An EOQ Model for Weibull Deteriorating Items with. Power Demand and Partial Backlogging

Transcription

1 . J. oemp. Mah. Scieces, Vol. 5, 00, o. 38, A EOQ Moel for Weibull Deerioraig ems wih Power Dema a Parial Backloggig. K. ripahy* a L. M. Praha ** *Deparme of Saisics, Sambalpur Uiversiy, Jyoi Vihar Sambalpur-76809,ia c.ripahy@yahoo.com **Deparme of Mahemaics, P.K.A.ollege of Egg. hakarke, Bargarh , ia lalishivbaba@gmail.com Absrac his paper a EOQ moel is evelope for Weibull eerioraig iem wih power ema paer i which shorages are allowe a parially backlogge. he backloggig rae is ake o be iversely proporioal o he waiig ime for he ex repleishme. his moel we have cosiere Weibull wo parameer eerioraio a power ema paer. A he e a umerical example is provie o illusrae he problem a sesiiviy aalysis have bee carrie ou for showig he effec of variaio i he parameers. Mahemaics Subjec lassificaio: 90B05 Keywors: Power Dema Paer, Weibull Deerioraio, Parial Backloggig, Shorages. roucio Deerioraio plays a impora role i may iveory sysems. Deerioraio is efie as ecay or amage i he qualiy of he iveory. some subsaces like foos, rugs, pharmaceuicals, a raio acive subsaces eerioraio akes place urig he ormal sorage perio of he uis a cosequely his loss mus be ake

2 896. K. ripahy a L. M. Praha io accou whe aalyzig he sysem. Whe he iems of he commoiy are kep i sock as a iveory for fulfillig he fuure ema here may be he eerioraio of iems akes place i he iveory sysem. Researchers i he fiel of iveory corol have suggese various moels akig io cosieraio iffere emas a eerioraios. Daa a Pal [5] ivesigae a iveory sysem wih power ema paer for iems wih variable rae of eerioraio. veory moel of eerioraig iems wih ime proporioal backloggig rae have bee suie by hag [3], hag [] a Dye []. Joi pricig a repleishme policy for eerioraig iveory was suie by Wee [5] a Yag [4]. Sock epee a parial backloggig i iveory moel was suie by Wu a Ouyag [8], eg a Yag [6], a Sigh [3]. Whe he shorage occurs, some cusomers are willig o wai for back orer a ohers woul ur o buy from oher sellers. Park [7] a Wag [] suie shorages a parial back loggig of iems. Permissible elay i paymes was suie by Sigh [4], Balkhi a Bekherouf [6]. Fiie rae of repleishme was suie by Roychouhury [9]. Opimal pricig a lo sizig uer coiios of perishabiliy a parial back loggig was evelope by Aba [0]. Hollier a Mak [] have evelope a iveory repleishme policy for eerioraig iems i a ecliig marke. he prese work a eermiisic iveory moel wih power ema paer is evelope i which he eerioraio is a Weibull wo parameer isribuio. Shorages are allowe a parially backlogge i his moel. We have show he effec ue o chages i various parameers by akig a suiable umerical example a a sesiiviy aalysis.. Assumpio a Noaio his paper he followig assumpios a oaios have bee use. i. is he iveory level a ay ime, 0 a S is he iiial iveory level. ii. is he life ime of iems a θ is he Weibull wo parameer eerioraio. Where 0 < <, > 0 are calle scale a shape parameer respecively. iii. D is he ema rae a ay ime such ha D, where is a posiive cosa, may be ay posiive umber, is he plaig horizo.

3 A EOQ moel for Weibull eerioraig iems 897 iv.,,, 3 a 4 eoe he se up cos, iveory carryig cos, eerioraio cos per ui ime, shorage cos for backlogge iems a he ui cos of los sales respecively. All of he cos parameers are posiive cosas. v. here is o repleishme or repair of eeriorae iems akes place i a give cycle. vi. he lea ime is zero. vii. A sigle iem is cosiere over he fixe perio which is subjec o weibull eerioraio rae. viii. Deerioraio akes place afer he life ime of iems. ix. he repleishme akes place a a ifiie rae. / x. Shorages are allowe a backloggig rae is whe iveory a is i shorage. he backloggig parameer a is posiive cosa a 0 < a <<. 3. Formulaio of he Moel A he begiig of each cycle he oal amou of iveory prouce or purchase is assume as Q. Le he iiial iveory be S. Due o marke ema iveory level graually ecreases urig he perio 0, a i becomes zero a ime. Shorages ake place i he perio, which is parially backlogge. he relae figure- of he moel is as follows. Fig.-

4 898. K. ripahy a L. M. Praha he iffereial equaios goverig he iveory level a ay ime urig he 0, are give as follows, cycle D, 0 θ D, D a, 3 he bouary coiios are, 0 S, 0 4 he soluio of equaio is S, 0 5 akig he firs wo erms of he expoeial series a he iegraig we ge he soluio of equaio as / / / / e / 6 Now akig he firs wo erms of he expoeial series a eglecig he erm coaiig he equaio 6 becomes / / / / /, Similarly he soluio of equaio 3 is 7 / / / / a a, / 8 From equaio 5 a 7 S ca be fou ou as

5 A EOQ moel for Weibull eerioraig iems 899 S / / / / / S / / / / /, 9 Usig equaio 9 i equaio 5 we ge 0, / / / / / / /, 0 Durig perio, 0 oal umber of uis holig H is 0 H Usig equaio 0 a equaio 7 we ge 0 / / / / / / / H / / / / / alculaig furher we ge H / / / / / / oal amou of eeriorae iems D, urig he perio, 0 is D θ

6 900. K. ripahy a L. M. Praha / / / / / D egraig his, eglecig he erm coaiig or higher egree of i as 0 << <, we ge D / / / / oal amou of shorage uis S urig he perio, 0 is give as a a S / / / / / a a a a / / / / / / 3 oal amou of los sales L urig he perio, 0 is give by L D a a / / / / 4 oal average cos of he sysem per ui ime is give by ] [ 4 3 L S D H K / / / / / / / / / 3 / / / / a

7 A EOQ moel for Weibull eerioraig iems 90 a / a / / a a 4 / / / / 5 Opimal value of ca be fou ou by solvig he followig equaio k 0 / / / / / / / / [ ] [ a a a ] 3 / / a he miimum oal average cos per ui ime is obaie for hose values of for which K > 0 7 By solvig equaio 6 he value of ca be obaie a he from equaio 9 a 5, he opimal value of S a K ca be fou ou respecively. 4. Numerical Example Le us ake, a,,,,,, 0.,, 0., 4, 60, 0.4,, 00,,0, 4,8 i heir [ ] [ ],,, 3, 4 respecive proper uis. he we ge , S 4.76, K abulaio a Sesiiviy Aalysis We ow examie he sesiiviy aalysis of he opimal soluio of he moel for chages i,, a a parameer values associae wih he sysem i he followig able 5.. We chage oe parameer a a ime keepig he oher parameers

8 90. K. ripahy a L. M. Praha uchage. Sesiiviy aalysis is performe by chagig he parameers,, a a by -50%, -40%, -30%, -0%, -0%, 0%, 0%,30%, 40%,50% oe by oe i he moel ha are give i he followig able. he iiial aa of all parameers have bee ake from he above umerical example. able 5. Parameer % chage S K a From he above able we ca coclue he followig:

9 A EOQ moel for Weibull eerioraig iems 903 A. Effecs of various parameers wih he icreme of scale parameer a he iveory perio ecreases. b iial iveory level also ecreases. c oal average cos of he sysem icreases. B. Effecs of various parameers wih he icreme of shape parameer a Variable chages occur i he iveory perio. b iial iveory level also chages variably. c oal average cos of he sysem ecreases.. Effecs of various parameers wih he icreme of back loggig parameer a a here is a icrease i iveory perio. b iial iveory level also icreases. c oal average cos of he sysem icreases. D. Effecs of various parameers wih he icreme of life ime parameer a here is a icrease i iveory perio. b iial iveory level also icreases. c oal average cos of he sysem icreases. 6. oclusio he prese work we have evelope a iveory moel wih power paer ema wih Weibull eerioraio rae. Shorages have bee allowe a compleely backlogge i his moel. his ype of power paer ema requires a iffere policy ha he coveioal policy base o geeral Weibull paer. cases where large porio of ema occurs a he begiig of he perio we use > a whe i is large a he e we use, 0 < <. Similarly a correspos o cosa ema a isaaeous ema respecively. Behaviors of iffere parameers have bee illusrae hrough he umerical example a sesiiviy aalysis. Refereces [].Y Dye, L.Y Ouyag, a.p Hsieh, Deermiisic iveory moel for eerioraig iems wih capaciy cosrai a ime-proporioal backloggig rae, Europea Joural of Operaioal Research, 78 3, 007, [] G. Pamaabha, a P. Vra EOQ moels for perishable iems uer sock epee sellig rae, Europea Joural of Operaioal Research, 86, 995, 8-9. [3] H.J hag, a,.y. Dye, A EOQ moel for eerioraig iems wih ime varyig ema a parial backloggig, Joural of he Operaioal Research Sociey, 50, 999, 76-8.

10 904. K. ripahy a L. M. Praha [4] H.L. Yag, A compariso amog various parial backloggig iveory losize moels for eerioraig iems o he basis of maximum profi, eraioal Joural of Proucio Ecoomics, 96, 005, 9-8. [5] H.M.Wee, Joi pricig a repleishme policy for eerioraig iveory wih ecliig marke, eraioal Joural of Proucio Ecoomics, Volume 40, ssues -3, 995, Pages [6] J.. eg, a H.L Yag, Deermiisic ecoomic orer quaiy moels wih parial backloggig whe ema a cos are flucuaig wih ime, Joural of he Operaioal Research Sociey, 555, 004, [7] K.S. Park, veory moels wih parial backorers, eraioal Joural of Sysems Sciece, 3, 98, [8] K.S.Wu, L.Y. Ouyag, a,..yag A opimal repleishme policy for o-isaaeous eerioraig iems wih sock epee ema a parial backloggig, eraioal Joural of Proucio Ecoomics, 0, 006, , [9] M. Roychowhury, a K.S hauhuri, A orer level iveory moel for eerioraig iems wih fiie rae of repleishme, Opsearch, 0, 983, [0] P.L. Aba, Opimal pricig a lo sizig uer coiios of perishabiliy a parial backloggig, Maageme Sciece, 48, 996, [] R.H. Hollier, a K.L. Mak, veory repleishme policies for eerioraig iems i a ecliig marke, eraioal Joural of Proucio Ecoomics,, 983, [] S.P. Wag, A iveory repleishme policy for eerioraig iems wih shorages a parial backloggig, ompuers a Operaioal Research, 9, 00, , [3] S.R Sigh, a.j.sigh, A EOQ iveory moel wih Weibull isribuio eerioraio, ramp ype ema a parial backloggig, ia Joural of Mahemaics a Mahemaical Scieces, 3, 007, 7-37, [4].J Sigh., S.R. Sigh a. Sigh, Perishable iveory moel wih quaraic ema, parial backloggig a permissible elay i paymes, eraioal Review of Pure a Applie Mahemaics, 008, [5].K. Daa. a A.K Pal, Orer level iveory sysem wih power ema paer for iems wih variable rae of eerioraio, ia Joural of Pure a Applie Mahs. 9, 988, [6] Z.. Balkhi, a L.Bekherouf, A proucio lo size iveory moel for eerioraig iems a arbirary proucio a ema rae, Europea Joural of Operaioal Research, 9, 996, Receive: February, 00