becomes C A D H were H > because > >. e problem s to fnd te optmal values of and suc tat n s mnmzed. ang te frst partal dervatves of wt respect to and [ ] respectvely. We obtan C A D H H and c A H. 5 By eamnng te second order suffcent condtons t can be easly verfed tat s not a conve functon of. However for fed and
11 s concave n [ ] because A / H / <. erefore for fed and te mnmum total epected annual cost wll occur at te end ponts of te nterval [ ]. On te oter and for a gven value of [ ] te mnmum value of wll occur at te pont wc satsfes and smultaneously. Furtermore we can sow tat for any gven value of [ ] s postve defnte at te statonary pont see Append for a detal proof. By settng Equatons and equal to zero we obtan A C H D H 6 and. 7 Substtutng Equaton 7 nto Equaton 6 leads to A C S D S 8 were H S.
12 Usng te Newton-Rapson metod to fnd te optmal soluton of suc tat 8 olds. erefore we can establs te followng algortm to fnd te optmal value of and. Algortm Step. For eac n and a gven q and ence te value of can be found drectly from te standard normal dstrbuton table compute from Equaton 8. If go to Step; oterwse let and go to Step. Step. Compute from Equaton 7 and ten compare wt. If If > s feasble ten go to Step. s not feasble. Set and calculate te correspondng value of from Equaton 6 ten go to Step. Step. For eac compute te correspondng total epected annual cost n. Step. Fnd Mn... n * *. If * Mn... n * * * ten s te optmal soluton. Hence we obtan te optmal target level s R * * * * * D te * * optmal bacorder rate s and te optmal orderng cost A * A * follows.
13 . Numercal eamples In order to llustrate te above mentoned soluton procedure let us consder an nventory system wt te data used n Ouyang and Cuang []: D 6 unts per year A $ per order $ per unt per year $5 per unt 7 unts per wee te protecton nterval demand follows a normal dstrbuton and te lead tme as tree components wt data sown n able. Insert able Eample. We assume tat lead tme and orderng cost reductons act dependently wt te followng relatonsp see Cen et al.[] and Cu []: A A A δ wc mples A a b were δ > a A A and b. We attempt to solve te cases wen te upper bound δ δ of te bacorder rate.8 δ and q. n ts stuaton te value of safety factor can be found drectly from te standard normal dstrbuton table and s.5. Applyng te Algortm procedure yelds te results as tabulated n able. From ts table optmal nventory polcy for eac case of δ can easly be found by comparng and tus we summarze tese n able. Furtermore n order to see te effect of lead
14 tme reducton wt nteracton of orderng cost we lst te results of fed orderng cost model settng A $ per order.e. tae δ n te same table. From te results sown n able t reveals tat as te value of δ decreases te larger savngs of total epected annual cost are obtaned comparng te result wt fed orderng cost model. And t s nterestng to observe tat decreasng te value δ wll result n a decrease n te total epected annual cost te revew perod te bacorder prce dscount and te target level. Insert able Insert able Eample. e data are te same as n Eample. We assume tat te lead tme and orderng cost reductons act dependently wt te followng relatonsp see Cen et al.[]: A A τ ln wc mples A f g ln were τ < A f τ A ln and g τ A >. We solve te cases wen te upper bound of te bacorder rate.8 τ..5.8 and q. t mples.5. Applyng te smlar Algortm procedure yelds te results as tabulated n able. From ts table te optmal nventory polcy for eac case of τ can be
15 found by comparng and tus we summarze tese n able 5. Moreover n order to observe te relatonsps between lead tme and orderng cost we lst te results of fed orderng cost model.e. tae τ n te same table. From te results sown n able 5 we see tat as te value of τ decreases te larger savngs of total epected annual cost are obtaned comparng te result wt fed orderng cost model. On te oter and decreasng te value τ wll result n a decrease n te total epected annual cost te revew perod te bacorder prce dscount and te target level. Insert able Insert able 5 5. Concludng Remars s paper modfes te wor of Ouyang and Cuang s [] model wt varable bacorder prce dscount and te reducton of lead tme accompanes a decrease of orderng cost. at s we mnmze te total epected annual cost by optmzng te revew perod te bacorder prce dscount and te lead tme. Under te assumpton tat te protecton nterval demand s normally dstrbuted an algortm procedure of fndng te optmal solutons s establsed. e results of numercal 5
16 6 eamples ndcate tat wen te reducton of lead tme accompanes a decrease of orderng cost te larger savngs of total epected annual cost can be realzed. In future researc on ts problem t would be nterestng to deal wt a med stocastc nventory model wt te dstrbuton free case were only te mean and standard devaton of protecton nterval demand are nown and fnte. Append Proof of s postve defnte at te statonary pont. For fed [ ] we frst obtan te Hessan matr H as follows H. en we proceed by evaluatng te prncpal mnor of H at statonary pont. ang te frst partal dervatve of wt respectve to and and settng te obtaned results equal to zero results n C A D H H A and
17 7. A From A we obtan D C A H H A or equvalently C A D / / H H. A From A t results n. A5 Net we obtan te second order partal dervatves as follows. ζ / A6 were ζ [ ] C A H H H. > A7 and
18 8 by A5. A8 If we let η C A H H. A9 en from A t mples > η /. A erefore from A6 and A we obtan η ζ > [ ] > H C A A and tus > H. Furter te second prncple mnor of H s H [ ] > H C A
19 9 > H H [ ] by equaton 7 >. erefore t s clear to see tat for fed [ ] s postve defnte at te statonary pont. Acnowledgements s researc was support by te Natonal Scence Councl of awan under rant NSC 9--E--5. References [] Ben-Daya M. and Raouf A. Inventory models nvolvng lead tme as decson varable Journal of te Operatonal Researc Socety Vol.5 pp [] Cen C.K. Cang H.C. and Ouyang.Y. A contnuous revew nventory model wt orderng cost dependent on lead tme Internatonal Journal of Informaton and Management Scences Vol. No. pp.-. [] Cu P. P. Economc producton quantty models nventory nvolvng lead tme
20 as a decson varable Master tess Natonal awan Unversty of Scence and ecnology 998. [] Harga M. and Ben-Daya M. Some stocastc nventory models wt determnstc varable lead tme European Journal of Operatonal Researc Vol. pp [5] Jonson.A. and Montgomery D.C. Operatons Researc n Producton Plannng Scedulng and Inventory Control Jon Wley & Sons New Yor 97. [6] ao C.J. and Syu C.H. An analytcal determnaton of lead tme wt normal demand Internatonal Journal of Operatons & Producton Management Vol. pp [7] Montgomery D.C. Bazaraa M.S. and Keswan A.I. Inventory models wt a mture of bacorders and lost sales Naval Researc ogstcs Vol. pp [8] Moon I. and Cos S. A note on lead tme and dstrbutonal assumptons n contnuous revew nventory models Computers and Operatons Researc Vol.5 pp [9] Naddor E. Inventory System Jon Wley New Yor 996. [] Ouyang.Y. and Cuang B.R. A perodc revew nventory-control system wt varable lead tme Internatonal Journal of Informaton and Management Scences Vol. No. pp.-. [] Ouyang.Y. Ye N.C. and Wu W.S. Mture nventory model wt bacorders and lost sales for varable lead tme Journal of te Operatonal Researc Socety Vol.7 pp [] Pan C.H. and Hsao Y.C. Inventory models wt bac-order dscounts and
21 varable lead tme Internatonal Journal of System Scence Vol. pp [] Slver E.A. and Peterson R. Decson Systems for Inventory Management and Producton Plannng Jon Wley New Yor 985. [] aylor III B.W. Introducton to Management Scence st edton Hall New Jersey 999. [5] ersne R.J. Prncples of Inventory and Materals Management Nort Holland New Yor 98.
22 able ead tme data ead tme Normal Mnmum Unt crasng Component duraton duraton cost b days a days c $/day
23 able Soluton procedures of Eample n wees δ C A R $ $
24 * * able Summary of te optmal soluton of Eample n wees δ * A * * * * * * * R Savng % $ $ Note: Savng s based on te fed orderng cost model.e. δ.
25 able Soluton procedures of Eample n wees τ C A R
26 ** ** able 5 Summary of te optmal soluton of Eample n wees τ ** ** A ** ** ** ** ** ** R Savng %. $ $ Note: Savng s based on te fed orderng cost model.e. τ. 6
27 Autors Informaton ang-yu Ouyang s a Professor n te Department of Management Scences & Decson Mang at amang Unversty n awan. He earned s B.S. n Matematcal Statstcs M.S. n Matematcs and P.D. n Management Scences from amang Unversty. Hs researc nterests are n te feld of Producton/Inventory Control Probablty and Statstcs. He as publcatons n Journal of te Operatonal Researc Socety Computers and Operatons Researc European Journal of Operatonal Researc Computers and Industral Engneerng Internatonal Journal of Producton Economcs IEEE ransactons on Relablty Sany a Metra Producton Plannng & Control Journal of te Operatons Researc Socety of Japan Opsearc Journal of Statstcs & Management Systems Journal of Interdscplnary Matematcs Internatonal Journal of Informaton and Management Scences Internatonal Journal of Systems Scence Yugoslav Journal of Operatons Researc e Engneerng Economst Matematcal and Computer Modellng Appled Matematcal Modellng Internatonal Journal of Advanced Manufacturng ecnology and Journal of lobal Optmzaton. raduate Insttute of Management Scences amang Unversty amsu ape Hsen awan 5 R.O.C. Emal: langyu@mal.tu.edu.tw el: et. 75 Bor-Ren Cuang s an assstance professor n te Department of Internatonal rade Department at a Hwa Insttute of ecnology n awan. Hs researc nterest les n te feld of te analyss of nventory systems. He as publsed artcles n Computers & Industral Engneerng Computers & Operatons Researc Internatonal Journal of Systems Scence Journal of te Operatons Researc Socety 7
28 of Japan Producton Plannng & Control and Yugoslav Journal of Operatons Researc. Department of Internatonal rade a Hwa Insttute of ecnology Hsncu awan 5 R.O.C. Emal: 79@mal.tu.edu.tw Yu-Jen n s an assocate professor n te Holstc Educaton Center at S. Jon s Unversty n awan. He earned s B.S. and M.S. n matematcs and P.D. n Management Scences from amang Unversty n awan. Hs researc nterests are n te feld of producton/nventory control. He as publsed artcles n Yugoslav Journal of Operatons Researc Journal of te Cnese Insttute of Industral Engneers Internatonal Journal of Informaton and Management Scences Journal of Statstcs & Management Systems amsu Oford Journal of Matematcal Scences and Journal of Informaton & Optmzaton Scences Holstc Educaton Center S. Jon s Unversty amsu ape Hsen awan 5R.O.C. Emal: lyj@mal.sju.edu.tw el: et
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