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

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1 Joual of Matematis ad Statistis 7 (: -9, ISS Siee Publiatios Peiodi Review Pobabilisti Multi-Item Ivetoy System wit Zeo Lead Time ude Costait ad Vayig Holdig Cost aglaa Hassa El-Soday Tid Statistiia, Geeal Depatmet of atioal Aouts, Cetal Agey fo Publi Mobilizatio ad Statistis, Caio, Egypt Abstat: Poblem statemet: Tis study teats te pobabilisti safety stok -items ivetoy system avig vayig oldig ost ad zeo lead-time subjet to liea ostait. Appoa: Te expeted total ost is omposed of tee ompoets: te aveage puase ost; te expeted ode ost ad te expeted oldig ost. Results: Te poliy vaiables fo tis model ae te umbe of peiods ad te optimal maximum ivetoy level Coespodig Auto: aglaa, H. El-Soday, Tid Statistiia, Geeal Depatmet of atioal Aouts, Cetal Agey fo Publi Mobilizatio ad Statistis, Caio, Egypt. m ad te miimum expeted total ost. Colusio/Reommedatios: e a obtai te optimal values of tese poliy vaiables by usig te geometi pogammig appoa. A speial ase is dedued ad a illustative umeial example is added. Key wods: Pobabilisti safety stok multi-item, zeo lead-time, vayig oldig ost, ostaied pobabilisti ivetoy system, adom vaiable, demad flutuatios, geometi pogammig teiques, otogoal oditios. ITRODUCTIO I may situatios demad is pobabilisti sie it is a adom vaiable avig a kow pobability distibutio. All eseaes ave studied uostaied pobabilisti ivetoy models assumig te oldig ost to be ostat. Hadley ad iti (963 ad Taa (997 ad Be-Daya (999 ave examied uostaied pobabilisti ivetoy poblems. Fabyky ad Baks (967 studied te pobabilisti sigle-item, sigle soue ivetoy system wit zeo lead-time, usig te lassial optimizatio. Also Haii ad Abou-El-Ata (995; 997 ad Kotb (998 ivestigated te ostaied detemiisti ivetoy models usig a geometi pogammig appoa. Reetly, Abou-El-Ata ( ad Fegay (5 itodued te pobabilisti multiitem ivetoy system wit zeo lead time ude ostaits ad vayig ode ost, usig geometi pogammig appoa. Te aim of tis study is to ivestigate te pobabilisti safety stok multi-item, sigle soue ivetoy model wit zeo lead-time ad vayig oldig ost. Te developed models ae te pobabilisti safety stok multi-item, sigle soue ivetoy model wit zeo lead-time ad vayig oldig ost ude te expeted ode ost ostait ad te pobabilisti safety stok multi-item, sigle soue ivetoy model wit zeo lead-time ad vayig oldig ost ude te expeted vayig oldig ost ostait. Te optimal umbes of peiods, te optimal maximum ivetoy levels m ad te miimum expeted total ost ae obtaied. Also a speial ase is dedued ad a illustative umeial example is added. Model developmet: Te followig otatios ae adopted fo developig ou model: p o = Te puase ost of te t item, = Te ode ost of te t item pe yle ( = Te vayig oldig ost of te t item pe peiod, wi takes te fom C ( = β wee, > ad β> ae ostat eal umbes seleted to povide us te best estimatio of te ost futio. H = Te expeted level of ivetoy eld pe t yle α = Te positive value epesetig apat of time fo safety stok x = A adom vaiable epeset te demad of te t item duig te yle

2 J. Mat. & Stat., 7 (: -9, F(x = Te pobability desity futio of te demad x E(x = Te expeted value of te demad x xu = x f(x dx xl wee, x u ad x l ae te maximum value ad miimum value of te demad x, espetively D = Te aual demad ate of te t item pe peiod E(D = Te expeted aual demad D m = Te maximum ivetoy level of te t item = Te umbe of peiod, yle, of te t item (a deisio vaiable ad a eview of te stok level of te t item is made evey peiod K o K = Te limitatio o te expeted ode ost = Te limitatio o te expeted vayig oldig ost E(PC = Te expeted aual puase ost E(HC = Te expeted aual oldig ost E(OC = Te expeted aual odeig ost E(TC = Te expeted total ost futio Te model aalysis: oside a ivetoy poess i wi a eview of te stok level is made evey peiods, =,,,. A amout is odeed so tat te stok level is etued to its iitial positio desigated by: m, =,,,. To avoid sotage duig peiods we must maitai a safety stok absobig demad flutuatios. Also, tis is doe maitaiig te quatity m = x u fo ay yle. Hee te esultig safety stok, E(D a, meet te exeed demads yle. Te system is epeseted gapially i Fig.. Te expeted aual total ost is omposed of tee ompoets: te expeted puase ost, te expeted vayig oldig ost ad te expeted ode ost, i.e.: ( C H E(HC = = wee, H is te aveage ivetoy give by: E(x H = m Sie, E(x = E(D, te: E(D H = m Te Optimizatio of te deisio vaiables ad m a be pefomed if we assume tat te maximum demad duig te yle, x u, is elated to te expeted demad duig te yle as: xu = E(x g( = E(D g( wee, g( is a elatioal futio, so we get: g( H = E(D oside te ase we g( is give by: + α g( = Te: E(D H = + E(D α E(TC = E(PC + E(OC + E(HC wee te expeted aual puase ost is give by: E(PC = E(D = p ad te expeted aual odeig ost is give by: o E(OC = = ad te expeted aual vayig oldig ost is give by: 3 Fig. : Safety stok fo peiodi eview ivetoy system

3 J. Mat. & Stat., 7 (: -9, by: Te te expeted vayig oldig ost is give E(D E(HC = C ( + E(D α = β+ E(D = + E(D α = wee, E(D a is te safety stok equied to absob demad flutuatios duig te ivetoy yle. Te expeted total ost is te give by: β+ o E(D E(TC = pe(d + + = + E(D α ] ( Ou objetive is to detemie te optimal umbe of peiods tat miimize te expeted total ost fo te followig two models: Model (I: Safety stok fo Pobabilisti Peiodi Review Multi- Item Ivetoy System wit Zeo Lead Time ad Vayig Holdig Cost ude te Expeted Ode Cost Limitatio Coside te elevat expeted total ost (, te estitio o te expeted ode ost is: o Ko ( = Te tems E(D ad = p E(D α ae = ostats ad a be postpoed witout ay effet ad te te expeted total ost a be witte as: β+ o E(D E(TC = + = Subjet to: (3 o (4 = K o Applyig te geometi pogammig teiques to Eqs.3 ad 4, te elaged pedual futio a be give by: wee, = j, < j <, j =,,3, =,,..., ae te weigts ad a be ose to yield te omal ad te otogoal oditios as follows: + = β+ =, =,,3,..., ( 3 Solvig te above equatios, we get: β+ + = = β+ β+ 3 ad 3 (6 Substitutig fom Eq.6 ito Eq.5, te dual futio is give i te fom: 3 β ( β+ o ( 3 E(D β+ β+ β+ o = 3 ( 3 kw o 3 g( = (7 β+ + Takig te logaitm of bot sides of Eq. 7: β+ 3 l g( 3 = l ( β+ o l ( β+ 3 = β l ( β+ E(D l ( + 3 β+ o + w3 l lw3 Ko To fid te optimal value of 3 wi miimize g( 3, take te fist deivative of l g( 3 wit espet to 3 ad put it equal to zeo, as follows: dlg( 3 = l ( β+ o l ( β+ 3 d3 β+ + l ( β+ E(D l ( + 3 β+ + = o l l w3 ko Simplifyig Eq. 8, we obtai: ( β+ β+ β + o 3 E(D E(D f( 3 = o o Koe o Koe + = β+ 3 β+ 3 ( 3 3 (8 G( = 3 o β+ E(D o = k o 3 l 3 o E(D o ( β+ 3 = k o 3 = (5 4 Let: A E(D o = o Koe β+

4 Te, we obtai: β+ 3 β ( f( = + + A β+ A= (9 ee: ( f( = β+ A< f( = β A> i meas tat tee exist a oot 3 (,. Ay metod su as te tial ad eo metod ould be used to alulate tis oot. Howeve we sall fist veify tat alulated fom Eq. 9 maximize g( 3. Tis 3 is doe by sowig tat te seod deivative is always egative: d lg( 3 d3 + ( β+ ( β+ 3 = < + ( β+ ( + w 3 3 Tus te oot 3 alulated fom Eq.9 maximize te dual futio g( 3. Hee te optimal solutio is j,j=,,3, wee 3 is te solutio of te Eq.9 ad, ae alulated by substitutig te value of 3 i Eq. 6. To fid te optimal umbe of peiods, use te followig elatios due to Duffi ad Peteso (974 teoem as follows: o = g( 3 β+ E(D = g( 3 Solvig tese equatios, te optimal umbe of peiods pe yle is give by: o β+ = E(D 3 ( + 3 ( β+ J. Mat. & Stat., 7 (: -9, ( Hee te optimal maximum ivetoy level is give by: +α m = E(D β+ ( + 3 ( β+ o = E(D + E(D α E(D 3 ( 5 Substitutig te value of te ostat tems, we get: i Eq.3 afte addig E(D ( β+ β+ 3 pe(d + o + o ( + 3 β+ o E(D ( β+ + 3 = E(D ( β+ 3 E(D α mi E(TC = + ( Model (II: Safety stok fo Pobabilisti Peiodi Review Multi- Item Ivetoy System wit Zeo Lead Time ad Vayig Holdig Cost ude te Expeted Vayig Holdig Cost Limitatio Coside te elevat expeted total ost (, te estitio o te expeted vayig oldig ost is: β+ E(D K = Te tems E(D ad = p E(D α ae = ostats ad a be postpoed witout ay effet ad te te expeted total ost a be witte as: β+ o E(D E(TC = + = Subjet to: β+ = K (3 E(D (4 Applyig te geometi pogammig teiques to Eq.3 ad 4, te elaged pedual futio a be give by: G( = 3 o β+ E(D β+ E(D = k3 3 o E(D E(D (+ 3 ( β+ = k 3 = (5 wee, j j,, j,,3,,,..., = < < = = ae te weigts ad a be ose to yield te omal ad te otogoal oditios as follows:

5 J. Mat. & Stat., 7 (: -9, + = + β+ =, =,,3,..., ( ( 3 Solvig te above equatios, we get: ( β+ ( + ( β+ 3 = ad = β+ β+ 3 (6 Substitutig fom Eq. 6 ito Eq. 5, te dual futio is give i te fom: ( β+ ( + β+ ( 3 3 ( β+ β+ β+ o ( β+ E(D = ( ( 3 ( 3 g( 3 = β+ + β+ E(D K3 3 Takig te logaitm of bot sides of Eq.7: (7 ( β+ ( + 3 l g( 3 = l ( β+ o l ( β+ ( + 3 = β+ β+ ( 3 + l E(D ( β+ l ( ( β+ 3 β+ E(D + 3 l l3 K To fid te optimal value of 3 wi miimize g( 3, take te fist deivative of l g( 3 wit espet to 3 ad put it equal to zeo, as follows: dlg( 3 β+ l o l 3 d3 β+ ( ( ( = β+ β+ + β+ l E(D ( β+ l ( ( β+ 3 β+ E(D + l l 3 = K Simplifyig Eq. 8, we obtai: β+ β+ o ( 3 E(D Ke β+ E(D f( = + + Let: o E(D β+ = ( β+ E(D Ke β+ β+ (8 Te, we obtai: β+ 3 β+ β+ β ( 3 f( = + + A β+ A= (9 ee: f( = A< f( = + Aβ> i meas tat tee exist a oot 3 (,. Ay metod su as te tial ad eo metod ould be used to alulate tis oot. Howeve we sall fist veify tat alulated fom Eq. 9 maximize g( 3. Tis 3 is doe by sowig tat te seod deivative is always egative: ( β+ ( β+ + d lg( 3 ( β+ ( + 3 ( β+ ( ( β+ 3 = < d Tus te oot 3 alulated fom (9 maximize te dual futio g( 3. Hee te optimal solutio is j,j=,,3, wee 3 is te solutio of (9 ad, ae alulated by substitutig te value of 3 i Eq.6. To fid te optimal umbe of peiods, use te followig elatios due to Duffi ad Peteso (974 teoem as follows: = g( o 3 β+ E(D = g( 3 Solvig tese equatios, te optimal expeted umbe of peiods pe yle is give by: ( ( β+ 3 ( β+ ( + o E(D 3 β+ = ( Hee te optimal maximum ivetoy level is give by: A β+ o E(D = β+ ( E(D Ke β+ 6 o ( β+ ( 3 β+ m = E(D + E(D α E(D ( β+ ( + 3 (

6 J. Mat. & Stat., 7 (: -9, Table : te paametes of te tee items Item paametes Item Item Item 3 E(D o p.. 4. Speial ase: e dedue a speial ase of ou models as follows. Fo Model (I, let β =, = ad K o te C ( = ad 3. Also, fo Model (II, let β =, = ad K te C ( = ad 3. Te Eq. ad beome: o = (3 E(D Also, Eq. ad beome: = E(D + E(D α (4 E(D o m Also, Eq. ad beome: Fig. : Te elatio betwee ad β, K o = mi E(TC = E(D + E(D + E(D α (5 p o Fig. 3: Te elatio betwee Substitutig te value of te ostat tem, we get: mi E(TC = + ad β, K = E(D pe(d + o o β+ o 3 E(D β+ = + E(D α i Eq. 3 afte addig ( β+ ( + 3 ( 3 ( ( β+ ( + 3 E(D β+ β+ β+ ( 7 Tis is a pobabilisti sigle-item ivetoy model witout ay estitio ad ostat osts, wi agee wit te model of maitaiig stok to absob demad flutios (Fabyky ad Baks, 967 A illustative example: Coside te ivetoy paametes give i Table, we will fid te optimal ivetoy dotie by detemiig te miimum expeted total ost we: Te system is pobabilisti peiodi eview multiitem ivetoy system ude te expeted ode limitatio K o = Te system is pobabilisti peiodi eview multiitem ivetoy system ude te expeted vayig oldig ost limitatio K = Also assume tat a = 5 ad. β. Usig te esults of ou models, te optimal expeted umbe of peiods pe yle, te optimal maximum ivetoy level ad te miimum expeted total ost mi E(TC a be summaized i te followig Table ad 3. Te solutio of te poblem may be detemied moe eadily by plottig mi E(TC agaist β ad agaist β fo te two ivetoy models i te followig Fig. -5.

7 Table : Te optimal solutio, K o = β m J. Mat. & Stat., 7 (: -9, m 3 m3 mi E(TC Table 3: Te optimal solutio, K = β m m 3 m3 mi E(TC MATERIALS AD METHODS Te aim of tis study is to ivestigate te peiodi eview pobabilisti multi-item ivetoy system wit zeo lead time we te oldig ost is a vayig futio of te ivetoy yle. Te geometi pogammig appoa is used to detemie te optimal ivetoy yle ad te optimal maximum ivetoy level wi miimize te expeted total ost ude te expeted ode ost ostait ad ude te expeted oldig ost ostait. RESULTS AD DISCUSSIO Fig. 4: Te elatio betwee mi E(TC ad β, K o = Te basi esults of tis apte ae. Te miimum aual expeted total ost ude te expeted ode ost ostait is give by: E(D ( β+ β+ 3 pe(d + o + o ( + 3 β+ o E(D ( β+ + 3 = E(D ( β+ 3 E(D α mi E(TC = + Fig. 5: Te elatio betwee mi E(TC ad β, K = 8 Ad miimum aual expeted total ost ude te expeted vayig oldig ost ostait is give by:

8 J. Mat. & Stat., 7 (: -9, mi E(TC = + E(D pe(d + o o β+ o 3 E(D β+ = + E(D α ( β+ ( + 3 ( 3 ( ( β+ ( + 3 E(D β+ β+ β+ At te ed of tis pape, a speial ase of peviously publised wok is added. Also a umeial illustative example is added wit some gaps by usig Matematia pogam. COCLUSIO e ave evaluated te optimal umbe of peiods, =,,, te we dedued te miimum expeted total ost mi E(TC of te osideed safety stok pobabilisti multi-item ivetoy model. e daw te uves ad mi E(TC agaist β, wi idiate te values of ad β tat gives te miimum value of te expeted total ost of ou umeial example. REFERECES Abou-El-Ata, Fegay, H.A. ad M.F. El-akeel,. Pobabilisti multi-item ivetoy model wit vayig ode ost ude two estitios: A geometi pogammig appoa. Itl. J. Podut. Eo., 83: 3-3. Be-Daya, M. ad M. Haiga, 999. Some stoasti ivetoy models wit detemiisti vaiable lead time. Eu. J. Opeat. Res., 4: 5. Duffi, R.J. ad E.L. Peteso, 974. Costaied miima teated by geometi meas. estigouse Si. Pape, 64: Fabyky,.J. ad J. Baks, 967. Pouemet ad Ivetoy Systems: Teoy ad Aalysis. Reiold Publisig Copotio, USA. Fegay, H.A., 5. Peiodi eview pobabilisti multi-item ivetoy system wit zeo lead Time ude ostaits ad vayig ode ost. Am. J. Applied Si., : 3:7. Hadley, G. ad T.M. iti, 963. Aalysis of Ivetoy Systems. Eglewood Cliffs,.J. Petie-Hall. Haii, A. M. A. ad M.O. Abou-El-Ata, 997. Multiitem podutio lot size ivetoy model wit vayig ode ost ude a estitio: A geometi pogammig appoa. Podut. Pla. Cotol, 8: Haii, A.M.A. ad M.O. Abou-El-Ata, 995. Multiitem podutio lot-size ivetoy model wit vayig ode ost ude a estitio: A Geometi pogammig appoa. Podut. Pla. Cotol., 6: Kotb, K.A.M., 998. Te esoues ostaied sigleitem ivetoy poblem wit demad-depedet uit ost. Podut. Pla. Cotol, 9: Taa, H.A., 997. Opeatios Resea. 6t Ed., Petie-Hall, IC, Eglewood Cliffs, J, USA. 9