AN EOQ MODEL FOR THREE PARAMETER WEIBULL DETERIORATING ITEM WITH PARTIAL BACKLOGGING

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1 LogForum 03, 9, 35-4 > Scienific Journ of Logisics < hp:// p-issn e-issn X AN EOQ MODEL FOR THREE PARAMETER WEIBULL DETERIORATING ITEM WITH PARTIAL BACKLOGGING L.M. Prdhn, C. K. Triphy Siicon Insiue of Technoogy, Smpur, Indi, Smpur Universiy, Smpur, Indi ABSTRACT. Bckground: Business orgnisions re fcing o of compeiion during hese dys. To wihsnd he compeiion nd o remin in he fron row, n enerprise shoud hve opimum profie pn for his usiness. Reserchers in recen yers hve deveoped vrious invenory modes for deerioring iems considering vrious prcic siuions. Pri ckogging is considery new concep inroduced in deveoping vrious modes for Weiu deerioring iems. Mehodoogy: In his pper n invenory mode hs een deveoped considering hree prmeer Weiu deeriorion of singe iem wih pri ckogging. Here demnd re is considered o e consn nd ed ime is zero. During he sock ou period he ckogging re is vrie nd is dependen on he engh of he wiing ime for he nex repenishmen. Resus nd concusion: Opim order quniy nd o vrie cos during cyce hs een derived for he proposed invenory mode considering hree prmeer Weiu deerioring iem wih pri ckogging. The resus oined in his pper re iusred wih he hep of numeric exmpe nd sensiiviy nysis.. Key words: EOQ, Weiu deeriorion, pri ckogging. INTRODUCTION I is genery oserved in vrious mrkes nd super mrkes h he demnd re is usuy infuenced y he moun of he sock eve. When usiness runs ou of sock we consider i s negive spec, u in cerin siuions his shorge my cuy prove enefici for he usiness, ecuse ckogging of demnd ows him o order rger o size nd hence ows for rger cyce invenory nd reduces cos for him. Pri ckogging is funcion of wiing ime where s compee ckogging is independen of wiing ime. Looking vrious siuions reserchers hve sudied in his direcion nd hve deveoped vrious invenory modes. Opim pricing nd osizing under condiions of perishiiy nd pri ckordering ws deveoped y Ad [996] nd Ad [00]. Chng nd Dye [999] sudied n EOQ mode for deerioring iems wih ime vrying demnd nd pri ckogging. Then Dye, C. [007-], deveoped mode of opim seing price nd o-size wih vrying re of deeriorion nd exponeni pri ckogging. A deerminisic invenory mode for deerioring iems wih cpciy consrin nd ime proporion ckogging re ws sudied y Dye [007] e.. Then Hung, K., [0] deveoped n invenory mode wih generized ype demnd, deeriorion nd ck order res. Erier Prk, K.S. [98] proposed n invenory mode wih pri ck orders nd Jn [996] e. sudied n EOQ mode for iems wih Weiu disriuion deeriorion, shorges nd rended demnd. Copyrigh: Wyższ Szkoł Logisyki, Poznń, Posk Ciion: Prdhn L.M., Triphy C.K., 03, An EOQ mode for hree prmeer weiu deerioring iem wih pri ckogging. LogForum 9, 35-4 URL: hp:// Acceped:..0, on-ine:

2 Prdhn L.M., Triphy C.K., 03, An EOQ mode for hree prmeer weiu deerioring iem wih pri ckogging. LogForum LogForum 9, URL: hp:// Ouyng e. [006] hve esished mode considering opim ordering poicy for deerioring iems wih pri ckogging under permissie dey in pymens. Shh, N. nd Shuk, K., [009] sudied deerioring invenory mode for wiing ime pri ckogging. An EOQ invenory mode wih Weiu disriuion deeriorion, rmp ype demnd nd pri ckogging ws sudied y Singh, S.R. nd. Singh, T.J. [007]. Then Singh, S.R. nd Singh, C., [008] esished Perishe invenory mode wih qudric demnd, pri ckogging nd permissie dey in pymens. Skouri, e. [009] deveoped n invenory mode wih rmp ype demnd re, pri ckogging nd Weiu deeriorion re. Teng e. [007], mde comprison eween wo pricing nd o-sizing modes wih pri ck ogging nd deeriored iems. Triphy C.K. nd Prdhn L.M., [00], deveoped n EOQ mode for Weiu deerioring iems wih power demnd nd pri ck ogging. In h pper hey hve owed shorges which re priy ckogged. Triphy, C. K., nd Mishr, U., [00] hve sudied n invenory mode wih ime dependen iner deerioring iems wih pri ckogging. Wu nd Cheng, [005] esished n invenory mode for deerioring iems wih exponeni decining demnd nd Pri ck ogging. In he presen pper n economic ordered quniy mode hs een deveoped considering hree prmeer Weiu deeriorion where shorges re owed nd re priy ckogged. The hoding cos nd demnd re re ssumed o e consn for his mode. In secion ssumpions nd noions required for he deveopmen of he mode re given. The opimum cyce ime, hoding cos, opim ordered quniy nd o verge cos of he mode re derived in he Secion 3. An iusrive numeric exmpe, sensiiviy nysis nd concusion re given in secion 4, 5 nd 6 respecivey. BASIC ASSUMPTIONS AND NOTATIONS The foowing re he ssumpions required for deveopmen of he mode:. The mode des wih singe iem.. Demnd re for he produc is known nd consn. 3. Pnning horizon is infinie. 4. Led ime is zero. 5. Once uni of he produc is produced, i is vie o mee he demnd. 6. The ckogging re is vrie nd is dependen on he engh of he wiing ime for he nex repenishmen. For he negive invenory he ckogging re is B T ; > 0 denoes he ckogging prmeer. 7. Deeriorion re is hree prmeer Weiu The noions h re empoyed here: A : ordering cos per order. : consn demnd re. C : Purchse cos per uni. h : Invenory hoding cos per uni per uni ime. θ : Weiu hree prmeer deeriorion re uni/uni ime, θ, where 0 < <<, > 0, nd 0 < <, where is ced sce prmeer is ced shpe prmeer nd is ced he ocion prmeer. : Bckordered cos per uni shor per ime uni. : Cos of os ses per uni. : The ime which he invenory eve reches zero, 0. : The engh of period during which shorges re owed, 0. T : The engh of cyce ime, i.e. T. I : The eve of posiive invenory ime, 0 I : The eve of negive invenory ime, IM : Mximum invenory eve of he produc during[ 0,T ]. IB : Mximum ckordered unis during sock ou period. 36

3 Prdhn L.M., Triphy C.K., 03, An EOQ mode for hree prmeer weiu deerioring iem wih pri ckogging. LogForum LogForum 9, URL: hp:// Q : Order quniy during cyce of enght, i.e. Q IM IB. TC : To verge cos per ime uni. MATHEMATICAL MODEL The inii invenory eve or he mximum eve of invenory I 0 IM, decreses due o he comined effec of demnd nd deeriorion during he ime [ 0, ]. Thus, he invenory eve of he produc ime over he period [ 0, ] cn e represened y he foowing differeni equion di θ I d, 0 Using he vue of θ, 0 < <<, > 0 nd 0 < < ced he sce, shpe nd ocion prmeer respecivey. di I d, 0 Invenory eve reches o zero ime. Afer h shorges occur. During he inerv[, ], he invenory eve depends on demnd nd frcion of demnd is ckogged. The se of invenory during [ ], cn e represened y he differeni equion, di d, Here he oundry condiions re I I 0 Equion is iner differeni equion. Is inegring fcor is given y d e e. Hence he souion of equion cn e wrien s e e d I c, where c, is he consn of inegrion. Since, 0 < << so king he firs wo erms from he series expnsion of he exponeni funcion nd hen inegring we ge I e c Using he given oundry condiion I 0 in he ove we ge he required souion of equion s I e I e Agin king he firs wo erms from he series expnsion of he exponeni funcion nd negecing he higher power of h is he power greer hn or equ o, in he ove he souion of equion cn e rewrien s I, 0 3 Simiry he souion of equion cn e wrien s I n{ } c Using he oundry condiion I 0 in he ove, required souion of equion is I [ n{ } n ], 4 The mximum posiive invenory is IM I 0 y, IB 5 The mximum ckordered unis re given I [ n{ } n ] n 6 37

4 Prdhn L.M., Triphy C.K., 03, An EOQ mode for hree prmeer weiu deerioring iem wih pri ckogging. LogForum LogForum 9, URL: hp:// 38 The order size during [ ],T 0 is IB IM Q n Q 7 Ordering cos per cyce is A OC 8 Invenory hoding cos per cyce is d I h IHC 0 d h 0 h 9 Bckordered cos per cyce is d I BC [ ] n n d [ ] n 0 Cos due o os ses per cyce is given y, d LS [ ] n Purchse cos per cyce is, Q C PC n C Therefore he o verge cos per ime uni is [ ] PC LS BC IHC OC TC h A n n n C 3 The necessry condiion for he o verge cos o e minimized is 0 TC h A n n n C 0 C h 4 nd 0 TC h A n n

5 Prdhn L.M., Triphy C.K., 03, An EOQ mode for hree prmeer weiu deerioring iem wih pri ckogging. LogForum LogForum 9, URL: hp:// C Provided, n n 0 C 5 TC TC TC > 0, for oined vue of,. 6 The equions 4 nd 5 re highy non iner. They cn e soved y mhemic-5. sofwre for given se of known prmeers. The oined vues of nd mus sisfy equion 6 o minimize he o cos per ime uni of he invenory sysem. To iusre hese we hve given numeric exmpe nd sensiiviy nysis in he foowing secions. NUMERICAL EXAMPLE Le us consider n invenory sysem wih he foowing prmeric vues in heir proper unis. [ A, C, h,,,,,,, ]. [ 3000,00, 0.4, 0, 60,, 8, 0.08,4, 0.4] Using hese vues in equion 4 nd 5 we ge,. 066 nd respecivey. Puing he opimum vues of nd in equion 7 nd 3 we ge, Q 3.79 nd TC respecivey. performed y chnging he vues of he prmeers from -50% o 50%, one y one in he mode which re given in he foowing e-. From he e- we cn concude he foowing: The opim ime increses s A,,,, increses u i decreses s, h,,, C increses. Simiry he opim ime increses s C, h, A,, increses u i decreses s,,,, increses. Agin he opim ordered quniy Q per cyce increses s A,,,, increses u i decreses s C,h,, increses. The, o verge cos TC of he sysem increses s A, C, h,,,,,, increses u i decreses s increses. CONCLUSION In he presen pper n economic ordered quniy mode hs een deveoped for n iem wih hree prmeer Weiu deeriorion where shorges re owed nd re priy ckogged. The opim cyce ime, opim ordered quniy per cyce nd o opim cos hs een derived for he mode. Sensiiviy nysis shows how he differen prmeers ffec he opim cyce ime, ordered quniy per cyce nd o opim cos. I cn e concuded h o minimise he o cos, i is required o minimize he ordering cos, purchse cos, hoding cos, ck ordered cos, he cos of os ses per uni, he demnd re, ckogging prmeer, sce prmeer nd shpe prmeer, wheres we need o mximise he vue of he ocion prmeer. SENSITIVITY ANALYSIS For sudy of sensiiviy nysis e us chnge one prmeer ime keeping he oher prmeers unchnged. The origin vues of he prmeers for sensiiviy nysis hve een ken from he exmpe given in secion 4 ove. Sensiiviy nysis is 39

6 Prdhn L.M., Triphy C.K., 03, An EOQ mode for hree prmeer weiu deerioring iem wih pri ckogging. LogForum LogForum 9, URL: hp:// Prmeer %chnge Q Te. sensiiviy nysis Te. Aniz wrżiwości A C h TC REFERENCES Ad, P.L., 00, Opim price nd order-size for reseer under pri ckogging, Compuers nd Operion Reserch, 8, Ad, P.L., 996, Opim pricing nd osizing under condiions of perishiiy nd pri ckordering, Mngemen Science, 4 8, Chng, H. nd Dye, C., 999, An EOQ mode for deerioring iems wih ime vrying demnd nd Pri ckogging, Journ of he Operion Reserch Sociey, 50, Dye, C., 007-, Deermining opim seing price nd o-size wih vrying re of 40

7 Prdhn L.M., Triphy C.K., 03, An EOQ mode for hree prmeer weiu deerioring iem wih pri ckogging. LogForum LogForum 9, URL: hp:// deeriorion nd exponeni pri ckogging, Europen journ of Operion Reserch, 8, Dye, C., Ouyng, L. nd Hsieh, T., 007, Deerminisic invenory mode for deerioring iems wih cpciy consrin nd ime proporion ckogging re, Europen journ of Operion Reserch, 78 3, Hung, K., 0, An invenory mode wih generised ype demnd, deeriorion nd ck order res, Europen journ of Operion Reserch, 08 3, Jn, A., Giri, R. nd Choudhury, K., 996, EOQ mode for iems wih Weiu disriuion deeriorion, shorges nd rended demnd, Inernion journ of Producion Economics., 3, Ouyng, L.Y, Teng, J.T. nd Chen, L.H, 006, Opim ordering poicy for deerioring iems wih pri ckogging under permissie dey in pymens, Journ of Go Opimizion, 34, Prk, K.S., 98, Invenory mode wih pri ck orders, Inernion journ of sysems Science, 3, Shh, N. nd Shuk, K., 009, Deerioring invenory mode for wiing ime pri ckogging, Appied Mhemic Sciences, 3, Singh, S.R. nd Singh, T.J., 007, An EOQ invenory mode wih Weiu disriuion deeriorion, rmp ype demnd nd pri ckogging, Indin Journ of Mhemics nd Mhemic Sciences, 3, Singh, T.J., Singh, S.R. nd Singh, C., 008, Perishe invenory mode wih qudric demnd, pri ckogging nd permissie dey in pymens, Inernion Review of Pure nd Appied Mhemics, Skouri, K., Konsnrs, S. nd Gns, I., 009, Invenory modes wih rmp ype demnd re, pri ckogging nd Weiu deeriorion re, Europen journ of Operion Reserch, 9, Teng, J., Ouyng, L. nd Chen, L., 007, A comprison eween wo pricing nd osizing modes wih pri ck ogging nd deeriored iems, Inernion journ of Producion Economics, 05, Triphy C.K. nd Prdhn L.M., 00, An EOQ mode for Weiu deerioring iems wih power demnd nd pri ck ogging, Inernion journ of Conemporry Mhemic Sciences, 538, Triphy, C. K., nd Mishr, U., 00. An invenory mode wih ime dependen iner deerioring iems wih pri ckogging, Inernion Journ of Mhemic. Anysis, 438, Wu, O. nd Cheng, 005, An invenory mode for deerioring iems wih exponeni decining demnd nd Pri ck ogging, Yugosv Journ of Operions Reserch, 5, MODEL EKONOMICZNEJ WIELKOŚCI PARTII DLA TRZECH PARAMETRÓW UTRATY WARTOŚCI WEIBULLA Z CZĘŚCIOWYMI ZALEGŁOŚCIAMI STRESZCZENIE. Wsęp: W oecnych czsch przedsięiorsw musz sprosć wieu wymgniom swinym prze konkurencyjny rynek. Ay pokonowć konkurencję pozosć iderem, przedsięiorswo powinno opierć się n opymnym pnie zyskowności swojej dziłności. W osnich ch nukowcy oprcowi wiee modei d sorymenów podegjących zużyciu prkycznie d kżdej wysępującej w rzeczywisości syucji. Częściowe zegłości są nową koncepcją wprowdzoną w rozwijnych modech Weiu. Meody: Mode zpsów sorymenu d kórego wysępują częściowe zegłości zosł oprcowny d rzech prmerów ury wrości Weiu. Złożono, że popy jes sły orz czs reizcji równ się zero. W okresie rków współczynnik zegłości jes zmienny i zeży od czsu oczekiwni n uzupełnienie zpsów. 4

8 Prdhn L.M., Triphy C.K., 03, An EOQ mode for hree prmeer weiu deerioring iem wih pri ckogging. LogForum LogForum 9, URL: hp:// Wyniki i wnioski: Opymn wiekość zmówieni orz cłkowiy kosz zmienny w czsie cyku zosły oprcowne d proponownego modeu zpsów uwzgędnijącego rzy prmery zużyci Weiu z częściowymi zegłościmi. Wyniki zosły zprezenowny przy pomocy przykłdu iczowego orz nizy wrżiwości.. Słow kuczowe: ekonomiczn wiekość prii, zużycie Weiu, częsciowe zegłości MODELL DER WIRTSCHAFTLICHEN LOSGRÖßE EINER BESTELLUNG FÜR DREI PARAMETER DES WEIBULL- WERTVERLUSTES MIT TEILHAFTEN RÜCKSTÄNDEN ZUSAMMENFASSUNG. Eineiung: Der Weewersmrk se heuzuge vor Unernehmen viee wichige Herusforderungen. Um die Konkurrenz zu üerhoen und die Führungsposiion zu ewhren, soen sich die Unernehmen uf den opimen Pn der Reniiä ihrer Beäigung süzen. In den ezen Jhren hen die Wissenschfer viee Modee für die Sorimene, die einem Verscheiß uneriegen, prkisch für jede in der Wirkichkei ufreende Siuion usgereie. Ds Mode mi den eihfen Rücksänden se ein neues, in den enwickeen Weiu-Modeen eingeführes Konzep dr. Mehoden: Ds Mode der Sorimensvorräe, ei denen die eihfen Rücksänden ufreen, wurde für die drei Prmeer des Weiu-Werveruses konzipier. Mn h ngenommen, dss die Nchfrge konsn ei und die Ausführungszei der Beseung dem Nu geich. In der Zei der ufreenden Defizie is der Koeffizien der Rücksände vrie und häng von der Wrezei eim Nchschu der Vorräe. Ergenisse und Fzi: Die opime Losgröße der Beseung sowie die vrien Gesmkosen innerh eines Zykus wurden für die Zwecke des vorgeschgenen Vorrsmodes, weches die drei Prmeer des Weiu-Werveruses mi eihfen Rücksänden erücksichig, usgereie. Die Ergenisse wurden nhnd eines Zheneispies und der Empfindichkeisnyse drgese. Codewörer: wirschfiche Losgröße, Weiu-Werverus, eihfe Rücksände Prdhn L.M. Deprmen of Mhemics Siicon Insiue of Technoogy Sson, Smpur-76800, Indi e-mi: [email protected] Triphy C. K. Deprmen of Sisics Smpur Universiy Jyoi Vihr Smpur-76809, Indi e-mi: [email protected] 4