Constrained Single Period Stochastic Uniform Inventory Model With Continuous Distributions of Demand and Varying Holding Cost

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1 Journal of Matmati Statiti (): , 6 ISSN Sin Publiation Contraind Singl Priod Stoati Uniform Invntory Modl Wit Continuou Ditribution of Dm Varying Holding Cot Hala, A. Frgany M. E. El-Saadani Dartmnt of Matmatial Statiti, Faulty of Sin, Tanta Univrity, Tanta, Egyt Dartmnt of Matmatial Statiti, Faulty of Eduation, Suz Canal Univrity, Suz, Egyt Abtrat: Ti ar driv t ingl riod toati invntory modl wit ontinuou ditribution of dm t olding unit ot i a funtion of t urad (ordrd) quantity. T objtiv i to find t otimal urad quantity wi minimiz t td total ot for t riod undr a rtrition on t td varying olding ot wn t dm during t riod follow t uniform, t onntial t Lala ditribution, uing t Lagrang multilir aroa. Som ial a ar ddud illutrativ numrial aml wit om gra ar addd. Ky word: Stoati invntory modl, zro lad tim, varying olding ot, ontinuou ditribution INTRODUCTION T ingl riod modl i onrnd wit t lanning ontrol of invntory itm for wi only on rlnimnt oortunity it. Unontraind ingl riod toati invntory ytm wit ontant unit ot tratd by Gould [], Namia [] Trin [3]. Alo, Silvr [4] amind t ingl riod modl wn t dm during t riod normally ditributd. Singl riod modl involving a numbr of itm onntd by on or mor ontraint an tak a varity of form. Hadly [5] tudid ontraind multil itm ingl riod invntory roblm wit ontinuou ditribution of dm ontant unit ot, uing t Lagrang multilir aroa. T robabiliti ingl itm ingl our zro lad tim invntory ytm wit intantanou uniform rat of dm ditribution av bn diud by Fabryky [6]. Rntly, Abou-El-Ata [7] Frgany [8] introdud a robabiliti ingl-itm invntory roblm wit varying ordr ot zro lad tim undr two rtrition, uing gomtri rogramming aroa. Ti ar invtigat t ingl riod toati uniform invntory modl tat onidring varying olding unit ot, a rtrition on t td olding ot t dm during t riod i a ontinuou rom variabl. Our objtiv i to dtrmin t otimal ura (ordr) quantity, wi minimiz t rlvant td total ot for t riod, wn t dm during tat riod follow t uniform, t onntial t Lala ditribution. Finally, om ial a, wi av bn rviouly ublid numrial illutrativ aml ar addd. Aumtion Notation: T following aumtion ar mad for dvloing t modl: T ytm i t ingl riod zro lad tim. Tat i, an itm i urad on only to atify t dm of a ifi riod of tim. T numbr of unit to ura for invntory at t bginning of t riod i (t diion variabl) t tok lvl dra at a uniform rat ovr t riod T dm during tat riod i ontinuou rom variabl wit known robability dnity funtion. Tr i no ordr (tu) ot. T following notation ar adotd for dvloing our modl: f() = T robability dnity funtion of t dm during t riod, E() = T td valu of, F() = T umulativ ditribution funtion of, R() = T rliability funtion, = T otimal urad quantity, ( ) =T robability of atifying all t dm during t riod if unit ar urad at t tart of t riod, ( > ) = T robability of ortag, Corronding Autor: Hala, A. Frgany, Dartmnt of Matmatial Statiti, Faulty of Sin, Tanta Univrity, Tanta, Egyt 334 = T ura unit ot, = T olding unit ot, C () = =T varying olding unit ot,, = T ortag unit ot, E(PC) = T td ura ot for t riod, E(HC) = T td olding ot for t riod, E(SC) = T td ortag ot for t riod, E(TC) K = T td total ot for t riod, = T limitation on t td varying olding ot for t riod.

2 J. Mat. & Stat. (): , 6 - (II) > - (I) < (Holding) (HOL (Sortag) Figur.: ingl riod uniform invntory modl Dvlomnt of t td total ot funtion: A gnral outlin for analyzing mot toati invntory roblm i t following: Dvlo an rion for t ot inurrd a a funtion of bot t rom variabl t diion variabl. Dtrmin t td valu of ti rion wit rt to t robability dnity funtion of dm. Dtrmin tat minimiz t rlvant td total ot. Tu, dnding on t amount dmd t quantity urad, t invntory oition aftr dm our may b itr oitiv (olding) or ngativ (ortag). T two ituation ar own in Figur.. Now, onidr tat: H(,) = T rom variabl rrnt t amount in invntory at t nd of t riod, S(,) = T rom variabl rrnt t ortag quantity at t nd of t riod. Tn, in t firt ituation in Figur w av: H(,)= S(,) = Alo, in t ond ituation of Figur. w gt: H(,)= H(, ) S(,) = ( ) for = for > for S(, ) = ( ) for > Trfor, t td ura, t td olding t td ortag ot for t riod rtivly ar givn by: E(PC) = f ()d = () E(HC)= ( )f ()d f ()d () ( ) E(SC) = f ()d (3) Hn, t td total ot for t riod i: E(TC)= ( ) f ()d f ()d ( ) f ()d (4) Our objtiv i to minimiz t funtion (4) ubjt to t following ontraint: E(HC) K (5) To olv ti roblm w an u t Lagrang multilir aroa. W introdu a Lagrang multilir λ form t funtion: L = E(TC) λ [ E(HC) K ] (6) wr λ = if E(HC) K <, λ > if E(HC) K =, λ E(HC) K = (7) tn [ ] 335

3 J. Mat. & Stat. (): , 6 T otimal valu rquir tat : L =, (8) wi giv: ( λ)( ) F() ( λ)( ) f () d ( λ) f ()d f () d R() =. Hn i t olution of t following quation: f () A F( ) ( B ) d C f ()d R( ) =, (9) ( λ)( ) wr A = ( λ)( ), B = ( λ) C =, by tting diffrnt valu of λ, it i oibl to dtrmin t otimum ura quantity tat mt t rquirmnt of t ontraining ondition E(HC). T itrativ ro rquir tat λ b t qual to zro mall inrmnt b addd to it until t roblm ontraint i mt for t diffrnt valu of ß. T otimal valu > obtaind from (9) minimiz (4), wr d E(TC) = ( ) F() d ( 3) f ()d ( ) f ()d f ()d >. Stard robability ditribution u a t uniform, t onntial t Lala ditribution ar frquntly aumd for t dm during t riod. T at olution of t otimal ura quantity an b drivd wn t dm follow t ditribution a follow: Dm during t riod follow t uniform ditribution Aum tat t dm during t riod follow t uniform ditribution wit man b, tn b E ( ) = F ( ) = R( ) =, b, b. Subtituting in quation (9), w gt : C b b A B ln ln wr: f ()d = d = b b b f () b d = d = ln b b ( ) b = Trfor, wn t dm during t riod uniformly ditributd t otimal invntory oliy i t olution of t following quation: b b A Bln ln b ( ) = () Wr A = ( λ)(4 3 ), 4 t minimum td total ot for t riod i: min E(TC) = b 3 b ( ) ln () 4 b Dm during t riod follow t onntial ditribution If t dm during t riod onntially ditributd wit man tard dviation, tn w av E() =, F() =, f ()d = d = F() R(), alo, Wr f () d = d = Γ(, ) R() = () a t (a, z) t dt Γ =, i t inomlt gamma z funtion. Subtituting in quation (9), t otimal ura quantity i t olution of t following quation: A C F( ) (B ) Γ(, ) (C ) R( ) =. (3) t minimum td total ot i givn by: mine(tc) ( )( ) = ( ) Γ, (4) 336

4 J. Mat. & Stat. (): , 6 Tabl : t otimal ura quantity tat minimiz t rlvant td total ot for t riod mt t ontraint E(HC) for t diffrnt valu of ß ß Uniform Ditribution Eonntial Ditribution Lala Ditribution min E(TC) min E(TC) min E(TC) Dm during t riod follow t Lala ditribution Aum tat t dm during t riod follow t lala ditribution wit man varian, tn w an gt t otimal ura quantity a follow: E() =, R() = V() =, f ()d ( )R(), > F() =, f () alo, d = Γ(, ), > (5) Subtituting in quation (9), t otimal ura quantity i t olution of t following quation: A F( ) C R( ) (B ) Γ(, ) C R( ) R( ) =. (6) And t minimum td total ot i: min E(TC) = ( )( ) 4 ( ) Γ,, (7) 4 Ti i unontraind ingl riod toati invntory modl wit uniform dm ontant unit ot, wi i agr wit t rult of Fabryky [6]. Ca : For quation (), lt λ = = C () =, A = B = C =. Tn t following rlation for an b obtaind: ln ( ) b b = (9) Ti i unontraind ingl riod toati invntory modl wit dm uniformly ditributd ontant unit ot,, wi i agr wit t rult of Taa [9]. Ca 3: For quation (3), lt = = C () =, A = B =, C =, tn t otimal ura quantity i: F( ) Γ (, ) = () Ti i unontraind ingl riod toati invntory modl wit dm onntially ditributd ontant unit ot. Ca 4: For quation (6), lt = = C () =, A = B =, C =,tn t otimal ura quantity i: Γ = F( ) (, ) Ti i unontraind ingl riod toati 4-Sial a: W ddu four ial a for our invntory modl wit ontant unit ot t modl a follow: uniform dm follow lala ditribution. Ca : For quation (9) lt λ = = C () =, A = B = C =, Illutrativ aml: Conidr t following ingl riod uniform invntory modl wit aramtr givn tn t otimal ura quantity givn by: a: =.5, =.5, = 5.5 K=. W will f () dtrmin t otimal ura quantity wn: F( ) d = (8) t dm during t riod uniformly ditributd, b= 5, 337 ()

5 J. Mat. & Stat. (): , 6 t dm during t riod onntially ditributd, =5, t dm during t riod i Lala ditributd, = 5 = Uing quation (), (3) (6) by tting diffrnt valu of, w an gt t otimal ura quantity tat minimiz t rlvant td total ot for t riod mt t ontraint E(HC) for t diffrnt valu of ß a own in Tabl : T olution of t roblm may b dtrmind mor radily by lotting mine(tc) againt ß for a ditribution of dm a t following Figur: 5 5 Uniform Eonntial Lala Figur: mine(tc) againt ß for a ditribution of dm CONCLUSION Ti artil driv t otimal olution for ontraind ingl riod toati uniform invntory modl tat onidring ontinuou ditribution of dm varying olding ot. W av valuatd tat minimiz t rlvant td total ot for t riod for a valu of ß. Alo, for our aml, at t am invntory aramtr w found tat t min E(TC) in t a of t dm uniformly ditributd i l tan t min E(TC) in t a wn t dm during t riod follow t onntial or t Lala ditribution for all diffrnt valu of ß. REFERENCES. Gould, F.J., G.D. En C.P. Smidt, 997. Introdutory Managmnt Sin. Prnti-Hall Intrnational, In.. Namia, S., 993). Prodution Oration Analyi. nd Ed. Irwin, In. 3. Trin, R.J., 994. Prinil of Invntory Matrial Managmnt. 4t Ed. Prnti-Hall, In. 4. Silvr, E.A. R. Ptron, 985. Diion Sytm for Invntory Managmnt Prodution Planning. Nw York, Jon Wily Son. 5. Hadly, G. T.M. Witin, 963. Analyi of Invntory Sytm. Englwood Cliff, N.J., Prnti-Hall. 6. Fabryky, W.J. J. Bank, 967. Prourmnt Invntory Sytm: Tory Analyi. Rinold Publiing Cororation, USA. 7. Abou-El-Ata, M.O., H.A. Frgany M.F. E- Wakl, 3. Probabiliti Multi-itm Invntory Modl Wit Varying Ordr Cot Undr Two Rtrition: A Gomtri Programming Aroa. Intl. J. Prodution Eonomi, 83: Frgany, H.A. M.F. El-Wakl, 4. Probabiliti Singl-Itm Invntory Problm wit Varying Ordr Cot Undr Two Linar Contraint. J. Egyt. Mat. So., : Taa, H., 997. Oration Rar. 6t Ed. Englwood Cliff, N.J., Prnti-Hall. 338