Optimal Operating Policies of EPQ Model for Deteriorating Items with Time Dependent Production having Production Quantity Dependent Demand

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1 IJIRS Ieaioal Joual fo Ioaie Reseach i Sciece & echology Volume 3 Issue May 07 ISSN (olie): Opimal Opeaig Policies of EPQ Model fo Deeioaig Iems wih ime Depede Poducio haig Poducio Quaiy Depede Demad D. K. Aua Kumai Depame of Mahemaics GI, GIAM Uiesiy, Visakhapaam Absac Ieoy models play a impoa ole i deemiig he opimal odeig policies. Much wok has bee epoed i lieaue egadig ieoy models wih fiie o ifiie epleishme. Bu i may pacical siuaios he epleishme is goeed by adom facos like pocueme, aspoaio, eiomeal codiio, aailabiliy of aw maeial ec., hece, i is eeded o deelop ieoy models wih diffee ypes of demad. I his pape we deelop ad aalyze a ieoy model wih he assumpio ha he demad is powe pae ad follows a expoeial disibuio. Howee, i some ohe ieoy sysems he demad is depede o poducio quaiy fo example i oil exploaio idusies, maufacuig idusies; he demad is a fucio of poducio quaiy. o aalyze his so of siuaios a ecoomic poducio quaiy (EPQ) model fo deeioaig iem is deeloped wih he assumpio ha he poducio ae is ime depede ad demad is depede o he poducio quaiy Q. I is fuhe assumed ha he demad ae is of he fom ( ) Q, 0, 0, 0, 0 whee, λ ad τ ae posiie cosas. his demad ae also icludes he cosa ae of demad, whe = 0. he isaaeous leel of ieoy a ay gie ime is deied hough diffeeial equaios. Wih suiable cos cosideaios he opimal odeig policies ae obaied. Keywods: EPQ model, Demad, Poducio ae, Poducio quaiy depede demad, Poducio schedulig I. INRODUCION Ieoy models ceae a lo of iees due o hei eady applicabiliy a aious places such as poducio pocesses, maufacuig aeas, busiess, admiisaio, waehousig, supply chai maageme, ad make yads ad so o. Mahemaical models poide he basic fame wok fo he aalysis of may pacical siuaios. I he same field Fade Heso defied ieoy model as he sock of goods kep fo fuue use. Ee hough he ieoies ae esseial ad hey poide a aleaie o poducio o puchase i fuue, hey also mea lockig up of capial of a eepise. Maiace of ieoies also cos much by way of expeses o soes, equipmes, pesoal isuace ec,. hus i he pocess excess ieoies become udesiable. Hece, ieoy cool ad maageme play a domia ole i he las fie decades ad much wok is epoed i lieaue egadig ieoy models wih aious assumpios. Ieoy models ae classified ude wo caegoies, such as ieoy models fo deeioaig iems ad ieoy models fo o-deeioaig iems. hey ae based accodig o he life ime of he commodiy, eihe fiie o ifiie. Deeioaig iems efe o he iems ha become decayed, damaged, eapoaie, expied, ialid, dealued ad so o hough ime (Wee IIM (993)). Deeioaig iems ca be classified io wo caegoies. he fis caegoy efes o he iems ha become decayed, damaged, eapoaed, o expied hough ime such as mea, egeables, fui, medicie, flowes, film, ec,. he ohe caegoy efes o he iems ha lose pa o oal alue hough ime, like compue chips, mobile phoes, fashio elaed iems ad seasoal goods, ec,. I classical ecoomic poducio quaiy (EPQ) models, i is cusomay o assume ha he poducio ae is pe-deemied ad iflexible. Bu i may pacical siuaios, he poducio ae is depede o seeal facos like aailabiliy of aw maeial, skill leels of he employees, machie life, qualiy equiemes, ec,. Hece, he poducio ae cao be cosideed as cosa houghou he peiod of poducio cycle. A ecoomic poducio quaiy (EPQ) model fo deeioaig iems wih demad as a fucio of poducio quaiy ad ime depede poducio ae is deeloped ad aalyzed wih shoages. he he opimal opeaig policies of he model ae deied. he sesiiiy of he model wih espec o cos ad paamees is also peseed. Howee, i some ohe ieoy sysems he demad is depede o poducio quaiy, fo example i oil exploaio idusies, maufacuig idusies; whee he demad is a fucio of poducio quaiy. o aalyze hese sos of siuaios, a ecoomic poducio quaiy (EPQ) model fo deeioaig iem is deeloped wih he assumpio ha he poducio ae is ime depede ad demad is depede o he poducio quaiy Q. I is fuhe assumed ha he demad ae is of he fom ( ) Q, 0, 0, 0, 0 whee, λ ad τ ae posiie cosas. his All ighs eseed by 35

2 Opimal Opeaig Policies of EPQ Model fo Deeioaig Iems wih ime Depede Poducio haig Poducio Quaiy Depede Demad (IJIRS/ Volume 3 / Issue / 007) demad ae also icludes he cosa ae of demad, whe = 0.Usig diffeeial equaios, he isaaeous sae of ieoy is deied. By miimizig he oal cos fucio, he opimal poducio quaiy, poducio dow ime ad poducio up ime ae deied wih shoages. he sesiiiy of he model wih espec o he paamees ad coss is sudied. II. NOAIONS he followig oaios ae used fo deelopig he model. : Deeioaio paamee. Q: Odeig quaiy i oe cycle A: Odeig cos C: Cos pe ui R: oal poducio : Idex paamee h: Ieoy holdig cos pe ui pe ui ime π: Shoages cos pe ui pe ui ime S: Sellig pice pe ui ad (+Q ): Demad ae III. ASSUMPIONS OF HE MODEL Fo deelopig he model he followig assumpios ae made he demad is kow ad poducio quaiy depede demad ae is (+Q ) he ae of poducio R () is ime depede ad follows a powe pae. R ( ), i.e., whee, is he oal poducio ad is he idex paamee. Lead ime is zeo Cycle legh is kow ad fixed say Shoages ae allowed ad fully back logged A deeioaed ui is los, ad hee is o epai o eplaceme of he deeioaed ui he life ime of he commodiy is a adom ad follows a expoeial disibuio, he he isaaeous ae of deeioaio is. IV. PRODUCION LEVEL INVENORY MODEL WIH SHORAGES I his model he sock leel is zeo a ime = 0. he sock leel iceases duig he peiod (0, ) due o excess poducio afe fulfillig he demad ad deeioaio. he poducio sops a ime whe he sock leel eaches S. he ieoy deceases gadually due o demad ad deeioaio i he ieal (, ). A ime he ieoy eaches zeo ad he back odes ge accumulaed duig he peiod (, 3). A ime 3 he poducio agai sas ad fulfils he backlog afe saisfyig he demad duig ( 3, ), ad he ieoy leel has iceased. he schemaic diagam epeseig he isaaeous sae of ieoy is gie i Figue. Fig..: Schemaic diagam epeseig he ieoy leel. All ighs eseed by 36

3 Opimal Opeaig Policies of EPQ Model fo Deeioaig Iems wih ime Depede Poducio haig Poducio Quaiy Depede Demad (IJIRS/ Volume 3 / Issue / 007) Le I() be he ieoy leel of he sysem a ime (0 ). he diffeeial equaios goeig he isaeeous sae of ieoy oe he cycle of leh ae d V I() I() ( Q ), 0 d I() I() ( Q ), d I() ( Q ), 3 () () (3) d I( ) ( Q ), 3 wih he iiial codiios I( ) = S, I( ) = 0, I() = 0 Solig he diffeeial equaios () o (4) we ge he o had ieoy a ime as u u u 0 I( ) e ( Q )e du e du S e, ( ) ( Q ) ( Q ) I( ) e S, (4) (5) (6) I() ( Q )( ), (7) 3 he sock loss due o deeioaio i he ieal (0, ) is I( ) ( Q )( ), (8) L ( ) R ( ) ( Q ) I( ), heefoe he sock loss due o deeioaio i he cycle of legh is Poducio quaiy Q i he cycle of legh is L () = ( Q ). Q. 3 Fom he equaios (5) ad usig he iiial codiio I (0) = 0 we ge he alue of S as u u u S e ( Q ) e du e du. 0 0 Fom equaio (6) ad usig he codiio I ( ) = 0 we ge ( ) e S ( Q ) ( Q ). Subsiue he alue of S fom equaio (0) i equaio () ad o simplificaio we ge i ems of as log = x( ) (say) ( Q ) akig = 3 i he equaios (7) ad (8) ad equaig hese, we ge (9) (0) () () All ighs eseed by 37

4 Opimal Opeaig Policies of EPQ Model fo Deeioaig Iems wih ime Depede Poducio haig Poducio Quaiy Depede Demad (IJIRS/ Volume 3 / Issue / 007) ( Q ) 3. Subsiuig he alue of fom he equaio () i equaio (3) we ge (3) 3 i ems of, y ( ). 3 ( Q ) whee y( ) x ( ) ad x ( ) is as gie i equaio () Le K (,, 3) be he oal cos pe ui ime. Sice he oal cos is sum of he se up cos, cos of he uis he ieoy holdig cos ad shoage cos, K (,, 3) becomes 3 A C Q h K (,, ) I() I() I() I() (4) (5) Subsiuig he alues of I () ad Q fom equaios (5), (6), (7),(8) ad (9) i equaio(5) we ge A C h e ( Q ) 3 3 K (,, ) = S e (e ) 3 ( e ) ( Q ) ( ) ( )( ) ( )( 3 ) ( Q ) (6) 3 ( ) ( ) Subsiuig he alues of S, ad 3 fom equaios(0), () ad (4) i equaios (6), K (,, 3) becomes K ( ) A C h ( Q ) x ( ) K ( ) = [ y( )] e (e ) ( ) 3 ( e ) ( Q )x ( ) ( ) ( ) ( ) ( ) ( 3 ) ( Q ) + [ y( )] x( ) (x( )) + [ y( )] [ y( )]. ( ) ( ) (7) OPIMAL POLICIES OF HE MODEL I his secio, he opimal policies of he ieoy sysem ae deied. o fid he opimal alues of, we miimize he oal cos pe ui ime wih dk ( ) d K ( ) 0 ad 0 espec o.he codiios fo opimal alue of ae, equaio (7)wih espec o ad equaig i o zeo we ge Diffeeiaig K () gie i All ighs eseed by 38

5 Opimal Opeaig Policies of EPQ Model fo Deeioaig Iems wih ime Depede Poducio haig Poducio Quaiy Depede Demad (IJIRS/ Volume 3 / Issue / 007) C x ( ) x ( ) ( Q ) z ( ) h e e e z ( ) x ( ) ( ) ( e e z ( ) e ) whee, ( Q ) x ( ) z ( ) e ( Q ) z ( ) x ( ) y( ) ( Q ) x ( ) y y( ) 0. (8) x ( ) ad y( ) ae as gie i equaio() ad(4) ad z ( ) = x ( ) Solig he equaio (8) fo usig he umeical mehods we ge he opimal alue of as *.he opimum ime 3 of 3 a which he poducio should be esaed is obaied by subsiuig he opimal alue of * i equaio (4), heefoe he opimal ime a which he poducio is o be saed is ( ) ( Q ) log 3 ( ) ( Q ). (9) he opimum poducio quaiy Q of Q i he cycle of legh is obaied by subsiuig he opimal alues of *ad 3* i equaio (9) heefoe he opimal poducio quaiy is ( Q ) * Q l. (0) ( ) ( Q ) V. NUMERICAL ILLUSRAION I his secio, le us coside he case of deiig opimal poducio quaiy, poducio dow ime ad poducio upime of a idusy. Hee, i is assumed ha he poduc is of deeioaig aue, shoages ae allowed ad fully backlogged. Fo demosaig he soluio pocedue of he model, he deeioaig paamee is cosideed o ay as 0., 0., 0., 0.3 ad 0.4, he alues of ohe paamees ad coss ae associaed wih he model. he subsiue he alues i he opimal poducio quaiy Q *. Poducio dowime, poducio upime ad opimal cos of poducio ae compued ad peseed i able. Fom able, i is obseed ha he deeioaio paamee ad poducio paamees hae a emedous ifluece o he opimal alues of he model. As he deeioaig paamee aious fom 0.0 o 0.4, he opimal poducio quaiy Q * deceases fom o 0.087, he opimal alue of he poducio dow ime deceases fom.60 o.565, he opimal alue of poducio upime deceases fom 4.60 o 3.93 ad he oal cos of poducio pe ui ime K iceases fom o uis, his icease is omial. able OPIMAL VALUES OF *, 3 *, Q * AND K C h π A * 3* Q* K All ighs eseed by 39

6 Opimal Opeaig Policies of EPQ Model fo Deeioaig Iems wih ime Depede Poducio haig Poducio Quaiy Depede Demad (IJIRS/ Volume 3 / Issue / 007) VI. SENSIIVIY ANALYSIS OF HE MODEL Sesiiiy aalysis is caied ou o exploe he effec of chages i paamees ad coss o he opimal policies by ayig each paamee (-5%, -0%, -5%, 5%, 0%, 5%) a a ime ad all paamees ogehe fo he model ude sudy. he esuls obaied ae peseed i able..he elaioship bewee he paamees, cos o he opimal alues of he poducio schedule ae show i Figue.. I is obseed ha he aiaio i he deeioaio paamee ad he demad paamees ad hae sigifica ifluece o opimal poducio quaiy Q *. As iceases, he poducio quaiy Q * is deceasig ad he poducio dow ime ad poducio upime ae also deceasig whe ohe paamees emai fixed. he poducio ae paamees hae sigifica ifluece o he opimal alues of he poducio quaiy Q* ad poducio dowime * ad poducio upime 3*. As iceases he alue of *, is deceasig, 3*, Q* ae iceasig. he decease i * is magial ad he icease i 3* is apid. able Sesiiiy Aalysis of he Model Wih Respec o Paamees ad Coss Vaiaio i Paamees Opimal Policies Peceage Chage i Paamees * * C Q* K * * h Q* K * * π Q* K * * Q* K All ighs eseed by 40

7 Opimal Opeaig Policies of EPQ Model fo Deeioaig Iems wih ime Depede Poducio haig Poducio Quaiy Depede Demad (IJIRS/ Volume 3 / Issue / 007) A * * Q* K * * Q* K * * Q* K * * Q* K * * Q* K * * Q* K All ighs eseed by 4

8 Opimal Opeaig Policies of EPQ Model fo Deeioaig Iems wih ime Depede Poducio haig Poducio Quaiy Depede Demad (IJIRS/ Volume 3 / Issue / 007) Wih Shoages Fig.. he gaphical epeseaio of sesiiiy aalysis of poducio quaiy depede demad wih shoages VII. CONCLUSION Poducio ieoy models play a domia ole i poducio schedulig ad esouce allocaio. I his pape, o deelop ieoy models fo deeioaig iems wih ime depede poducio haig poducio quaiy depede demad haig powe pae ad follows a expoeial disibuio. he models wee illusaed wih umeical examples ad sesiiiy aalysis of he models wih espec o cos ad paamees was also caied ou. Diffee ypes of poducio haig opimal ode quaiies, opimal up ime, opimal dow ime ad opimal sysem pofi wee obaied fo diffee choices of coss ad paamees. I ca be cocluded fom he umeical examples ad sesiiiy aalysis ha he poducio quaiy depede aue of poducio ae is haig sigifica ifluece o he opimal poducio quaiy ad poducio up-ime, poducio dow ime ad he demad paamees emedously ifluece he opimal alues of he oal poducio ae. he models ae deeloped by assumig ha he moey alue emais cosa houghou he peiod of ime. I is also possible o deelop he EPQ models discussed i his pape ude iflaio (ime alue of moey). he models deeloped fo sigle iem ca also be exeded o iclude muliple commodiies. REFERENCES [] Bhuia, A.K. ad Maii, M. (997) A ieoy model fo deeioaig iems wih sellig pice, fequecy of adeiseme ad liealy ime depede demad wih shoages, IAPQRas, Vol., [] C.. Chag, Y.J. Che,.R. sai ad S.J. Wu, Ieoy models wih sock ad pice depede demad fo deeioaig iems based o limied shelf space, Yugosla Joual of Opeaios Reseach, 0() (00), [3] B.C. Gii, S. Pal, A. Goswami ad K.S. Chaudhui, A ieoy model fo deeioaig iems wih sock depede demad ae, Euopea Joual of Opeaioal Reseach, 95(996), [4] Deb, M. ad Chaudhui, K.S. (986) A EOQ model fo iems wih fiie ae ofpoducio ad aiable ae of deeioaio, OPSEARCH, Vol.3, All ighs eseed by 4

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