Periodic Review Probabilistic Multi-Item Inventory System with Zero Lead Time under Constraints and Varying Order Cost

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1 Ameica Joual of Applied Scieces (8: 3-7, 005 ISS Sciece Publicatios Peiodic Review Pobabilistic Multi-Item Ivetoy System with Zeo Lead Time ude Costaits ad Vayig Ode Cost Hala A. Fegay Lectue of Mathematical Statistics, Faculty of Sciece, Tata Uivesity, Tata, Egypt Abstact: This study teats the pobabilistic safety stock -items ivetoy system havig vayig ode cost ad zeo lead-time subject to two liea costaits. The expected total cost is composed of thee compoets: the aveage puchase cost; the expected ode cost ad the expected holdig cost. The policy vaiables i this model ae the umbe of peiods ad the optimal maximum ivetoy level Q m ad the miimum expected total cost. e ca obtai the optimal values of these policy vaiables by usig the geometic pogammig appoach. A special case is deduced ad a illustative umeical example is added. Key wods: Pobabilistic model, zeo lead-time, safety stock, multi-item, vayig ode cost, geometic pogammig ITRODUCTIO I may situatios demad is pobabilistic sice it is a adom vaiable havig a kow pobability distibutio. All eseaches have studied ucostaied pobabilistic ivetoy models assumig the odeig cost to be costat ad idepedet of the umbe of peiods. Hadley, et al [4] ad Taha [6], has examied ucostaied pobabilistic ivetoy poblems. Fabic ad Baks [3] studied the pobabilistic sigleitem, the sigle souce ivetoy system with zeo leadtime, usig the classical optimizatio. Also Haii ad Abou-El-Ata [5] deduced the detemiistic multi-item poductio lot size ivetoy model with a vayig ode cost ude a estictio: a geometic pogammig appoach. Recetly Abou-El-Ata, et al [] studied the pobabilistic multi-item ivetoy model with vayig ode cost ude two estictios: a geometic pogammig appoach. The aim of this study is to ivestigate the pobable safety stock multi-item, sigle souce ivetoy model with zeo lead-time ad vayig ode cost ude two costaits, oe of them of the expected holdig cost ad the othe o the expected cost of safety stock. The optimal amout of peiods, the optimal maximum Q ivetoy levels m ad mi E (TC ae obtaied. Also special case is deduced ad a illustative umeical example is added. Model developmet: The followig otatios ae adopted fo developig ou model: C p The puchase cost of the th item, C o ( The vayig ode cost of the th item pe cycle The holdig cost of the th item pe peiod Coespodig Autho: Hala A. Fegay, Lectue of Mathematical Statistics, Faculty of Sciece, Tata Uivesity, Tata, Egypt 3 C h I x F(x E (x D E (D Q m υ The expected level of ivetoy held pe th cycle A adom vaiable epeset the demad of the th item duig the cycle The pobability desity fuctio of the demad x The expected value of the demad x xu x f (x dx, whee x u ad xl xl ae the maximum value ad miimum value of x The aual demad ate of the th item pe peiod The expected aual demad D The maximum ivetoy level of the th item The umbe of peiods, cycle,of the th item (a decisio vaiable ad a eview of the stock level of the th item is made evey peiod The positive value epesetig a pat of time fo safety stock K The limitatio o the expected holdig cost K The limitatio o the expected safety stock cost E (TC The expected total cost fuctio. The model aalysis: Coside a ivetoy pocess i which a eview of the stock level is made evey peiod,,,,. A amout is odeed so that the stock level has etued to its iitial positio desigated by: Q m,,,,. To avoid shotage duig.

2 Am. J. Applied Sci., (8: 3-7, 005 be: g( υ Hece, the followig fom gives the expected holdig cost pe peiod: Fig. : Ivetoy system with safety stock Peiods we must maitai a safety stock absobig demad fluctuatio. Also, this is doe maitaiig the quatity Q m x u fo ay cycle. Hece the esultig safety stock, D v, meets the exceed demads cycle. The peiodic ivetoy system is exhibited gaphically as show i Fig.. The expected aual total cost is composed of thee compoets: the expected puchase cost the expected ode cost ad the expected holdig costs as follows: E(TC E(PC E(OC E(HC, C ( o E(PC Cp, E(OC, Ch I E(HC hee: E(x I Qm E(x The: [ ] C Q E(HC h m Ch [ υ] E(HC The ode cost pe uit is a vayig fuctio of the expected umbe of peiods,, which takes the followig fom: C O ( Co β, whee, C o > 0 ad 0.5 β < ae costats eal umbes selected to povide us the best estimatio of the cost fuctio. Ou objective is to miimize the elevat expected aual total cost fuctio, accodig to the pevious assumptios of the model: C C υ ( β p o E(TC C h C h i.e. Ude the followig costaits: Ch K Ch υ K ( The cost of safety stock isuace is give by the last tem i the equatio (, i the safety pocess a amout is held i excess of the expected equiemet as isuace agaist the isk of a stakeout. The tems C E( D p ad C E( D υ ca be posted without h ay effect. The the miimum expected total cost ca be witte as: The Optimizatio of the decisio vaiables ad Q m ca be pefomed if we assume that the maximum demad duig the cycle, x u, is elated to the expected demad duig the cycle as: xu E(x g( g( C whee, g ( is a elatioal fuctio which coside to K 4 β Ch mi E(TC Co (3 Subject to: h ChE(x υ ad K (4

3 Am. J. Applied Sci., (8: 3-7, 005 Applyig the geometic pogammig techiques to the equatio (3 ad (4, the elaged pedual fuctio could be witte i the followig fom: β 3 o h h h C C C C E(x υ G( K K C o Ch Ch K3 ChE(x υ K 4 4 ( β 3 4 whee, j, 0 < j <, j,,3, 4,,,, ae the weights ad ca be chose to yield the omal ad the othogoality coditios as follows: (5 4 Similaly: l g( 3, 4 { l( βco l ( 3 4 } β β 4 C ( β β β ( h l l 3 4 ChE(x υ l l 4 0 β K (9 Simplifyig the equatio (8 ad (9 ad multiplyig them, we get: C E(x υ (0 h 3 4 KK e ( β 0,,,,. 3 4 Solvig the above equatios, we get: The, we obtai: f ( a A 4β 3β β j j j j j j b d A b 0 j j j ( 3 4 β 3 4 ad,,,,. β β (6 Substitutig fom (6 ito (5, the dual fuctio is give i the fom: 3 4 β 3 4 ( β C o β ( βch β g( 3, ( β C C E(x 4 h h υ K3 K4 Takig the logaithm of both sides of (7: lg( 3, 4 [ 3 4 ] { l ( Co l ( 3 4 } β β Ch ( β [ β 3 4 ] l l [ β 3 4 ] β C C E(x υ l l l l h h K K (7 To calculate 3 ad 4 which maximize g( 3, 4, equate the fist patial deivatives of l g( 3, 4 with espect to 3 ad 4 espectively to zeo as follows: l g( 3, 4 { l( βco l ( 3 4 } β β 3 C ( β β β [ ] h l l 3 4 Ch l l 3 0 β K (8 hee: C E(x υ A, 5 h KKe C C β o h B, Ch Ke β Ch υ Ch C K e Co, j 3 a j, β, j 4 B, j 3 bj ad C, j 4 B ( β, j 3 d j C, j 4 It could be easily poved that fj(0 < 0 ad fj(>0, j 3, 4 ad this is meas that thee exists a oot j ε (0,,j 3,4. Ay method such as the tial ad eo, could be used to calculate these oots. ow to veify that ay oot 3 ad 4 calculated fom equatios ( maximize g( 3, 4 espectively. Applyig the followig coditios: l g(, < 0 β 3 4 β l g(, < 0 β 3 4 β

4 Am. J. Applied Sci., (8: 3-7, 005 l g(, > β 3 4 β Hece: l g(, l g(, l g(, < 0 β 3 4 β Thus, the oots 3 ad 4 calculated fom equatios ( maximize the dual fuctio g ( 3, 4. j, j,,3, 4, Hece the optimal solutio is whee 3, ae the solutio of ( ad, ae calculated by substitutig the values of 3, i expessio (6. To fid the optimal umbe of peiods, use the followig elatios due to Duffi ad Peteso s theoem [] as follows: C g(, β o 3 4 Table : The paametes of thee items Items Item Item Item 3 Paametes E (D C h C o C p Also assumig that υ 5, K 0000, K 000 ad 0.5 β < Solutio: Table : The esults usig the Mathematica pogam 3 β mi E(TC C h g(, 3 4 Solvig these equatios, the optimal expected umbe of peiods pe cycle is give by: Ch 3 β 4 Co 3 4 { } { β } ( Fig. : The Relatio betwee ad β The: β h 3 4 β ( { } o { β 3 4} C Qm υ C (3 Substitutig the value of i equatio (3 afte addig the costat tems, we get: { } mi E(TC Cp C o { β 3 4 } β β ChCo 3 4 β C h { 3 4} h C C h Co { β 3 4 } υ 6 Special case: Let β0, ad K, K C O ( C o costat, 3, 4 0 ad /. This is a pobabilistic sigle-item ivetoy model without ay estictio ad costat costs, which agee with the model of maitaiig stock to absob demad fluctuatios [3], the equatios (, (3 ad (4 become: C C υ o o,qm Ch Ch mi E ( TC C E( D C C E( D C p h o h (4 E( D υ

5 Am. J. Applied Sci., (8: 3-7, 005 REFERECES Fig. 3: The Relatio betwee mie (TC ad β A illustative example: Let us fid the optimal expected umbe of peiods ad the miimum expected total cost mi E (TC fo the pevious model of peiodic eview pobabilistic multi-item ivetoy system with zeo lead time ude costaits ad vayig ode cost, o the data of Table. Also, by usig the feelace pogam we ca daw the elatio betwee, mi E (TC agaist β as show i Fig. ad 3 espectively. Mi E (TC. Abou-El-Ata, Fegay, H.A. ad M.F. El-akeel, 00. Pobabilistic multi-item ivetoy model with vayig ode cost ude two estictios: A geometic pogammig appoach. Itl. J. Poduct. Eco., 83: Duffi, R.J. ad E.L. Peteso, 974. Costaied miima teated by geometic meas. estighouse Scietific pape, 64: Fabycky,.J. ad J. Baks, 967. Pocuemet ad Ivetoy Systems: Theoy ad Aalysis. Reihold Publishig Copoatio, USA. 4. Hadley, G. ad T.M. hiti, 963. Aalysis of Ivetoy Systems. Eglewood Cliffs,.J. Petice-Hall. 5. Haii, A.M.A. ad M.O. Abou-El-Ata, 995. Multi-item poductio lot-size ivetoy model with vayig ode cost ude a estictio: A Geometic pogammig appoach. Poduct. Pla. Cotol., 6: Taha, H.A., 997. Opeatios Reseach. 6th Ed. Petice-Hall, IC, Eglewood Cliffs, J, USA. COCLUSIO e have evaluated the optimal expected umbe of peiods,,,,, the we deduced the miimum expected total cost mi E (TC of the cosideed safety stock pobabilistic multi-item ivetoy model. e daw the cuves ad mi E (TC agaist β, which idicate the values of ad β that give the miimum value of the expected total cost of ou umeical example. 7