A Production Model for Time Dependent Decaying Rate with Probabilistic Demand
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1 ISSN (ONLINE: 5-758, ISSN (PRIN: Volume-7, Issue-3, May-Jue 7 Iteatoal Joual of Egeeg ad Maagemet Reseach Page Numbe: 4-47 A Poducto Model fo me Depedet Decayg Rate wth Pobablstc Demad P.K. Gupta Depatmet of Statstcs, D.A.V. (P.G. College, Muzaffaaga, UP, INDIA ABSRAC I ths study, we developed a poducto vetoy system fo decayg tems wth pobablstc demad ude flato. Decayg ate s depedet o tme. Patally backlogged shotages ae allowed wth tme depedet backloggg ate. he backloggg ate s expoetal deceasg fucto of tme. he expecteotal cost wth ufom demad ad dscete uts ad cotuous uts has bee take. Keywods-- Pobablstc demad, flato, patal backloggg I. INRODUCION I busess wold, tme s a pheomeo, whch affects eveythg aou. Iflato vey clealy s a cocept closely elateo tme. Iflato s that state dsequlbum, whch a expaso of puchasg powe teds to cause, o t s the effect of cease the pce level. Dug flato, thee s too much cuecy elato to the physcal volume of the busess beg doe. Iflato s omally assocated wth hgh pces, whch cause decle the puchasg powe o the value of moey. Sce the 97 s eegy css, flato has become a embedded featue of the ecoomy though out the wold, ad lage scale flato ates wee ot may dustes. I ths decto, the poee eseach, Buzacott (975 was the fst popoet fo developg a EOQ model wth flato subject to dffeet types of pcg polcy. Chada ad Bahe (985 dscussed some vetoy systems wth the effects of flato ad tme value of moey. I 99, Datta ad Pal cosdeed a model wth lea tme depedet demad ate ad shotages to vestgate the effects of flato ame value of moey o odeg polcy ove a fte tme hozo. Haga (995 modfed Datta ad Pal s(99 model by elaxg the assumpto of equal vetoy cayg tme dug each epleshmet cycle ahe mathematcal fomulato Haga ad Be-Daya (996 exteded Haga(995 by emovg the estcto of equal epleshmet cycle ad povdewo soluto pocedues wth o wthout shotages. Chag (4 suggested a model fo deteoatg tems wth flatoay evomet ude a stuato whch the supple povdes the puchase, a pemssble delay of paymets f the puchase odes a lage quatty. Yag, H. L. (4 cosdeed a two waehouses vetoy models fo deteoatg tems wth shotages ude flato. Moo et al. (5 pesetehe deteoato to ameloato whe the evomet was flatoay. Jagg et al. (6 dscussed a optmal ode polcy fo deteoatg tems wth flato duced demad. I ths pape, they pesetehe optmal vetoy epleshmet polcy of deteoatg tems ude flatoay codtos usg a DCF appoach ove a fte plag hozo. I the same yea, Yag (6 used a two waehouses vetoy models fo deteoatg tems ude flato wth patal backloggg ad costat demad ates. Sgh, S.R. et al. (9 dscussed a vetoy system fo peshable tems wth stock depedet demad ame depedet patal backloggg. I the model, costat holdg cost has bee take. pathy, C.K. ad Msha, U. ( developed a detemstc vetoy model fo deteoatg tems wth costat demad. Shotages wee allowed wth costat patal backloggg ate the study. Sghal et al. (4 peseted a pobablstc vetoy model fo webull deteoatg tems wth flexblty ad elablty cosdeato. Sghal et al. (6 developed a stochastc patal backloggg vetoy models fo deteoatg tems wth tme depedet demad ad volume elastcty. I the peset pape attempts have bee made to vestgate a EPQ model assumg the exstece of a pobablstc demad, tme depedet deteoato ate. Shotages ae allowed wth patal backloggg. he backloggg ate s expoetal deceasg fucto of tme. Iflato s also take ths study. We also cosdeehe expecteotal cost wth ufom demad ad dscete uts ad cotuous uts. 4 Copyght 7. Vadaa Publcatos. All Rghts Reseved. II. ASSUMPIONS AND NOAIONS ASSUMPIONS. Model s developed o mult-tem poducts.. Demad s ufomly ove the peod ad a fucto of tempeatue that follows as pobablty dstbuto.
2 ISSN (ONLINE: 5-758, ISSN (PRIN: Shotages ae allowed ad patally backlogged. 4. Leame s zeo. 5. he tme hozo s fte. 6. Deteoato ate s take as tme depedet. NOAIONS. K Poducto ate pe ut tme.. Demad ove the peod. 3. Q Ivetoy level of th tem. 4. p Sellg pce pe ut of th tem. HE MODEL ( I ths model, we cosde, demads ae empeatue follows pobablty dstbuto ove peod. Hee, c pj a j, j ( p Whee, a ( p j j, j s costat ( p c ( p p j j, j ( p 5. Duato of the poducto cycle. 6. bt Quadatc ate of deteoato. t 7. B e - Backloggg ate. 8. C h Holdg cost pe th tem pe ut tme. 9. C s Shotage cost pe th tem pe ut tme.. Iflato ate C Ls lost sale cost.,... that depeds upo tempeatue ad sellg pce of th tem. magal espose of th cosumpto to a chage. (tempeatue magal espose of th cosumpto to a chage j j, j he ato of the aveage sellg pce of (j =, -, + tems to the sellg pce of ( p th tem ( s costat that depeds upo the choce of the cosumes. III. MODELING AND ANALYSIS I ths study, we have developewo cases: Case I: Whe shotages do ot occu: I ths case, the tal vetoy of the cycle s zeo ad poducto stats at the vey begg of the cycle. As poducto cotues, vetoy begs to ple up cotuously afte meetg demad ad deteoato. Poducto stops at tme t. he accumulated vetoy s just suffcet eough to accout fo demad ad [ t, ]. he stataeous deteoato ove the teval vetoy level Q (t at ay tme s goveed by the followg equatos: 4 Copyght 7. Vadaa Publcatos. All Rghts Reseved.
3 ISSN (ONLINE: 5-758, ISSN (PRIN: Ivetoy level S (t btq (t K, t t ( (t btq (t,t t ( Q (, Q (t S Wth bouday codtos, Solutos of equato (, ( ae: (t Q 3 bt bt (K t e t t 6 43 Copyght 7. Vadaa Publcatos. All Rghts Reseved. (3 b bt (t t ( b 3 3 Q (t S e (t t (t t e 6 (4 Peset woth holdg cost occus dug (, s: t t ( t t h t t H C e Q (t e Q (t t b t b t b t b t C h [(K [ ] S[( t (t t b 3 3 t ( t ( t t ( t ( t ] 3 b t ( t ( t t ( t ( t t 3 3 b t 5 5 ( t ( t ( t ( t b t b t ( t ( t ( t ( t t b 4 t 4 4 t ( t ( t t ( t ( t 6 4 me (5
4 ISSN (ONLINE: 5-758, ISSN (PRIN: Peset woth deteoato cost occus dug (, s: t t (t t D bte Q (t bte Q (t t bt b t b t b t b t (K [ ] [b t ( t b t ( t ( t ( t t b t b ( t ( t ( t ( t b t b t ( t ( t ( t ( t t t 3 3 b t t 5 5 b ( t ( t ( t ( t b t 4 4 S b b t ( t b ( t ( t ( t ] (6 3 4 Case II: Whe shotages occu: I ths case, the tal vetoy of the cycle s zeo ad poducto stats at the vey begg of the cycle. As poducto cotues, vetoy begs to ple up cotuously afte meetg demad ad deteoato. Poducto stops at tme t. he accumulated vetoy s just suffcet eough to accout fo demad ad deteoato ove the teval [ t, t ]. Shotage stats afte t wth the cocept of patal backloggg ad each to maxmum shotage level at tme t 3. Poducto estats afte t 3 to fulfll the backlog demad ahe cycle eds wth zeo vetoy. he stataeous vetoy level Q (t at ay tme s goveed by the followg equatos: Ivetoy level S t 3 t t (t btq (t K, t t (7 (t btq (t, t t t (8 R me 44 Copyght 7. Vadaa Publcatos. All Rghts Reseved.
5 ISSN (ONLINE: 5-758, ISSN (PRIN: (t B, t t t3 (9 (t K, t3 t ( Wth bouday codtos (, Solutos of these equatos ae: Q Q ( t S, Q ( t, Q ( t3 3 bt ( bt Q (t (K t e t t ( 6 b (t t Q (t S e (t t bt b ( 3 3 (t t e ( t t t ( 6 t t Q ( t e e, t t t3 R (3 Q ( t R ( K ( t t3, t3 t (4 Peset woth holdg cost occu dug (o, t s: t t t ( t t H Ch e Q ( t e Q ( t t t bt t b t bt b t Ch ( K [ ] b 3 3 S ( t t t ( t t ( t t 3 t ( t t ( t t ] t ( t t ( t t b t 3 3 t ( t t ( t t ( t t ( t t b t 5 5 b t ( t t ( t t ( t t ( t t b t t ( t t ( t t t ( t t ( t t 3 6 b 4 t 4 4 t ( t t ( t t ] (5 6 4 Peset woth deteoato cost occus dug (, t s: t t t ( t t D bt e Q( t bt e Q( t t bt b t t b ( a t b t b t ( K [ b ] [b t (t t (t b t t (t t (t t Copyght 7. Vadaa Publcatos. All Rghts Reseved.
6 ISSN (ONLINE: 5-758, ISSN (PRIN: t b t b (t t (t t (t t (t t b t b t (t t (t t (t t (t t t t 3 3 b t t 5 5 b (t t (t t (t t (t t ] b t 4 4 S ( b b t ( t t b ( t t ( t t ( t t ] 3 4 Peset woth shotages cost occus dug ( t, s: t3 ( t t ( t3 t S s t t3 C [ e Q ( t e Q ( t ] (6 t ( t e ( t3 ( t e 3 [ { t t Cs e e e e } ( e R { ( ( ( ( ( }] t t3 e K t3 e e K Peset woth lost sale cost occus dug ( t, t 3 s: t3 t3 L CL s ( B e t t 3 t3 t CL s [ e {( t3 t ( e e } (8 CASE :- UNIFORM DEMAND AND DISCREE UNIS s a..v. wth pobablty P(, s. t. P( ad P ( heefoe, the total expected aveage cost s: Ea c( [ H P( d H P( d D P( d D P( d S P( d L P( d ] (7 Whee H, H, D, D, S, L ae gve by equatos (5, (5, (6, (6, (7 ad (8 espectvely. Sce, Q Q c a ( p j, j c pj j, j Q (say a ( p p j 46 Copyght 7. Vadaa Publcatos. All Rghts Reseved.
7 ISSN (ONLINE: 5-758, ISSN (PRIN: he, Q Also, Q a c j, j ( p ad Q, p j CASE : UNIFORM DEMAND AND CONINUOUS UNIS Whe uceta demad s estmated as a cotuous adom vaable, the cost equato of the vetoy volves tegals stead of summato sgs. he dscete pot pobablty P ( ae eplaced by the pobablty dffeetal f ( fo small teval. I ths case, ( d Ea c( f ad f ( [ H f ( d H f ( d D f ( d D f ( d S f ( d L f ( d ]. he total expected aveage cost dug peod (, s, whee P, P, H, H, D, D, S, L ae gve by equato (5, (5, (6, (6, (7 ad (8 espectvely. IV. CONCLUSION A poducto vetoy model fo tme depedet deteoatg ate has bee developed. he demad ate s stochastc because eal stuato, thee ae may commodtes whose demad s ot ceta. Shotages ae allowg wth patal backloggg. he evomet of whole study has bee take as flatoay. We hope that ths study has the scope of dect applcato the vey pactcal stuatos. REFERENCES [] Buzacott, J. A. (975, Ecoomcs ode quattes ude flato, Opeatoal Reseach Quately, 6, [] Chada, M.J. ad Bahe, M. J. (985, he effects of flato ame value of moey o some vetoy systems, Iteatoal Joual of Poducto Reseach, 3, [3] Chag, C..(4, A EOQ model wth deteoatg tems ude flato whe supple cedts lkeo ode quatty, Iteatoal Joual of Poducto Ecoomcs, 88, [4] Datta,. K. ad Pal, A. K. (99, Effects o flato ame value of moey o a vetoy model wth lea tme depedet demad ate ad shotages, Opseach, 7, [5] Haga, M. A. (995, Effects of flato ame value of moey o a vetoy model wth tmevayg demad ate ad shotage, Euopea Joual of Opeatoal Reseach, 8, 5-5. [6] Haga. M. A. ad Be-Daya, M. (996, Optmal tme vayg lot szg models ude flatoay codtos, Euopea Joual of Opeatoal Reseach, 89, [7] Jagg, C.K. et al. (6, Optmal ode polcy fo deteoatg tems wth flato duced demad, Iteatoal Joual of Poducto Ecoomcs, 3, [8] Moo, I., G, B. C., Ko, B.(5, Ecoomcs ode quatty models fo ameloatg deteoatg tems ude flato ame dscoutg, Euopea Joual of Opeatoal Reseach, 6, [9] Sgh, S.R. et al. (9, A ode level vetoy model fo peshable tems wth stock depedet demad ad patal backloggg, Iteatoal Joual of Computatoal ad Appled Mathematcs, 4,, 9 8. [] Sghal, S., Sgh, S.R., Sgh, D. (4. A pobablstc vetoy model fo webull deteoatg tems wth flexblty ad elablty ude the effect of flato. Poceedgs of st Iteatoal Scece & echology Cogess (Elseve, [] Sghal, S., Sgh, S.R. ad Gupta, P.K. (6. Stochastc patal backloggg vetoy models fo deteoatg tems wth tme depedet demad ad volume elastcty, Iteatoal Joual of Agcultual ad Statstcal Sceces,,,, [] pathy, C.K. ad Msha, U. (, A vetoy model wth tme depedet lea deteoatg tems wth patal backloggg, Iteatoal Joual of Math. Aalyss, 4, 38, [3] Yag, H. L. (4, wo waehouse vetoy models fo deteoatg tems wth shotages ude flato, Euopea Joual of Opeatoal Reseach, 57, [4] Yag. H. L. (6, wo waehouse patal backloggg vetoy models fo deteoatg tems ude flato, Iteatoal Joual of Poducto Ecoomcs, 3, Copyght 7. Vadaa Publcatos. All Rghts Reseved.
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