nernaional Journal of Applie Engineering esearch SSN 097-456 Volume, Number 9 08 pp. 76-767 esearch nia Publicaions. hp://www.ripublicaion.com An nvenory Moel wih ime-varying Deman for Non- nsananeous Deerioraing ems wih Maximum Life ime A. K. Malik, Poras Vei, Saish Kumar * Deparmen of Mahemaics, B.K. Birla nsiue of Engineering & echnology, Pilani, ajashan, nia. Deparmen of ompuer Science an Engineering, B.K. Birla nsiue of Engineering & echnology, Pilani, ajashan, nia. Deparmen of Mahemaics, D.N. PG ollege, Meeru, U.P., nia. orresponing auhor Absrac We evelop he invenory moel for non-insananeous eerioraing ems. Here eman is aken o be ime varying. Generally, mos of he propose moels researcher consiere he consan eerioraion rae bu in acual siuaions, maximum iems eeriorae ue o finish of heir maximum life ime. he main iea inegrae for he moeling of he propose invenory moel is: ime-varying eman, ime varying sales revenue, an sensiiviy analysis of parameers wih profi funcion. o fin he opimal oal profi an he opimal orer quaniy, a numerical example is provie o emonsrae he pracical usage of he mahemaical resul an sensiiviy analysis of he opimal oal profi funcion wih respec o consrains is carrie ou. Keywors: nvenory Moel, ime-varying eman, Maximum life ime an ime-varying Sales revenue cos. NODUON n he recen years, he problem of eerioraion in he invenory moels has receive consierable aenion. Harris [] evelope he firs EOQ Economic Orer Quaniy invenory moel, which was generalize by Wilson [] who invene a formula o fin economic orer quaniy. Whiin [] iscusse he invenory moel wih he fashionable proucs eerioraing a he en of he sorage perio. Suy of eerioraing invenory moel began wih Ghare an Schraer [4] who evelope an invenory moel for an exponenially ecaying invenory an also esablishe he EOQ invenory moel wih fix eerioraion rae an wihou shorages. over an Philip [5] exene Ghare an Schraer s invenory moel an presene an EOQ moel wih variable rae of eerioraion by consiering a woparameer Weibull isribuion. his invenory moel was furher moifie by Philip [6] by assuming a hree-parameer Weibull isribuion. Misra [7] consiere an EPQ invenory moel wih eerioraing iems an assuming he wo ype of eerioraion rae consan an variable form. Afer ha he ifferen ypes of orer-level invenory moels for eerioraing iems a a consan eeriorae rae were iscusse [8-] Dave [] presene invenory orering policies for eerioraing iems. Kang an Kim [] suie on he price an proucion invenory moel wih eerioraing iems. Aggarwal an Jaggi [4] presene an invenory moel wih orering policies for eerioraing iems. aafa [5] gave a survey of he exising lieraure on consanly eerioraing invenory moel. Some of he recen works in his fiel have been one by hung an ing [6], Goyal an Ganasekaran [7]. hang an Dye [0] propose a moel wih parial backlogging an ime-varying eman. n he recen research work, he relaionship beween ime an eerioraing rae is a common phenomenon assume in invenory moels. his ype of siuaion, we have several scenarios; incluing linear eerioraing rae Bhunia an Maii [8], Mukhopahyay e al. [4] an eerioraing rae is consier oher ime funcion Aba []. Wee [9] iscusse a eerioraing invenory moel for quaniy iscoun an pricing uner he parial backorering. Shah e al. [], Goyal an Giri [] gave recen rens of moelling in eerioraing iem invenory. n realiy, no all kins of invenory iems eeriorae as soon as hey obaine by he reailer. During he fresh ime of prouc/iem, he original qualiy is mainaine because of no eerioraion in prouc. Ouyang e al. [6] calle his occurrence as non-insananeous eerioraion, an hey evelope a non-insananeous invenory moel for eerioraing iems uner he permissible elay in paymens. Manna an hauhuri [5] presene an EOQ invenory moel having eman rae of ramp ype for ime epenen eerioraion rae an uni proucion cos uner he shorages. Zhou an Gu [7] gave an opimal proucioninvenory policy for eerioraing iems. Liao [8] gave an economic orer quaniy EOQ invenory moel for non-insananeous receip wih exponenial eerioraing iem uner he wo level rae creis. Li e al. [9] suggese a review on eerioraing invenory suy. n his suy hey compare wih he exan reviews of aafa [5] an Goyal []. his research paper propose some new key parameers which shoul be assume in he eerioraing invenory suies. Singh an Malik [] consiere an Opimal orering policy invenory moel wih linear eerioraion an exponenial eman uner he wo sorage capaciies. Sana [0] examine opimal selling price an lo size invenory moel wih ime varying eerioraion uner he parial backlogging. Singh an Malik [] evelope an sock epenen eman invenory moel wih wo sorages 76
nernaional Journal of Applie Engineering esearch SSN 097-456 Volume, Number 9 08 pp. 76-767 esearch nia Publicaions. hp://www.ripublicaion.com capaciy for non-insananeous eerioraing iems. Sarkar [4] propose an EOQ invenory moel wih permissible elay in paymens an ime epenen eerioraion rae. n his paper he eal wih an EOQ moel for finie replenishmen rae in which eman an eerioraion rae boh are ime-epenen. Se e al. [] invesigae an invenory moel of wo-warehouse consiering quaraically increasing eman an ime varying eerioraion. Gupa e al. [5] iscusse an opimal orering policy wih sock-epenen eman invenory moel for noninsananeous eerioraing iems. Sarkar an Sarkar [6] iscusse an improve invenory moel for ime varying eerioraion wih sock-epenen eman uner he parial backlogging. Sarkar an Sarkar [7] evelope an economic quaniy moel wih probabilisic eerioraion rae in he proucion sysem. Ghoreishi e al. [8] consiere an opimal pricing an orering policy for non-insananeous eerioraing iems uner he inflaion. Sheng e al. [9] evelope an ime varying eerioraing iems having invenory moel uner he rae crei. Sarkar e.al. [40] presene an invenory moel wih qualiy improvemen an backorer price iscoun uner conrollable lea ime. Malik e. al. [4] propose an invenory moel wih noninsananeous an ime-varying eerioraing ems. Mos researchers on invenory moels o no consier simulaneously he phenomena wih non-insananeous an ime varying eerioraion, because hese phenomena are no uncommon in real life siuaions, so we incorporae hem in our moel. A numerical example, graphical images wih sensiiviy analysis are provie o he suy of he effec of changes in he associae parameers on he opimal soluion. NOAONS AND ASSUMPONS For eveloping he invenory moel, we use he following noaions an assumpions: Noaions + : he eman rae, o : he orering cos per orer, h : he invenory holing cos per uni ime per uni iem, p : he purchasing cos per uni iem, : he eerioraing cos per uni iem, S -S : he sales revenue cos per uni iem, θ : he ime varying eerioraion rae a ime,, where 0 θ, L : maximum invenory level : maximum life ime of he eerioraing iem, : he invenory level a he ime [0, ] wih no eerioraion in he prouc. : he invenory level a he ime [, ] wih he eerioraion in he prouc sars. : he ime inerval in which he prouc has no eerioraion i.e., fresh prouc ime, : he orering cycle P : he oal profi per uni ime of he invenory sysem. Assumpions Deman is he linear increasing funcion of ime. is assume ha in his invenory moel he ime uraion [0, ], he prouc has no eerioraion i.e., fresh prouc ime. Afer he inerval [0, ], on-han invenory eerioraes an here is no repair or replacemen of he eeriorae unis. he eerioraion funcion epens upon ime, where is he maximum life ime of he iem. Shorages are no allowe. he Lea ime is zero or negligible. MAHEMAAL MODEL n he ime-inerval [0, ], he invenory level ecreases ue o eman rae, an no eerioraion uring his perio of ime. Afer he ime, he eman an eerioraion sar simulaneous an sop whenever he invenory reuces o zero level. he invenory levels a any ime uring he ime inerval [0, ] can be represene by he ifferenial equaions given below:.. Wih he bounary coniions, respecively. By solving hese ifferenial equaions, we ge he invenory level as follows: L. log log. 4 76
nernaional Journal of Applie Engineering esearch SSN 097-456 Volume, Number 9 08 pp. 76-767 esearch nia Publicaions. hp://www.ripublicaion.com 764 Assuming ha coninuiy of a =, i follows from equaions an 4 ha L log log 5 he opimum profi per cycle conains he following values: he orering cos per cycle is O= 0. 6 he eerioraion cos per cycle is cycle is given by D= log log log log log. 7 he purchasing cos per cycle is given by P= L P * P log log. 8 he sales revenue cos per cycle is given by S= S 0 s s s s. 9 he holing cos per cycle is given by H= h 0 L h log 4 log 6. 0 hus he opimum profi P per cycle per uni ime is given by ] [ P D H O S P. o obain he opimal cycle ime * we have o maximize he oal presen value of profi. o maximize he P, we have o iffereniae wih respec o an equae i o zero 0 P. an 0 P. he profi funcion is highly non-linear; herefore i canno invenory sysem is maximize by a evelope soluion proceure. Also a numerical example, graphical esign wih he sensiiviy analysis is uilize o illusrae his evelope invenory moel. SOLUON POEDUE Sep. npu he value of he parameers o, h, p, s,,, D, in equaion ; Sep. Now using he equaion obain an from he relaion obain P; Sep. Subsiuing he value of in he equaion an check he opimal profi P. f saisfie hen go o sop oherwise repea he process from sep o.
nernaional Journal of Applie Engineering esearch SSN 097-456 Volume, Number 9 08 pp. 76-767 esearch nia Publicaions. hp://www.ripublicaion.com NUMEAL EXAMPLE AND SENSVY ANALYSS o emonsrae he above soluion proceure, we ake he following example: o=500, S =0, S =.5, p=0, h=0.0, =0.08, =0, =0.5, =800 an =50. Using opimizaion echniques we obain he opimal orering cycle *=4.4007 an corresponing P*=805.94 an L*=465.79 are evaluae an abulae in able for varying moel s parameers: S S p h o hange in L * P* P 000.590 45.06 9865.585-998.489 800.9007 465.79 805.94-7.66 600 4.867 480.68 679.957-5.4506 6.5 4.699 55.5 8895.5609-687.5045 50.9007 465.79 805.94-7.66 7.5.5 07.5 785.05-85.85 7.5 5.4445 6074.70 8408.4605-495.960 0.9007 465.79 805.94-7.66.5.4709 77.4 7878.769-9.086 75 6.78 746.9 98.69-499.748 0.9007 465.79 805.94-7.66 65.50 744.4 66.07-476.468.875.604 95.44 79599. -797.0746.5.9007 465.79 805.94-7.66.5 4.084 460.6 864.68-66.4948 50.589 86.9 56.85-468.896 0.9007 465.79 805.94-7.66 90 6.768 8069.0 46.809-409.05 0.5 4.07 457.5 8050.45-77.5057 0.0.9007 465.79 805.94-7.66 0.075.60 95.97 809. -680.88 875.976 497.4 8067.764-7.6959 500.9007 465.79 805.94-7.66 5.87 4.74 8047.6096-708.9079 0.0.900 465. 8050.949-7.5090 0.08.9007 465.79 805.94-7.66 0.06.90 466.47 805.6-7.44 From he above able we represen he following resuls:. When he eman rae D increases, hen he oal profi increases.. When he maximum life ime an sales revenue cos S increase, hen he oal profi increases.. When he purchasing cos p, holing cos h, orering cos o are increase an eerioraing cos are increases, hen oal profi ecreases. 765
nernaional Journal of Applie Engineering esearch SSN 097-456 Volume, Number 9 08 pp. 76-767 esearch nia Publicaions. hp://www.ripublicaion.com he following graphs Fig. an Fig. show he relaion beween he oal profi P an he ime uraion an. Figure : oal Profi w.r.. an Figure : Graph represenaion of oal Profi w.r.., an ONLUSONS n mos of he evelope invenory moels researcher have iscusse wih he consan eerioraion rae. Bu in acual coniions, maximum proucs/iems eeriorae ue o expiraion of heir maximum life ime. Accoring o auhor s, such form of view for ime varying eerioraion funcion of he ime wih assuming non-insananeous iems, has no ye been propose. A numerical example an sensiiviy analysis are presene o he suy of he effec of changes in he relae parameers on he opimum funcion. Furher, a fuure research irecion, in he suy for invenory moel wih variable eman, sock epenen eman, price an muli value eman, variable holing cos, inflaion, probabilisic eman an eerioraion, parial backlogging, wo sorages, reliabiliy an rae crei. EFEENES [] Harris F. W. Operaions an os. A. W. Shaw ompany, hicago, 48-54 95. [] Wilson. H. A Scienific ouine for Sock onrol. Harvar Business eview, 6-8 94. [] Whiin.M., heory of nvenory Managemen. Princeon Universiy Press, Princeon, NJ 957. [4] Ghare P.M. an Schraer G.F., An invenory moel for exponenially eerioraing iems. Journal of nusrial Engineering 4, 8-4 96. [5] over. P. an Philip G.., 97. An EOQ moel for iems wih Weibull isribuion eerioraion. AE ransacions 5, -6 97. [6] Philip G.., A generalize EOQ moel for iems wih Weibull isribuion eerioraion. AE ransacions 6, 59-6 974. [7] Mishra.B., Opimum proucion lo-size moel for a sysem wih eerioraing invenory. nernaional Journal of Proucion esearch Sociey, 495-505 975. [8] Shah Y. K. an Jaiswal M.., An orer level invenory moel for a sysem wih consan rae of eerioraion. Opsearch 4, 74-84 977. [9] Aggarwal S. P., A noe on an orer-level invenory moel for a sysem wih consan rae of eerioraion. Opsearch 5, 84 87 978. [0] Dave, U., an Pael, L.K.,, Si policy invenory moel for eerioraing iems wih ime proporional eman. Journal of Operaional esearch Sociey, 7-4 98. [] oy chowhury M. an hauhuri K.S., An orer level invenory moel for eerioraing iems wih finie rae of replenishmen. Opsearch 0, 99-06 98. [] Dave U., An orer level invenory moel for eerioraing iems wih variable insananeous eman an iscree opporuniies for replenishmen. Opsearch, 44-49 986. [] Kang, S., an Kim, A., A suy on he price an proucion level of he eerioraing invenory sysem. nernaional Journal of Proucion esearch, 899-908 98. [4] Aggarwal, S.P. an Jaggi,.K., Orering policy for ecaying invenory. nernaional Journal of Sysem Science 0, 5-55 989. [5] aafa, F., Survey of lieraure on coninuously eerioraing invenory moel. Journal of he Operaional esearch Sociey 4, 7-7 99. [6] hung KJ, ing PS., A heurisic for replenishmen for eerioraing iems wih a linear ren in eman. Journal of Operaional esearch Sociey 766
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