International Journal of Industrial Engineering Computations

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1 Intnational Jounal of Industial Engining Computations 5 ( Contnts lists availabl at GowingScinc Intnational Jounal of Industial Engining Computations hompag: A nw modl fo dtioating itms with inflation und pmissibl dlay in paymnts R.P. ipathi a* and Manoj Kuma b a Dpatmnt of Mathmatics, Gaphic Ea Univsity, Dhadun (U.K. India b Dpatmnt of Mathmatics, Shivalik Collg of Engining, Dhadun (U.K. India C H R O N I C L E A B S R A C Aticl histoy: Rcivd Sptmb Rcivd in Rvisd Fomat Apil 6 4 Accptd Apil 8 4 Availabl onlin Apil 9 4 Kywods: Invntoy modl Exponntial dmand at Dtioation at Inflation Inflation is an impotant facto influncing taditional conomic od quality modls. Makting statgy dpnds on inflation du to public dmand and availability of th matials. his pap psnts an optimal invntoy policy fo dtioating itms using xponntial dmand at und pmissibl dlay in paymnts. Mathmatical modl has bn divd und two cass: cas I: cycl tim is gat than o qual to pmissibl dlay piod, cas II: cycl tim is lss than pmissibl dlay piod by considing holding cost as a function of tim. Numical xampls and snsitivity analysis a givn to flct th numical sults. Mathmatica softwa is usd fo finding optimal solutions. 4 Gowing Scinc Ltd. All ights svd. Intoduction h objctiv of many invntoy managmnt poblms is to dal with th minimization of invntoy caying xpnditus (Donalson, 977. hus, it is ncssay to dtmin th optimal invntoy lvl as wll as optimal tim of plnishmnt of invntoy to mt any futu dmand. Among th classical invntoy managmnt modls, th a many cass on solving th optimal od quantity by ignoing th typ of paymnt. At psnt scnaio, it is obsvd that suppli offs a ctain fixd piod to sttl th account fo stimulating tails dmand. Duing th cdit piod whn th paymnt is mad, som itms can b sold and vnu can b accumulatd to an intst. his pap invstigats invntoy modl fo dtioating itms with xponntial tim dpndnt dmand at. In cnt yas, dtioating invntoy modls hav bn widly studid. In al lif situations, it is obsvd that dmand fo a paticula poduct can b influncd by intnal facto such as inflation, pic and availability. h chang in th dmand is sponsibl fo th chang in invntoy is commonly fd to as dmand lasticity. Gnally, invntoy modl consids a cas in which dpltion of invntoy is causd by dmand at and dtioation. Most of th itms dtioat ov tim and this phnomna plays ssntial ol fo dcision making in modn oganization. Duing th past fw dcads, many sachs hav dvlopd invntoy modls fo dtioating itms. * Cosponding autho. tipathi_p@diffmail.com (R.P. ipathi 4 Gowing Scinc Ltd. All ights svd. doi:.567/j.ijic.4.4.6

2 66 h analysis of dtioating modl is discussd by Gha and Schad (96 with constant at of dtioation. In th ali piod, sachs hav discussd vaious dmand pattns fitting th stag of poduct lifcycl. Rsh t al. (976 and Donaldson (977 considd th situation and linaly tim vaying dmand and stablishd an algoithm to dtmin th optimal numb of plnishmnts and timing. Hny (976 futh gnalizd th dmand at by considing a concav dmand function incasing with tim. Hais (9 fist intoducd th basic conomic od quantity (EOQ modl. Sval intsting sach paps a associatd with dtioation, which a basd on constant dmand without any dtioation function (.g. Sachan, 984; Dav & Patl, 98; Goyal & Gii, ; Liao & Haung, ; Sana, ; Lin t al., ; Saka, ; tc.. Covt and Phillip (97 xtndd Gha and Schad s modl by considing vaiabl at of dtioation. Data and Pal (988 dvlopd an EOQ modl by intoducing a vaiabl dtioation at and pow dmand pattn. Chung and ang (994 dtmind th plnishmnt schduls fo dtioating itms with tim popotional dmand. Howv, in al lif, th dmand may incas o dcas in th cous of tim. Many sachs considd th vaying dmand (.g. Sana, ; Sana & Chaudhui, 8; Donaldson, 977; Goyal, 986; Khana & Chaudhui, ; Silv & Mal, 969; Hainga, 996; Goyal t al., 986. Vaious sachs hav dvlopd th inflanatoy ffcts on th invntoy policy. Liao t al. ( dvlopd an invntoy fo initial stock dpndnt consumption at whn a dlay in paymnt is pmissibl. Hou (6 divd an invntoy modl fo dtioating itms with stock dpndnt consumption at and shotags und inflation and tim discounting ov a finit planning hoizon. Buzacolt (975 dvlopd an invntoy modl with inflation. Vat and Padmanabhan (99 dvlopd an invntoy modl und a constant inflation at and initial stock dpndnt consumption at. Datta and Pal (99 dvlopd a modl with lina tim dpndnt dmand at and shotags to invstigat th ffcts of inflation with tim valu of mony on oding policy ov a finit tim hoizon. Rcntly, ng t al. ( dvlopd an EOQ modl with tad cdit financing fo non dcasing dmand and fundamntal thotical sults obtaind. Saka ( dvlopd an EOQ modl with dlay in paymnt fo tim vaying dtioation at and obtaind a function fo maximization of pofit. Picing and lot sizing policis fo dtioating itms with patial backlogging und inflation was psntd by Hsih and Dy ( by considing picing and lot sizing policis fo dtioating itms with patial backlogging und inflation. In this pap, dmand at is xponntial tim dpndnt and holding cost is tim dpndnt. h psnt modl is discussd by using tuncatd aylo s sis. h conditions fo convxity of optimality a obtaind and numical xampls and snsitivity analysis a givn. h st of th pap is oganizd as follows: In th nxt sction assumptions and notations a givn. In sction mathmatical fomulations with maximization of total invntoy cost is givn. In sction 4 numical xampls fo th cass I and II a givn. In sction 5 snsitivity analysis with vaious paamts is givn to validat th invntoy cost function. Finally conclusion and futu sach dictions a givn in th last sction 6.. Assumptions and Notations h mathmatical modl of invntoy fo dtioating itms is basd on th following assumptions: (i Dmand at is xponntial and Inflation is constant (ii Shotags a not allowd and lad tim is zo

3 R.P. ipathi and M. Kuma / Intnational Jounal of Industial Engining Computations 5 (4 67 (iii Duing th pmissibl dlay piod th sals vnu gnatd is dpositd in an intst baing account. At th nd of th tad cdit piod th custom pays off all units odd and bgins paying fo th intst chagd on th itms in stock. (iv h is no pai o plnishmnt of dtioatd itms duing th givn cycl. (vii Holding cost is tim dpndnt i.. h( t ht, t h following notations a usd thoughout th manuscipt: h : Holding cost p unit t p( t p : Instantanous slling pic p unit p : Slling pic p unit at t = C : Puchasing pic p unit at t = t C( t C : Instantanous oding cost p od C : Oding cost p od at t = H I c I : Lngth of finit planning hoizon : Intst chagd : Intst and p annum by th tail : Optimum lngth of cycl tim, H = n Q I(t R : Allowabl dlay piod duing sttlmnt of th account : Optimum oding quantity : Constant at of inflation : Instantanous lvl of invntoy A : Dmand at, A : Dmand at t = : Constant dtioation at ( < O C : Oding cost C D : Cost of dtioation I HC : Invntoy holding cost Z, Z : otal cost fo cas and spctivly. Mathmatical Fomulation h lvl of invntoy I(t dcass gadually mainly to mt dmands and du to dtioation. hus, th vaiation of invntoy with spct to tim can b dscibd by th following diffntial quation:

4 68 di( t I( t A dt t, t ( h solution of quation is givn by A I( t ( ( and od quantity A Q ( ( Sinc I(t is a piodic function with piod hnc w hav I(k + t = I(t, thfo, A I( k t ( (4 Oding Cost: Instantanous oding cost p od O C n kt ( H C C 4 k h numb of dtioatd units = h cost of total dtioatd units Q A CD C AC Invntoy Holding Cost I HC is givn by A t dt ( (5 (6 (7 n Ah k ( t I HC k C. tdt = H AhC ( ( (8 Dpnding on th custom s choic and th lngth of cycl tim two possibl cass a takn into account: Cas I Optimal cycl tim is gat than th pmissibl dlay tim, th intst chagd IC duing th piod [, H] is givn by IC I n c k C( k I( k t dt H cc ( ( (9 Intst and IE duing th piod [, H] is givn by

5 R.P. ipathi and M. Kuma / Intnational Jounal of Industial Engining Computations 5 (4 69 IE n I (. P k A k H ( t dt P ( C D HC otal Invntoy Cost Z O C I IC IE H H ( ( H cc P H AhC ( C ( AC 4 ( ( ( ( Diffntiating Z ( with spct to two tims w gt H Z H cc ( C( AC ( P ( H ( H H Z H 6 cc ( 6 P ( ( C ( ( ( Z Optimal (minimum solution is obtaind by solving, AC C ( C H Cas II ( H AC ( H cc ( ( I ( c H P ( H yilds ( (4 In this cas, th tail pays th pocumnt cost to th suppli pio to xpiation of th dlay piod povidd by th suppli. Hnc, th intst chagd IC will b zo. Sinc cycl tim is lss than pmissibl dlay tim, th intst and IE duing [, H] is givn by IE n I P( k A dt ( A k H ( dt P I A (5 ( C D HC otal Invntoy Cost in this cas Z O C I IE H H H AhC ( P ( Z C ( AC 4 ( (6 Diffntiating patially (6 w..t. two tims yilds

6 7 Z H H P ( C ( AC Z C 6 P ( H H ( 4 Z Optimal (minimum solution is obtaind by solving, yilds th following AC C ( H P ( H C Not: uncatd aylo s sis in xponntial tms i. Eqs. (5-7 fo finding closd fom solution. 4. Numical Exampl Lt A,.. to.5,., ( H H, h., I., C 5, I c.5, P, 5,, 45, 6,75 and 9days (7... tc. is usd fom Optimal solution fo diffnt valus of paamts associatd with modl fo is givn in abl as follows, abl h sults of optimal solution fo diffnt valus of > (Days Paamts (yas Q(units Z($ (yas Q(units Z($ (yas Q(units Z($ C Numical Exampl Lt A,.,. to.5,., H, h., I., 5, 4, 55, 7, 85 and days Optimal solution fo diffnt valus of paamts associatd with modl fo

7 R.P. ipathi and M. Kuma / Intnational Jounal of Industial Engining Computations 5 (4 7 abl h sults of optimal solution fo diffnt valus of < (Days Paamts (yas Q(units Z(Dollas (yas Q(units Z(Dollas (yas Q(units Z(Dollas (yas Q(units Z(Dollas (yas Q(units Z(Dollas Snsitivity Analysis: Cas I Whn and a allowd to vay by using 5,,45,6,75and 9 valus of paamts as shown in abl as follows, days. W gt diffnt abl h sults of optimal solution fo diffnt valus of Paamts (Days (yas Q(units Z(Dollas (yas Q(units Z(Dollas (yas Q(units Z(Dollas (yas Q(units Z(Dollas Cas II: Whn and a allowd to vay by using = 5, 4, 55, 7, 85 and days. W gt diffnt valus of paamts as shown in abl 4 as follows,

8 7 abl 4 h sults of optimal solution fo diffnt valus of Paamts (Days (yas Q(units Z(Dollas (yas Q(units Z(Dollas (yas Q(units Z(Dollas (yas Q(units Z(Dollas (yas Q(units Z(Dollas Fom th abov tabls w conclud th following sults: Fom abl, w obsv that: Incas in sults dcas in, Q and incas in Z, kping, and constant and incas in sults incas in, Q and Z kping constant. Fom abl, w obsv that: Incas in sults dcas in, Q and Z kping constant and incas in sults incas in, Q and Z kping constant. Fom abl, w obsv that: Incas in dtioation at sults dcas in, Q and incas in Z kping constant. An incas in sults incas in and dcas in Q and Z kping and constant. Fom abl 4, w obsv that: Incas in sults dcas in, Q and incas in Z kping constant. An incas in sults incas in, Q and Z. 6. Conclusion In this pap, an invntoy modl has bn dvlopd fo dtioating itms und pmissibl dlay in paymnts. Optimal solutions w obtaind fo both cass i.. cas I and II. Numical xampls and snsitivity analysis hav bn psntd to obtain optimal cycl tim and optimal total avag cost p unit tim. h snsitivity analysis is quit snsitiv to th managial point of viw. h poposd modl can b xtndd in sval ways fo instanc w may consid th dmand at as quadatic tim dpndnt o stock dpndnt pattns as wll as discount dmand. W could xtnd th

9 R.P. ipathi and M. Kuma / Intnational Jounal of Industial Engining Computations 5 (4 7 modl fo non dtioating dmand function to stock dpndnt dmand function. In addition, w could gnat th modl to allow shotags, finit capacity and oths. Rfncs Buzacott, J. A. (975. Economics od quantitis with inflation. Opational Rsach Quatly, 6, Covt, R.P., & Phillip, G.C. (97. An EOQ modl with Wibull distibution dtioation. IE anspotations, 5, 6. Chung, K.J., & ing, P.S. (994. On plnishmnt schdul fo dtioating itms with tim popotional dmand. Poduction Planning and Contol, 5(4, Dutta,.K., & Pal, A.K. (99. Effcts on inflation and tim valu of mony on an invntoy modl with lina tim dpndnt at and shotags. Euopan Jounal of Opational Rsach, 5, 6. Dav, U., & Patl, L.K. (98. (,S i policy invntoy modl fo dtioating itms with tim popotional dmand. Jounal of Opation Rsach Socity,, 7 4. Donalson, W. A. (977. Invntoy plnishmnt policy fo a lina tnd in dmand. An analytical solution. Opational Rsach Quatly, 8, Donaldson, W.A. (977. Invntoy plnishmnt policy fo a lina tnd in dmand, an analytical solution. Opations Rsach Quatly, 8, Dutta,.K., & Pal., A.K. (988. Od lvl invntoy systm with pow dmand pattn fo itms with vaiabl at of dtioation. Indian Jounal of Pu and Applid Mathmatics 9, 4 5. Gha P.M., & Schad, S.K. (96. A modl fo xponntially dcaying invntoy. Jounal of Industial Engining, 4 (5, 8 4. Goyal, S.K. (986. On impoving plnishmnt policis fo al tnd in dmand. Eng. Cost pod. Econ., Goyal, S.K., Kusy, M., & Soni, R. (986. A not on conomic od intvals fo an itm with lina tnd dmand Eng. Costs Pod. Econ., Hais, F.W. (9. How many pats to mak at onc. Factoy Mag. Manag, Hny, R.J. (976. Invntoy plnishmnt policy fo incasing dmand. Jounal of th Opational Rsach Socity, (7, Hainga, M. (996. Optimal EOQ modls fo dtioating itms with tim vaying dmand. Jounal of th Opational Rsach Socity, 47, Hou, K.L. (6. An invntoy modl fo dtioating itms with stock dpndnt consumption at and shotags und inflation and tim discounting. Euopan Jounal of Opational Rsach, 68, Hsih,.P., & Dy, C.Y. (. Picing and lot sizing policis fo dtioating itms with patial backlogging und inflation. Expt Systms with Applications, 7, Khana, K.S., & Chaudhui. (. A not on an od lvl invntoy modl fo a dtioating itm with tim dpndnt quadatic dmand. Comput and Opations Rsach,, Goyal, S.K., & Gii, B.C. (. Rcnt tnds in modling of dtioating invntoy. Euopan Jounal of Opational Rsach. 4, 6. Lin, J.J., & Haung K.N. (. Dtioating Invntoy modl dtioating itms with tad cdits financing and capacity constaints. Computational and Industial Engining,59, Lin, Y.H., Lin. C., & Lin, B. (. On conflict and coopation in a two chlon invntoy modl fo dtioating itms. Computs and Industial Engining, 59, 7 7. Liao, H.C. sai, C.H., & Su, C.. (. An invntoy modl fo dtioating itms und inflation whn a dlay in paymnt is pmissibl. Intnational Jounal of Poduction Economics, 6, 7 4. Rsh, M., Fidman, M., & Babosa, L.C. (976. On a gnal solution of th dtministic lot siz poblm with tim popotional dmand. Opations Rsach 4,

10 74 Sachan, R.S., (984. On (,S i. policy invntoy modl fo dtioating itms with tim popotional dmand. Jounal of Opation Rsach Socity, 5, 9. Sana, S.S. (. Optimal slling pic and lot siz with vaying dtioation and patial backlogging. Applid Mathmatics and Computation, 7, Saka, B. (. An EOQ modl with dlay in paymnts and tim vaying dtioation at. Mathmatical and Comput Modling (Aticl in pss. Saka, B. (. An EOQ modl with dlay in paymnts and tim vaying dtioation at. Mathmatical and Comput Modling, 55, Sana, S., & Chaudhui, K.S. (8. A dtministic EOQ modl with dlays in paymnts and pic discount offs. Euopan Jounal Opational Rsach, 84, Silv, E.A., & Mal, H.C. (969. A simpl modification of th EOQ fo th cas of vaying dmand at. Poduction and Invntoy Managmnt,, ng, J.., Min, J., & Pan, Q. (. Economic od quantity modl with tad cdit financing fo non dcasing dmand. Omga, 4, 8 5. Vat, P., & Padmanabhan, G. (99. An invntoy modl und inflation fo stock dpndnt consumption at itms. Intnational Jounal of Poduction Economics, 9, 79 8.

(, ) which is a positively sloping curve showing (Y,r) for which the money market is in equilibrium. The P = (1.4)

(, ) which is a positively sloping curve showing (Y,r) for which the money market is in equilibrium. The P = (1.4) ots lctu Th IS/LM modl fo an opn conomy is basd on a fixd pic lvl (vy sticky pics) and consists of a goods makt and a mony makt. Th goods makt is Y C+ I + G+ X εq (.) E SEK wh ε = is th al xchang at, E