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

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1 Journal of Matemati and Statiti (1): , 6 ISSN Siene Publiation Contrained Single Period Stoati Uniform Inventory Model Wit Continuou Ditribution of Demand and Varying Holding Cot 1 Hala, A. Fergany and M. E. El-Saadani 1 Deartment of Matematial Statiti, Faulty of Siene, Tanta Univerity, Tanta, Egyt Deartment of Matematial Statiti, Faulty of Eduation, Suez Canal Univerity, Suez, Egyt Abtrat: Ti aer derive te ingle eriod toati inventory model wit ontinuou ditribution of demand and te olding unit ot i a funtion of te uraed (ordered) quantity. Te objetive i to find te otimal uraed quantity wi minimize te eeted total ot for te eriod under a retrition on te eeted varying olding ot wen te demand during te eriod follow te uniform, te eonential and te Lalae ditribution, uing te Lagrange multilier aroa. Some eial ae are dedued and illutrative numerial eamle wit ome gra are added. Key word: Stoati inventory model, zero lead time, varying olding ot, ontinuou ditribution INTRODUCTION Te ingle eriod model i onerned wit te lanning and ontrol of inventory item for wi only one releniment oortunity eit. Unontrained ingle eriod toati inventory ytem wit ontant unit ot treated by Gould [1], Namia [] and Terine [3]. Alo, Silver [4] eamined te ingle eriod model wen te demand during te eriod normally ditributed. Single eriod model involving a number of item onneted by one or more ontraint an take a variety of form. Hadley [5] tudied ontrained multile item ingle eriod inventory roblem wit ontinuou ditribution of demand and ontant unit ot, uing te Lagrange multilier aroa. Te robabiliti ingle item ingle oure zero lead time inventory ytem wit intantaneou and uniform rate of demand ditribution ave been diued by Fabryky [6]. Reently, Abou-El-Ata [7] and Fergany [8] introdued a robabiliti ingle-item inventory roblem wit varying order ot and zero lead time under two retrition, uing geometri rogramming aroa. Ti aer invetigate te ingle eriod toati uniform inventory model tat onidering varying olding unit ot, a retrition on te eeted olding ot and te demand during te eriod i a ontinuou random variable. Our objetive i to determine te otimal urae (order) quantity, wi minimize te relevant eeted total ot for te eriod, wen te demand during tat eriod follow te uniform, te eonential and te Lalae ditribution. Finally, ome eial ae, wi ave been reviouly ublied and numerial illutrative eamle are added. Aumtion and Notation: Te following aumtion are made for develoing te model: Te ytem i te ingle eriod and zero lead time. Tat i, an item i uraed one only to atify te demand of a eifi eriod of time. Te number of unit to urae for inventory at te beginning of te eriod i (te deiion variable) and te tok level dereae at a uniform rate over te eriod Te demand during tat eriod i ontinuou random variable wit known robability denity funtion. Tere i no order (etu) ot. Te following notation are adoted for develoing our model: f() = Te robability denity funtion of te demand during te eriod, E() = Te eeted value of, F() = Te umulative ditribution funtion of, R() = Te reliability funtion, = Te otimal uraed quantity, ( ) =Te robability of atifying all te demand during te eriod if unit are uraed at te tart of te eriod, ( > ) = Te robability of ortage, Correonding Autor: Hala, A. Fergany, 1 Deartment of Matematial Statiti, Faulty of Siene, Tanta Univerity, Tanta, Egyt 334 = Te urae unit ot, = Te olding unit ot, C () = =Te varying olding unit ot, 1, = Te ortage unit ot, E(PC) = Te eeted urae ot for te eriod, E(HC) = Te eeted olding ot for te eriod, E(SC) = Te eeted ortage ot for te eriod, E(TC) K = Te eeted total ot for te eriod, = Te limitation on te eeted varying olding ot for te eriod.

2 J. Mat. & Stat. (1): , 6 - (II) > - (I) < (Holding) (Sortage) Fig. 1: Single eriod uniform inventory model Develoment of te eeted total ot funtion: A general outline for analyzing mot toati inventory roblem i te following: Develo an ereion for te ot inurred a a funtion of bot te random variable and te deiion variable. Determine te eeted value of ti ereion wit reet to te robability denity funtion of demand. Determine tat minimize te relevant eeted total ot. Tu, deending on te amount demanded and te quantity uraed, te inventory oition after demand our may be eiter oitive (olding) or negative (ortage). Tee two ituation are own in Figure. 1. Now, onider tat: H(,) = Te random variable rereent te amount in inventory at te end of te eriod, and S(,) = Te random variable rereent te ortage quantity at te end of te eriod. Ten, in te firt ituation in Figure 1 we ave: H(,)= and S(,) = Alo, in te eond ituation of Figure. 1 we get: H(,)= H(, ) and and S(,) = ( ) for = for > for S(, ) = ( ) for > Terefore, te eeted urae, te eeted olding and te eeted ortage ot for te eriod reetively are given by: E(PC) = f ()d = (1) E(HC)= ( )f ()d + f ()d () ( ) E(SC) = f ()d (3) Hene, te eeted total ot for te eriod i: E(TC)= + ( ) f ()d + f ()d ( ) + f ()d (4) Our objetive i to minimize te funtion (4) ubjet to te following ontraint: E(HC) K (5) To olve ti roblem we an ue te Lagrange multilier aroa. We introdue a Lagrange multilier λ and form te funtion: L = E(TC) + λ [ E(HC) K ] (6) were λ = if E(HC) K <, λ > if E(HC) K =, λ E(HC) K = (7) ten [ ] 335

3 J. Mat. & Stat. (1): , 6 Te otimal value require tat : L =, (8) wi give: + (1 + λ)(1 + ) F() (1 + λ)( + ) + 1 f () + d (1 + λ) 1 f ()d f () + d R() =. Hene i te olution of te following equation: f () A F( ) + ( B + ) d 1 C f ()d R( ) =, (9) (1 + λ)( + ) were A = (1 + λ)(1 + ), B = (1 + λ) and C =, by teting different value of λ, it i oible to determine te otimum urae quantity tat meet te requirement of te ontraining ondition E(HC). Te iterative roe require tat λ be et equal to zero and mall inrement be added to it until te roblem ontraint i met for te different value of ß. Te otimal value > obtained from (9) minimize (4), were d E(TC) 1 = ( + 1) F() + d ( + 3) + 1 f ()d (1 ) 1 + f ()d + f ()d >. Standard robability ditribution u a te uniform, te eonential and te Lalae ditribution are frequently aumed for te demand during te eriod. Te eat olution of te otimal urae quantity an be derived wen te demand follow tee ditribution a followe: Demand during te eriod follow te uniform ditribution Aume tat te demand during te eriod follow te uniform ditribution wit mean b, ten b E ( ) = F ( ) = R( ) = 1, b, b. Subtituting in equation (9), we get : C b b A + B ln + 1+ ln were: f ()d = d = b b and b f () 1 1 b d = d = ln b b b = + ( ) Terefore, wen te demand during te eriod uniformly ditributed te otimal inventory oliy i te olution of te following equation: b + 1 eb A1 + Bln + ln + b ( ) = (1) 1 Were A 1 = (1 + λ)(4 + 3 ), and 4 te minimum eeted total ot for te eriod i: min E(TC) = b + 3 b ( ) ln (11) 4 b Demand during te eriod follow te eonential ditribution If te demand during te eriod eonentially ditributed wit mean and tandard deviation, ten we ave E() =, F() = 1 e, f ()d = e d = F() R(), alo, Were f () 1 1 d = e d = Γ(, ) R() = e and (1) a1 t (a, z) t e dt Γ =, i te inomlete gamma z funtion. Subtituting in equation (9), te otimal urae quantity i te olution of te following equation: 1 + A C F( ) + (B + ) Γ(, ) + (C ) R( ) =. (13) te minimum eeted total ot i given by: mine(tc) ( + )( ) = + e ( ) + + Γ, (14) 336

4 J. Mat. & Stat. (1): , 6 Table 1: te otimal urae quantity tat minimize te relevant eeted total ot for te eriod and meet te ontraint E (HC) 1 for te different value of ß ß Uniform Ditribution Eonential Ditribution Lalae Ditribution min E(TC) min E(TC) min E(TC) Demand during te eriod follow te Lalae ditribution Aume tat te demand during te eriod follow te lalae ditribution wit mean and variane, ten we an get te otimal urae quantity a follow: 1 E() =, V() =, F() = 1 e, 1 R() = e and f ()d ( + )R(), > f () 1 alo, d = e Γ(, ), > (15) Subtituting in equation (9), te otimal urae quantity i te olution of te following equation: A F( ) + C R( ) + e (B + ) Γ(, ) 1 C R( ) + R( ) =. (16) And te minimum eeted total ot i: min E(TC) = ( + )( ) + + e 4 ( ) + + e Γ,, (17) 4 Ti i unontrained ingle eriod toati inventory model wit uniform demand and ontant unit ot, wi i agree wit te reult of Fabryky [6]. Cae : For equation (1), let λ = and = C () =, A1 = B = and C =. Ten te following relation for an be obtained: ln ( ) eb b = + (19) Ti i unontrained ingle eriod toati inventory model wit demand uniformly ditributed and ontant unit ot,, wi i agree wit te reult of Taa [9]. Cae 3: For equation (13), let λ = and = C () =, A = B =, and C =, ten te otimal urae quantity i: F( ) + Γ (, ) = () + Ti i unontrained ingle eriod toati inventory model wit demand eonentially ditributed and ontant unit ot. Cae 4: For equation (16), let λ = and = C () =, A = B =, and C =,ten te otimal urae quantity i: + Γ = + F( ) e (, ) Ti i unontrained ingle eriod toati 4-Seial ae: We dedue four eial ae for our inventory model wit ontant unit ot and te model a follow: uniform demand follow lalae ditribution. Cae 1: For equation (9) let λ = and = C () =, A = B = and C =, Illutrative eamle: Conider te following ingle eriod uniform inventory model wit arameter given ten te otimal urae quantity given by: a: =.5, =.5, = 15.5 and K=1. We will f () determine te otimal urae quantity wen: F( ) + d = (18) te demand during te eriod uniformly + ditributed, b= 5, 337 (1)

5 J. Mat. & Stat. (1): , 6 te demand during te eriod eonentially ditributed, =5, te demand during te eriod i Lalae ditributed, = 5 and = Uing equation (1), (13) and (16) by teting different value of λ, we an get te otimal urae quantity tat minimize te relevant eeted total ot for te eriod and meet te ontraint E(HC) 1 for te different value of ß a own in Table 1: Te olution of te roblem may be determined more readily by lotting mine(tc) againt ß for ea ditribution of demand a te following Figure: Uniform Eonential Lalae Figure: mine(tc) againt ß for ea ditribution of demand CONCLUSION Ti artile derive te otimal olution for ontrained ingle eriod toati uniform inventory model tat onidering ontinuou ditribution of demand and varying olding ot. We ave evaluated tat minimize te relevant eeted total ot for te eriod for ea value of ß. Alo, for our eamle, at te ame inventory arameter we found tat te min E(TC) in te ae of te demand uniformly ditributed i le tan te min E(TC) in te ae wen te demand during te eriod follow te eonential or te Lalae ditribution for all different value of ß. REFERENCES 1. Gould, F.J., G.D. Een and C.P. Smidt, Introdutory Management Siene. Prentie-Hall International, In.. Namia, S., 1993). Prodution and Oeration Analyi. nd Ed. Irwin, In. 3. Terine, R.J., Prinile of Inventory and Material Management. 4t Ed. Prentie-Hall, In. 4. Silver, E.A. and R. Peteron, Deiion Sytem for Inventory Management and Prodution Planning. New York, Jon Wiley and Son. 5. Hadley, G. and T.M. Witin, Analyi of Inventory Sytem. Englewood Cliff, N.J., Prentie-Hall. 6. Fabryky, W.J. and J. Bank, Prourement and Inventory Sytem: Teory and Analyi. Reinold Publiing Cororation, USA. 7. Abou-El-Ata, M.O., H.A. Fergany and M.F. E1- Wakeel, 3. Probabiliti Multi-item Inventory Model Wit Varying Order Cot Under Two Retrition: A Geometri Programming Aroa. Intl. J. Prodution Eonomi, 83: Fergany, H.A. and M.F. El-Wakeel, 4. Probabiliti Single-Item Inventory Problem wit Varying Order Cot Under Two Linear Contraint. J. Egyt. Mat. So., 1: Taa, H., Oeration Reear. 6t Ed. Englewood Cliff, N.J., Prentie-Hall. 338