A Deterministic Inventory Model For Weibull Deteriorating Items with Selling Price Dependent Demand And Parabolic Time Varying Holding Cost

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1 Inernionl Journl of Sof omuing nd Engineering IJSE ISSN: 3-37 Volume-5 Issue- Mrc 5 A Deerminisic Invenory Model For Weibull Deerioring Iems wi Selling rice Deenden Demnd And rbolic ime Vrying Holding os Viin Kumr Anum Srm.B.Gu Absrc- is er wi develomen of n invenory model wen deeriorion re follows Weibull wo wy rmeer disribuions. I is ssumed demnd re is funcion of selling rice nd olding cos is rbolic in erms of ime. In is models bo e cses wi sorge nd wiou sorge re ken ino considerion. Wenever sorge llowed is comleely bcklogged. o illusre e resul numericl exmles re given.e sensiive nlysis for e model s been erformed o sudy e effec of cnges e vlue of rmeers ssocied wi e model. Memics Subjec lssificion: - 9B5 Keywords: - EOQ model deerioring iems Weibull disribuion sorge rice deenden demnd rbolic olding cos. I. Inroducion ese dys resercer re ying more enion o e invenory model of deerioring iems. I s no esy o neglec e effec of deeriorion s i is relisic feure nd is very common in dily rouine life. So ve o consider i. Wee HM [993] is e firs one wo define deerioring iems refers o e iems become decyed dmged evorive exired invlid devluion nd so on by e ssing ime. rdiionlly i ws considered e iems cn reserve eir crcerisics wile ey ke sored in invenory. Bu i is no rue for ll. onsidering is fc now dys i s gre cllenge o conrol nd minin e invenory of deerioring iems for e decision mkers. e firs EOQ invenory model ws develoed by Hrris [95] wic ws furer generlized by Wilson [93] o obin formul for economic order quniy. Wiin [957] sudied e deeriorion of e fsion goods e end of e rescribed sorge eriod. Gre nd Scrder [963] develoed model for n exonenilly decying invenory. Misr [975] develo model wi Weibull deeriorion re wiou bckordering. Moving furer Dve nd el [98] were e firs o sudy deerioring invenory wi liner incresing demnd wen sorges re no llowed. is model ws furer generlized by Scn[98] wi llowed sorges. Furer work in is field s been done by ung nd ing [993]; Wee [995] sudied n invenory model wi deerioring iems. ng nd Dye [999] develoed n invenory model wi ime-vrying demnd nd ril bcklogging. Goyl nd Giri[] gve recen rends of modeling in deerioring iem invenory. Mnuscri Received on Februry 5. Viin Kumr De. of Memics BKBIE ilni Anum Srm Reserc Scolr Mewr Universiy Rjsn Indi.B. Gu De. of Memics BIS ilni ey clssified invenory models on e bsis of demnd vriions nd vrious oer condiions or consrins. Wi exonenil declining demnd nd ril bcklogging Ouyng nd eng [5] develoed n invenory model for deerioring iems. Almri nd Blki[7] sudied e effecs of lerning nd forgeing on e oiml roducion lo size for deerioring iems wi ime-vrying demnd nd deeriorion res. Dye [7] find n oiml selling rice nd lo size wi vrying re of deeriorion nd exonenil ril bcklogging. ey ssume frcion of cusomers wo bcklog eir orders increses exonenilly s e wiing ime for e nex relenismen decreses. Roy [8] develoed deerminisic invenory model wen e deeriorion re is ime roorionl. Demnd re is funcion of selling rice nd olding cos is ime deenden. Lio [8] gve n economic order quniy EOQ model wi non insnneous recei nd exonenil deerioring iem under wo level rde credis. reek e l. [9] develoed deerminisic invenory model for deerioring iems wi slvge vlue nd sorges. Skouri e l. [9] develoed n invenory model wi rm-ye demnd re ril bcklogging nd Weibull's deeriorion re. Misr nd Sing [] develoed deerioring invenory model for wiing ime ril bcklogging wen demnd nd deeriorion re is consn. riy nd Misr [] give e model for deerioring iems wi rice deenden demnd nd liner olding cos. Mndl [] gve n EOQ invenory model for Weibull-disribued deerioring iems under rm-ye demnd nd sorges. Hung [] gve n invenory model wi generlized-ye demnd deeriorion nd bckorder res. Misr nd Sing [] gve n invenory model for rm-ye demnd imedeenden deerioring iems wi slvge vlue nd sorges nd deerioring invenory model for imedeenden demnd nd olding cos wi ril bcklogging. In is er we exend e er of riy Misr[]. we develoed generlized EOQ model for deerioring iems were deeriorion re follows wormeer Weibull disribuion nd olding cos re exressed s rbolic funcions of ime nd demnd re considered o be funcion of selling rice. For e model were sorges re llowed ey re comleely bcklogged. Here we ve considered bo e cse of wi sorge nd wiou sorge in develoing e model. II. Assumions nd Noions e fundmenl ssumions of is model re s follows:- e deeriorion re is roorionl o ime 5 ublised By: Blue Eyes Inelligence Engineering & Sciences ublicion v. Ld.

2 A Deerminisic Invenory Model For Weibull Deerioring Iems wi Selling rice Deenden Demnd And rbolic ime Vrying Holding os b e deeriorion of unis follows e wo rmeer Weibull disribuion θ = were << is e se rmeer c Demnd re is funcion of selling rice. d Sorge wenever llowed re comleely bcklogged. e Holding cos er iem er ime- uni is ime deenden nd is ssumed o be = were >>. f Selling rice follows n incresing rend demnd re ossess e negive derivive rougou is domin were demnd re is f = > Memicl formulion nd soluions :- g is e comleeleng of cycle. Relenismen is insnneous nd led ime is zero. i e order quniy in one cycle is q. j A is e cos of lcing n order. k e selling rice er uni iem is. l is e uni cos of n iem. m e invenory olding cos er uni er uni ime is. n is e sorge cos er uni er uni ime.. o Invenory is deleed due o deeriorion nd demnd of e iem. A ime e invenory becomes zero nd sorge srs occurring. Le Q be e invenory level ime. e differenil equions o describe insnneous se over re given by dq Q = d dq d =. Wi = = Solving equion nd equion nd neglecing iger ower of we ge Q = And Q = Now sock loss due o deeriorion is given by:- D = Q d =..3 Ordering quniy is given by:- q = D d q = q = 53 ublised By: Blue Eyes Inelligence Engineering & Sciences ublicion v. Ld.

3 Inernionl Journl of Sof omuing nd Engineering IJSE ISSN: 3-37 Volume-5 Issue- Mrc 5 5 ublised By: Blue Eyes Inelligence Engineering & Sciences ublicion v. Ld. Holding cosis :- d Q H = = d Neglecing iger ower of we ge H= Now Sorge cos during e cycle:- = d Q S S =..5 From equion 3 nd 5. ol rofi er uni ime is given by = -- A q H S = A.6 Le = < < = A..7 is equion gives e rofi funcion. In order o mximize e rofi funcion. e necessry condiions for re given by

4 A Deerminisic Invenory Model For Weibull Deerioring Iems wi Selling rice Deenden Demnd And rbolic ime Vrying Holding os 55 ublised By: Blue Eyes Inelligence Engineering & Sciences ublicion v. Ld. = nd = We ge A =.. 8 And -- { } 3 3 = 9 Using e sofwre Memic-5. nd Ms. Excel from e equions 8 nd 9 we cn clcule e oimum vlue of * nd * simulneously nd e oiml vlue * of e verge ne rofi is deermined by 7 rovided ey sisfy esufficiency condiions for mximizing * re < <... And > = * nd = *.. If e soluion obined from e equions 8 nd 9 do no sisfy e sufficiency condiions nd we conclude no fesible soluion will be oiml for e se of rmeer vlues ken o solve equions 8 nd 9. Suc siuion will imly e rmeer vlues re inconsisen nd ere is some error in esimion. Numericl Exmle se-i wi Sorge Exmle-: Le A= = = =. =. =. =.3 =.95 =. Bsed on ese inu d e comued ouus re s follows: * = * =3.675 * = q=8.38 * =.8659 se-ii wiou sorge Exmle-:

5 Inernionl Journl of Sof omuing nd Engineering IJSE ISSN: 3-37 Volume-5 Issue- Mrc 5 Le A= = = =. =. =. =.3 =.95 =. * = * =3.5 * =6.377 q= * = III. Sensiive Anlysis nge in rmeers will ffec e resul lso. So in order o find cnge roer sensiive nlysis is erformed for is model. I is done by king % nd 5% vriion in rmeers. For is nlysis one rmeer is cnged nd oers remin e sme. Alying is conce ow e resul will flucue is sown in bles below. ble sows e resul wi sorge nd ble wiou sorge. ble- rmeers % cnge rofi Selling rice ime Ordering Quniyq A A H ublised By: Blue Eyes Inelligence Engineering & Sciences ublicion v. Ld.

6 A Deerminisic Invenory Model For Weibull Deerioring Iems wi Selling rice Deenden Demnd And rbolic ime Vrying Holding os Γ Sudy of e ble- revels e following:- i If vlue of rmeer A increse i will led o increse in * * q * nd decrese in *. ii If vlue of rmeer increse i will led o increse in * q * nd decrese in * *. iii If vlue of rmeer increse i will led o increse of * nd decrese in * * q *. iv If vlue of rmeer increse i will led o increse of * * nd decrese in* q *. v If vlue of rmeer incresei will led o increse of * * nd decrese in * q *. vi If vlue of rmeer increse i will led o increse of * nd decrese in * * q *. vii If vlue of rmeer incresei will led o increse of * nd decrese in * q * *. viii If vlue of rmeer increse i will led o increse of * nd decrese in *q * *. ix If vlue of rmeer increse i will led o increse of * * nd decrese in * q *. ble- rmeers % cnge rofi Selling rice ime Ordering Quniyq A A H ublised By: Blue Eyes Inelligence Engineering & Sciences ublicion v. Ld.

7 Inernionl Journl of Sof omuing nd Engineering IJSE ISSN: 3-37 Volume-5 Issue- Mrc Γ Sudy of e ble- revels e following:- i If vlue of rmeer A increse i will led o incresein* * q * nd decrese in *. ii If vlue of rmeer increse i will led o increse of * q * * nd decrese in *. iii If vlue of rmeer increse i will led o increse of * nd decrese in * * q *. iv If vlue of rmeer increse i will led o increse of * * nd decrese in * q *. v If vlue of rmeer increse i will led o increse of * nd decrese in * * q *. vi If vlue of rmeer increse i will led o increse of * nd decrese in * q * *. vii If vlue of rmeer increse i will led o increse of * nd decrese in *q * *. viii If vlue of rmeer increse i will led o increse of * * nd decrese in * q *. IV. onclusion In is er we develoed deerminisic invenory model for deerioring iems for wi sorge nd wiou sorge cses. e deerminisic demnd re is ssumed o be funcion of selling rice. Wenever sorge re llowed re comleely bcklogged nd olding cos is ssumed o be rbolic ime deenden. We cn mke good comrive sudy beween e resul of e wi sorge nd wiou sorge cse. In e numericl exmles i is found e oimum verge rofi in wiou sorge cse is more n of e wi sorge cse. From e bove model one cn clcule e oimum verge rofi mrgins for e sorge cse nd wiou sorge cse for e deerminisic invenory model wi vrying demnd re nd olding cos subjeced o e condiions. References. Hrris FW 95 Oerions nd cos. A. W Sw omny icgo. Wilson RH 93 A scienific rouine for sock conrol. Hrv Bus Rev 3: Wiin M 957 e eory of invenory mngemen nd ediion. rinceon Universiy ress rinceon. Gre M Scrder GF 963 A model for n exonenilly decying invenory. 5. Dve U el LK 98 Si olicy invenory model for deerioring iems wi ime roorionl demnd. Journl of Oerionl Reserc Sociey 3:37 6. ung KJ ing S 993 A eurisic for relenismen for deerioring iems wi liner rend in demnd. Journl of Oerionl Reserc Sociey :35 7. Wee HM 995 A deerminisic lo-size invenory model for deerioring iems wi sorges nd declining mrke. omuionl Oerion reserc : Abd L 996 Oiml ricing nd lo-sizing under condiions of erisbiliy nd ril bckordering. Mnge Sci :93 Abd L oiml rice nd order-size for reseller under ril bcklogging. omuionl Oerion reserc 8: ng HJ Dye Y 999 An EOQ model for deerioring iems wi ime vrying demnd nd ril bcklogging Journl of Oerionl Reserc Sociey 5:76 8. Goyl SK Giri B Recen rends in modeling of deerioring invenory. Euroen Journl of Oerion Reserc 3: 6. Ouyng W eng X 5 An invenory model for deerioring iems wi exonenil declining demnd nd ril bcklogging. Yugoslv Journl of Oerion Reserc5 : Almri AA Blki Z 7 e effecs of lerning nd forgeing on e oiml roducion lo size for deerioring iems wi ime vrying demnd nd deeriorion res. Inernionl Journl of roducion Economics 7: Dye Y Ouyng LY Hsie 7 Deerminisic invenory model for deerioring iems wi cciy consrin nd ime-roorionl bcklogging re. Euroen Journl of Oerion Reserc 783: Roy A 8 An invenory model for deerioring iems wi rice deenden demnd nd ime vrying olding cos. Adv Modeling O : Lio JJ 8 An EOQ model wi non-insnneous recei nd exonenildeerioring iem under wo-level rde credi Inernionl Journl of roducion Economics 3: ublised By: Blue Eyes Inelligence Engineering & Sciences ublicion v. Ld.

8 A Deerminisic Invenory Model For Weibull Deerioring Iems wi Selling rice Deenden Demnd And rbolic ime Vrying Holding os 6. Skouri K Konsnrs I crisos S Gns I 9 Invenory models wi rm ye demnd re ril bcklogging nd Weibull deeriorion re.euroen Journl of Oerion Reserc 9: Mndl B An EOQ invenory model for Weibull disribued deerioring iems under rm ye demnd nd sorges. Oserc 7: Misr VK Sing LS Deerioring invenory model wi ime deenden demnd nd ril bcklogging. Alied Memicl Sciences 7: riy K Misr U[] An invenory model for Weibull deerioring iems wi rice deenden demnd nd ime vrying olding cos.. Hung K- An invenory model wi generlized ye demnd deeriorion nd bckorder res. Euroen Journl of Oerion Reserc 83:39. Misr VK Sing LS Invenory model for rm ye demnd imedeenden deerioring iems wi slvge vlue nd sorges. Inernionl Journl of memics nd sisics 3D: ublised By: Blue Eyes Inelligence Engineering & Sciences ublicion v. Ld.