Inventory Model with Backorder Price Discount

Transcription

1 Vol. No. 7-7 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount Yu-Jen n Receved: Mar. 7 Frst Revson: May. 7 7 ccepted: May. 7 bstract The stochastc nventory model analyzed n ths paper eplore the problem that the lead tme and orderng cost reductons are nterdependent n the contnuous revew nventory model wth bacorder dscount. The objectve of ths paper s to mnmze the total related cost by smultaneously optmzng the order quantty lead tme and bacorder prce dscount. Moreover the lead tme demand s assumed to be normally dstrbuted. procedure of fndng the optmal soluton s developed. Furthermore the senstvty analyss s ncluded and two numercal eamples are gven to llustrate the results. Keywords: nventory lead tme bacorder prce dscount. Holstc Educaton Center St. John s Unversty - -

2 Vol. No. 7. Introducton In most of early lterature dealng wth nventory problems ether usng determnstc or probablstc models lead tme s vewed as a prescrbed constant or a stochastc varable whch therefore s not subject to control see e.g. Johnson and Montgomery 97 Naddor 9 Slver and Peterson 95. However ths may not be realstc. In some practcal cases lead tme can be shortened at an added crashng cost; n other words t s controllable. By shortenng the lead tme we can lower the safety stoc reduce the stocout loss and mprove the servce level to the customer so as to ncrease the compettve edge n busness. On the other hand n the real maret at the manufacturng level a supply stocout of an tem usually results n a bacorder whle at the retal level a stocout may result n a lost sale because the customer wll refuse the bacorder case and go to elsewhere to mae the purchase. Therefore when the nventory system unsatsfed demands occur durng the stocout perod how to provde the prce dscount from suppler such that the customers wllng to wat for the bacorder whch s very mportant. The prce dscount s a potental factor that may motvate the customers desre for bacorders. In general provded that a suppler could offer a prce dscount on the stocout tem by negotaton to secure more bacorders t may mae the customers more wllng to wat for the desred tems. In other words the bgger the dscount the bgger the advantage to the customers and hence a larger number of bacorder rato may result. Ths phenomenon reveals that as unsatsfed demands occur durng the stocout perod how to fnd an optmal bacorder rato through controllng a prce dscount from a suppler to mnmze the relevant nventory total cost s a decson-mang problem worth dscussng. In recent years several authors have presented varous nventory models wth lead tme reducton. Intally ao and Shyu 99 presented an nventory model n whch lead tme s a unque decson varable and the order quantty s predetermned. Ben-aya and Raouf 99 etended ao and Shyu s 99 model by allowng both the lead tme and the order quantty as decson varables. Ouyang et al. 99 generalzed Ben-aya and Raouf s 99 model by allowng shortages wth partal bacorders. Pan and Hsao revsed Ouyang et al. s 99 model to consder the bacorder dscount as one of the decson varables whle Chuang et al. etended Pan and Hsao model develop a mnma dstrbuton free procedure for nventory model wth bacorder dscount and varable lead tme. It s notced that the above papers Ben-aya and Raouf 99 Chuang et al. ao and Shyu s 99 Ouyang et al. s 99 Pan and Hsao are all focusng on the contnuous revew nventory model to derve the benefts from lead tme reducton and the orderng cost s treated as a fed constant. In a recent paper Ouyang et al. 999 proposed contnuous revew nventory model to study the effects of lead tme and orderng cost - -

3 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount reductons. We note that the lead tme and orderng cost reductons n Ouyang et al. s models 999 are assumed to act ndependently; however ths s only one of the possble cases. In practce the lead tme and orderng cost reductons may be related closely; the reducton of lead tme may accompany the reducton of orderng cost and vce versa. For eample accordng to Slver and Peterson 95 the mplementaton of electronc data nterchange EI may reduce the lead tme and orderng cost smultaneously. Therefore t s more reasonable to assume that orderng cost reducton vary accordng to dfferent lead tmes. nd then ther functonal relatonshp may be as lnear logarthmc eponental and the les. In ths paper s to nvestgate the effect of lead tme reducton on a contnuous revew nventory model ncludes the controllable bacorder prce dscount and the reducton of lead tme accompanes a decrease of orderng cost. We assume that the lead tme demand s normal dstrbuton and see to mnmze the total related cost by optmzng the order quantty bacorder dscount and lead tme smultaneously. The rest of ths paper s organzed as follows. In the net secton the notaton and assumptons are presented. The model that the lead tme demand has perfect nformaton s formulated n Secton. Two numercal eamples are provded to llustrate the proposed model and senstvty analyss of the optmal solutons wth respect to parameters are also ndcated n Secton and Secton 5 s a conclusons.. Notaton and assumptons The notaton used here are: = average demand per year = orgnal orderng cost = orderng cost per order < h = nventory holdng cost per unt per year = order quantty a decson varable r = reorder pont = fracton of the shortage that wll be bacordered.e. bacorder rato < = upper bound of the bacorder rato = bacorder prce dscount offered by the suppler per unt a decson varable = margnal proft.e. cost of lost demand per unt X = length of lead tme a decson varable = lead tme demand f X = the probablty densty functon p.d.f. of X wth fnte mean and standard devaton where denotes the standard devaton of the demand per unt - 5 -

4 Vol. No. 7 tme E = mathematcal epectaton = mamum value of and.e. = Ma{ } The assumptons of the model are:. The reorder pont r = epected demand durng lead tme safety stoc SS and SS = standard devaton of lead tme demand.e. r = where s the safety factor and satsfes P X > r = q q s gven to represent the allowable stocout probablty durng.. Inventory s contnuously revewed. Replenshments are made whenever the nventory level falls to the reorder pont r.. The lead tme conssts of n mutually ndependent components. The -th component has a mnmum duraton a and normal duraton b and a crashng cost per unt tme c. Further for convenence we rearrange c such that c c... c. n Then t s clear that the reducton of lead tme should be frst on component because t has the mnmum unt crashng cost and then component and so on. n. We let = b j and be the length of lead tme wth components j= crashed to ther mnmum duraton then can be epressed as = b b a = n; and the lead tme crashng cost C per cycle n j j= j= j j for a gven [ ] s gven by C = c c b a. j j j j= 5. The reducton of lead tme accompanes a decrease of orderng cost and s a strctly concave functon of.e. > and.. The suppler maes decsons n order to obtan profts. Therefore f the prce dscount s greater than the margnal proft then the suppler may decde aganst offerng the prce dscount. 7. urng the stocout perod the bacorder rato s varable and s n proporton to the prce dscount offered by the suppler per unt. The bacorder rate s defned as = where < and. - -

5 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount. Models formulaton In a recent study Pan and Hsao and Chuang et al. consder the nventory model wth bacorder dscount and varable lead tme. They assumed that the orderng cost s treated as a fed constant and ndependent of lead tme. In ths study we wll closely follow Pan and Hsao and Chuang et al. whch the lead tme demand follows the normal dstrbuton for the bacorder dscount on the contnuous revew nventory model nvolvng the orderng cost dependent on lead tme. Specfcally by assumptons -5 the total epected annual cost whch s composed of orderng cost nventory holdng cost stocout cost and lead tme crashng cost s epressed by = h r E X r [ ] E X r C where E X r s the epected demand shortage at the end of cycle. Further by assumpton 7 durng the stocout perod the bacorder rato s varable and s proporton to the prce dscount offered by the suppler per unt that s =. Therefore the bacorder prce dscount offered by the suppler per unt can be treated as a decson varable nstead of the bacorder rato. That s the objectve of cost functon s to mnmze the followng total epected annual cost. = h r E X r E X r C Moreover when the lead tme demand X follows a normal dstrbuton has a p. d. f. fx wth fnte mean and standard devaton by usng r = the epected demand shortage at the end of the r cycle E X r = r f d = where X [ Φ ] > φ φ and Φ denote the standard normal probablty densty functon p.d.f. and dstrbuton functon d.f. respectvely. Thus the total epected annual cost functon can be transformed nto followng formulaton.

6 Vol. No = h C G h where G = > because > >. In order to determne the optmal va lues of and respectvely such that n s mnmzed. Tang the frst partal dervatves of wth respect to and [ ] respectvely. We obtan: C G h = h = 5 and c G h h =. By eamnng the second order suffcent condtons SOSC t can be easly verfed that s not a conve functon of. However for fed and s concave n [ ] because = h G h <. Therefore for fed and the mnmum total epected annual cost wll occur at the end ponts of the nterval [ ]. On the other hand for a gven value of [ ] s conve n both and see ppend for a detal proof. Thus for fed [ ] the mnmum value of wll occur at

7 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount the pont whch satsfes = smultaneously. By settng equatons and 5 equal to zero we obtan G C = h and and = 7 = h respectvely. Substtutng equaton nto equaton 7 and smplfy we get C = h h. 9 It note that from 7 and G > the order quantty s greater than zero. Theoretcally for gven [ ] and whch depends on the allowable stocout probablty q and the p.d.f. f X from equatons and 9 we can obtan the optmal soluton such that the total epected annual cost has mnmum values. Therefore we develop the followng algorthm to fnd the optmal soluton for the order quantty prce dscount and lead tme. lgorthm Step. For each = n and a gven q and hence the value of can be found drectly from the standard normal dstrbuton table compute from equaton 9 and then determne from equaton. nd compare and. If s feasble then go to Step. If > s not feasble. Set = and calculate the correspondng value of from equaton 7 then go to Step

8 Vol. No. 7 Step. For each compute the correspondng total epected annual cost = n. Step. Fnd Mn =... n. If = Mn =... n then s the optmal soluton. Once we obtan the the optmal reorder pont s r = the optmal bacorder rato s = and the optmal orderng cost = follows.. Numercal eamples The numercal eamples gven below llustrate the above soluton procedure. We consder the contnuous revew nventory system wth the followng data used n Pan and Hsao Chuang et al. : = unts per year = $ per order h = $ per unt per year $5 per unt 7 unts per wee. Besdes we assume that the lead tme has three components wth data shown n Table. Table ead tme data ead tme Normal Mnmum Unt crashng Component duraton duraton cost b days a days c $/day Eample. We assume that lead tme and orderng cost reductons act dependently wth the followng relatonshp see Chen et al.: = λ whch mples = a b where λ > a = and b =. We attempt to solve the λ λ cases when the upper bound of the bacorder rato =. and the scalng parameter - -

9 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount λ = and q =. n ths stuaton the value of safety factor can be found drectly from the standard normal dstrbuton table and s.5. pplyng the lgorthm procedure yelds the results as tabulated n Table. From ths table the optmal nventory polcy can easly be fo und by comparng for = and thus we summarze these n Table. Furthermore n order to see the effect of lead tme reducton wth nteracton of orderng cost we lst the results of fed orderng cost model settng = $ per order.e. tae λ = n the same table. From the results shown n Table t reveals that as the value of λ decreases the larger savngs of total epected annual cost are obtaned comparng the result wth fed orderng cost model. nd t s nterestng to observe that decreasng the value λ wll result n a decrease n the total epected annual cost the order quantty the bacorder prce dscount. Table Soluton procedures of Eample n wees λ C r

10 Vol. No. 7 Besdes we further eamne the effects of changes n the system parameters h and on the optmal order quantty optmal bacorder dscount and mnmum total epected annual cost n Eample. When the upper bound of the bacorder rato =. and the scalng parameter λ = a senstvty analyss s performed by changng each of the parameters by 5% 5% 5% and 5% tang one parameter at a tme and eepng the remanng parameters unchanged. The results are shown n Table. Table Summary of the optmal soluton of Eample n wees λ r Savng % Note: Savng s based on the fed orderng cost.e. λ =. Table Effects of changes n the parameters of the nventory model of Eample % change n parameters % change h

11 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount On the bass of the results of Table the followng observatons can be made. and ncrease whle decreases wth an ncrease n the value of the model parameter h. The obtaned results show that and are hghly senstve whereas s lowly senstve to the changes n h. and ncrease whle decreases wth an ncrease n the value of the model parameter. Moreover and are hghly senstve whereas s lowly senstve to the changes n. and ncrease wth an ncrease n the value of the model parameter. Moreover and are moderately senstve whereas s lowly senstve to the changes n. v and ncrease wth an ncrease n the value of the parameter. Moreover and are moderately senstve whereas s hghly senstve to the changes n. Eample. The data are the same as n Eample and we assume that the lead tme and orderng cost reductons act dependently wth the followng relatonshp see Chen et al.: = µ ln whch mples = f g ln where µ < - -

12 Vol. No. 7 f = µ ln and g = µ. We solve the cases when the upper bound of the > bacorder rato =. and the scalng parameter µ = and q =.. Utlzng the smlar procedure as lgorthm we obtan the results as tabulated n Table 5. From ths table the optmal nventory polcy can easly be found by comparng for = and thus we summarze these n Table. Moreover n order to observe the relatonshps between lead tme and orderng cost we lst the results of fed orderng cost model.e. tae µ = n the same table. From the results shown n Table 5 we see that as the value of µ decreases the larger savngs of total epected annual cost are obtaned comparng the result wth fed orderng cost model. On the other hand decreasng the value µ wll result n a decrease n the total epected annual cost the order quantty the bacorder prce dscount. Table 5 Soluton procedures of Eample n wees µ C r Table Summary of the optmal soluton of Eample n wees - -

13 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount µ r Savng % Note: Savng s based on the fed orderng cost model.e. µ =. 5. Conclusons The prmary purpose of ths study nvestgates the nventory system wth varable lead tme and bacorder prce dscount and the reducton of lead tme accompanes a decrease of orderng cost. We see to mnmze the total epected annual cost by smultaneously optmzng the order quantty bacorder dscount or bacorder prce dscount and lead tme. Under the assumpton that the lead tme demand s normally dstrbuted an algorthm procedure of fndng the optmal solutons s establshed. Two numercal eamples results show that when the reducton of lead tme accompanes a decrease of orderng cost and comparng the fed orderng cost model ndcate that can acheve a sgnfcant amount of savng the total epected annual cost and senstvty analyss of the optmal solutons wth respect to parameters are also ndcated. In future research on ths problem t would be nterestng to deal wth a med stochastc nventory model wth the dstrbuton free case where only the mean and standard devaton of lead tme demand are nown and fnte. Moreover a possble etenson of ths wor may tae orderng cost as one decson varable. cnowledgements The author greatly apprecates the anonymous referees for several helpful comments and suggestons on an earler verson of the paper. ppend For gven value of we frst obtan the Hessan matr H s follows: H = mnor determnant of H. The frst prncpal mnor determnant of H s:. Then we proceed by evaluatng the prncpal - 5 -

14 Vol. No > = = C G H The second prncple mnor determnant of H s: H = = C G [ ] [ ] C = [ ] >. Therefore t s clear to see that for gven [ ] s a conve functon n.

15 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount References. Ben-aya M. and. Raouf 99 Inventory models nvolvng lead tme as decson varable Journal of the Operatonal Research Socety 5 pp Chuang B. R.. Y. Ouyang and Y. J. n mnma dstrbuton free procedure for med nventory model wth bacorder dscounts and varable lead tme Journal of Statstcs & Management Systems 7 pp Chen C. K. H. C. Chang and. Y. Ouyang contnuous revew nventory model wth orderng cost dependent on lead tme Internatonal Journal of Informaton and Management Scences pp. -.. Gallego G. and I. Moon 99 The strbuton Free Newsboy Problem: Revew and Etensons Journal of the Operatonal Research Socety pp Johnson.. and. C. Montgomery 97 Operatons Research n Producton Plannng Schedulng and Inventory Control New Yor: John Wley.. ao C. J. and C. H. Shyu 99 n analytcal determnaton of lead tme wth normal demand Internatonal Journal of Operatons & Producton Management pp Naddor E. 9 Inventory System New Yor : John Wley.. Ouyang. Y. C. K. Chen and H. C. Chang 999 ead tme and orderng cost reductons n contnuous revew nventory systems wth partal bacorders Journal of the Operatonal Research Socety 5 pp Ouyang. Y. N. C. Yeh and W. S. Wu 99 Mture nventory model wth bacorders and lost sales for varable lead tme Journal of the Operatonal Research Socety 7 pp Pan C. H. and Y. C. Hsao Inventory models wth bac-order dscounts and varable lead tme Internatonal Journal of Systems Scence pp Slver E.. and R. Peterson 95 ecson Systems for Inventory Management and Producton Plannng New Yor: John Wley