Constrained Probabilistic Continuous Review Inventory System with Mixture Shortage and Stochastic Lead Time Demand

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1 ISSN [PRINT] ISSN [ONLINE] Advances in in Natual Natual Science Science Advances Vol. 5, Vol. 6,No. No.4 1, 013, pp DOI: /j.ans Constained Poailistic Continuous Review Inventoy System with Mixtue Shotage Stoastic Lead Time Dem CONTENTS Potection of High Voltage Tansmission Lines of Canada fom the Ice y High-Fequency Electomagnetic [a], ; Hala A. Fegany[] Mona Waves F. El-Wakeel The Ideas Behind the Electomagnetic Atomic Theoy Highe Institute fo Computes, Infomation Management TenolSeveal, inventoy models with mixtue of ackogy Tanta, EGYPT. mona elwakeel@yahoo.com. odes lost sales wee poposed y Posne [] Depatment of Mathematical Statistics Faculty of Science, Tanta Uni[1], Montgomey et al. [], Roseneg [3], Kyung [4]. Yansouni Effects of Foundation on Jack-up Site Assessment vesity Tanta, EGYPT. Models halafegany@yahoo.com. Almost all the pevious esea woks used γ as a Coesponding Autho. [a] faction of unsatisfied dem that will e ackodeed the emaining faction 1 γ completely lost to model patial National Spatial Data Infastuctue in Botswana An Oveview ackodes. Since it is optimal to allow some stockouts if all Astact customes will wait γ = 1 it is optimal to eithe allow no stockouts o lose all sales if customes have no patience Effect of Extenal Electic Field upon Lowe Alkanols This pape deives the poailistic continuous eview ack- γ = 0. odes lost sales inventoy system when the ode cost Also, Rainowitz et al. [5] modeled a, inventoy is a function of the ode quantity. Ou ojective is to min- system using a contol vaiale, whi limits the maximum The Dynamic annual Collision Featues of Micoscopic Paticles Descied y the Nonlinea Södinge Equation imize the expected total cost unde a estiction on nume of ackodes allowed to accumulate duing a cycle. in the Nonlinea uantum Systems the expected annual holding cost when the lead time de- Also, Zipkin [6] shows that if dems occuing duing a m follows some continuous distiutions y using the stockout peiod ae lost sales athe than ackodes, the Lagangian method. Some pulished special cases ae d- optimal policy is to have eithe no stockouts o all stockouts. Study on Jilin Gowth with Tendsome Model educed an Povince s illustative Population numeical example Recently, Fegany El-wakeel [7,8] intoduced pogaphs is added. ailistic lostsales models with nomal distiution othe continuous distiution. Also, El-Wakeel [9] deived The Hamiltonian in Covaiant Theoy of Gavitation Key wods: Poailistic model; Mixtue shotage a poailistic inventoy ack-odes model with unifom inventoy system; Lost sales; Vaying ode cost; Holding distiution. cost; Safety stock; Lead time dem; Continuous In this pape, we assume that oth ackode lost distiutions. sales costs ae independent of the duation of the stockout γ is the ackode faction. Also, we deduced Mona F. El-Wakeel, & Hala A. Fegany 013. Constained Poailistic the model with vaying ode cost when the dem Continuous Review Inventoy System with Mixtue Shotage is a om vaiale, the lead-time is constant the Stoastic Lead Time Dem. Advances in Natual Science, 61, Availale fom: distiution of the lead time dem is known unde the DOI: holding cost constaint. The situation will e consideed in whi a single item is stocked to meet a poailistic dem. When the nume of units on h on ode eaes the eode point, action is initiated to pocue a eplenishment quantity. Received 3 Feuay 013; accepted 1 Ma 013 INTRODUCTION 1. ASSUMPTIONS AND NOTATIONS The two asic questions that any inventoy contol system has to answe ae when how mu to ode. Ove the yeas, hundeds of papes ooks have een pulished pesenting models fo doing this unde a wide vaiety of conditions assumptions. Most of authos have shown that if dem that cannot e filled fom stock is ackodeed o using the lost sales model. The following assumptions ae adopted fo developing ou model: The system is a continuous eview whi means that the dems ae ecoded as they occu the stock level is known at all times. AnSode quantity ofsize pe cycle is placed evey time the stock level eaes a cetain eode 9 Copyight Canadian Resea & Development Cente of Sciences Cultues

2 Constained Poailistic Continuous Review Inventoy System with Mixtue Shotage Stoastic Lead Time Dem point ae two decision vaiales. Thus, the assumptions ae: 1 The cycle is defined as the time etween two successive aivals of odes assume that the system epeats itself in the sense that the inventoy position vaies etween duing ea cycle. The aveage nume of cycles pe yea can e witten as n = D/ then the inventoy cycle is N = /D. 3 Thee is neve moe than a single ode outsting. The following notations ae adopted fo developing ou model: D = The aveage ate of annual dem, = A decision vaiale epesenting the ode quantity pe cycle, = A decision vaiale epesenting the eode point, N = The inventoy cycle, n = The aveage nume of cycles pe yea, L = The lead time etween the placement of an ode its eceipt, γ = A faction of unsatisfied dem thatwillackodeed, x = The continuous om vaiale epesents the dem duing L lead time dem, f x = The poaility density function of the lead time dem Fx its distiution function, x = The omvaiale epesents the net inventoy when the pocuement quantity aives if the lead time dem x, E x = ss = Safety stock = The expected net inventoy = 0 x f xdx = Ex ode cost, the expected holding cost the expected shotage cost as follows: ETotal Cost =EOde Cost EHolding Cost EShotage Cost I.e., ETC = EOC EHC EBC ELC.1 whee EOC = Co n = co β D = co Dβ 1. Ex 1 γ S EHC = H = = Ex 1 γ x f xdx.3 c γ D EBC = c n γ S = ELC = cl n 1 γ S = x f xdx.4 c 1 γ D x f xdx.5 Theefoe Ex E [TC, ] = co D c γ D cl D S 1 γ S.6 β 1 x f xdx Ou ojective is to minimize the expected annual H = The aveage on h inventoy = Max. on h total cost E [TC, ] unde the expected holding cost Min.on h/ = ss ss/ constaint:.7 Ex 1 γ S K = Ex x f xdx R = The eliaility function = 1 F = f xdx S = The expected value of shotages pe cycle = x f xdx To find the optimal values whi minimize equation.6 unde the constaint.7, we will use the Lagange multiplie tenique as follows: β 1 L,, λ = co D Ex c γ D cl D 1 γ S S [ ] Ex 1 γ S K.8 λ c = The ackodes cost pe cycle, co = The ode cost pe cycle, To find the optimal values can e found y Co = Co β = The vaying ode cost pe cycle, 0 < β < 1, whee eta is a constant eal nume selected to povide setting ea of the coesponding fist patial deivatives of equation.8 equal to zeo, then we otain: the est fit of estimated expected cost function. = The holding cost pe yea, A Bβ [γ M G G]S = 0.9 cl = The lost sales cost pe cycle, K = The limitation on the expected annual holding cost. A R =.10 γ M 1 γ G A wheea = a λ, B = 1 β co D, G=cl D M = S It is possile to develop the expected annual total cost whee c D. Clealy thee is no closed fom solution of equations it consisted of thee components: the expected vaying THE MATHEMATICAL MODEL Copyight Canadian Resea & Development Cente of Sciences Cultues 10

3 Mona F. El-Wakeel, & Hala A. Fegany 013. Advances in Natual Science, 61, 9-13 Thus we can otain the optimal values y solving equations espectively fo diffeent values of 3.1 Lead-time Dem Follows Unifom Distiution β vay λ until the smallest positive value is found su Assume that the lead-time dem follows the unifom that the constaint holds. Hence the minimum expected annual total cost is: distiution as follows: 3. LEAD-TIME DEMAND 1 f x = ; 0 x with Ex = R = 1 E [TC, ] = co D β 1 { ]} 1[ 1 1 γ e D e 3.8 [c γ cl 1 γ ] 3.1 By sustituting fom equation 3.4 into equations.9.10 then solving them simultaneously, we can otain 3.3 Lead-time Dem Follows Laplace Distiution the optimal ode quantity fom the following equation: Similaly, conside the lead-time dem follows Laplace 3 4 β 3 β distiution; we can otain the exact solution as follows: A 1 γ A 1 γ [γ M 1 γ G] S= A B1 γ A [γ M 1 γ G] β AB1 γ [γ M 1 γ G A] f x = 1 x µ e θ ; θ < x <, < µ <, θ >0 B [γ M 1 γ G A] = 0 3. then, Also, the optimal eode level is given y: R = γ M A 1 γ G γ M 1 γ G A θ µ 3.9 e θ, R 0, θ > 0 Sustituting fom equation 3.9 into equations then solving them simultaneously, we get: S = Thus we can otain the optimal values y solving equations espectively fo diffeent values of β vay λ until the smallest positive value is found su that the constaint holds. Hence the minimum expected annual total cost is: β 1 E [TC, ] = co D cl 1 γ D c D γ 3.4 A 1 γ 3 β A [γ M G G] β AB1 γ θ A [γ M G G] 1 β B [γ M G G] = = µ θ ln 1 γ γ M 1 γ G A 3.11 Thus we can otain the optimal values y solving equations espectively fo diffeent values of β vay λ until the smallest positive value is found su that the constaint holds. Hence the minimum 3. Lead-time Dem Follows Exponential Distiu- expected annual total cost is: tion E [TC, ] = co D β 1 Lead-time dem follows Exponential distiution: 1 γ θ µ θ e c µ h f x = e ; x 0, 0, θ > 0 then, [γ c 1 γ cl ] Dθ µ e θ S = e, Similaly, sustituting fom equation 3.5 into equations 4. SPECIAL CASES.9.10 solving them simultaneously, we get: Case 1: Let γ = 0, β = 0 K Co = co λ = 0. Thus equations.9.10 ecome A 1 γ 3 β A [γ M 1 γ G] β 1 β D co cl S AB1 γ A [γ M 1 γ G] = R = cl D B [γ M 1 γ G] = [ ] 1 G γ M G = ln 1 γ A This is unconstained lostssales continuouseview inventoy model with constant units of cost, whi ae the same 3.7 esults as in Hadley [10]. 11 Copyight Canadian Resea & Development Cente of Sciences Cultues

4 Constained Poailistic Continuous Review Inventoy System with Mixtue Shotage Stoastic Lead Time Dem Tale 1 The Optimal Solutions the Min ETC fo Ea Distiution at γ = 0.7 β Unifom Distiution minetc Exponential Distiution minetc Laplace Distiution minetc Case : Let γ = 1, β = 0 K Co = co i Unifom distiution with f x = 1/50, 0 x 50 λ = 0. Thus equations.9.10 ecome units. ii Exponential distiution with = units. D co c S iii Laplace distiution with µ = 15 θ = 0 units. = R = Fom the aove example, we have the following c D paametes values: D = 1600, co = $400, = $10, c = This is unconstained ackodes continuous eview inven- $600, γ = 0.7, cl = $000 K = $8500. By solving toy model with constant unit costs, whi ae the same the pevious deduced equations fo ea distiution at esults as in Hadley [10]. diffeent values of β, we otain Tale 1. Fom the data given in Tale 1, we can daw the optimal Note: When γ = 1, β = 0 K Co = co values of against fo all distiutions as shown in the λ = 0. following Figues 1. Equations give unconstained simple model with constant units of cost the lead-time dem follows the Unifom distiution, whi ae the same esults as in Faycky, W., et al [11]. Equations will e the fom of unconstained continuous eview model with constant units of cost the lead-time dem follows the Exponential distiution, whi ae agee with esults of Hillie [1]. Equations give unconstained continuous eview model with constant units of cost the leadtime dem follows the Laplace distiution, whi ae the Figue 1 The Optimal Values of Against β same esults as in Nahmias [13]. 5. AN ILLUSTRATIVE EXAMPLE The cosmetics depatment of a lage depatment stoe has ecently intoduced a constained, system with vaying ode cost mixed shotages to contol many items in the depatment. A paticula type of expensive pefume has an annual dem ate equals 1600 units. The cost of placing an ode amounts to $4000 the inventoy holding cost is $10. This paticula pefume is not easy to otain elsewhee, hence dems occuing when the stoe is out of stock ae patially ackodeed. The management estimates that 70% of unsatisfied dem will e ackodeed with ackode cost equals $600 the emaining dem will e lost with cost $000. Thee is a estiction that the aveage holding cost is eithe less than o equal $8500 pe yea lead-time is S the pocuement constant. Detemine when the lead time dem has the following distiution: Copyight Canadian Resea & Development Cente of Sciences Cultues Figue The Optimal Values of Against β CONCLUSION This pape deducing ou poailistic, model with mixed shotage when lead-time dem follows Unifom, 1

5 Mona F. El-Wakeel, & Hala A. Fegany 013. Advances in Natual Science, 61, 9-13 Exponential Laplace distiutions. Fo su distiutions, we can evaluate the solution of fo ea value of β λ whi yields ou expected holding cost constaint then otain the minimum expected total cost. Fom the pevious example, we can deduce that the least minetc otained when the lead-time dem follows Laplace distiution its optimal minetc will e at β = 0.1. [6] [7] [8] REFERENCES [1] Posne, M., & Yansouni, B A class of inventoy models with custome impatience. Naval Resea Logistics uately, 193, [] Montgomey, D. C., Bazaaa, M., & Keswani, A. K Inventoy models with a mixtue of ackodes lost sales. Naval Resea Logistics uately, 0, [3] Roseneg, D A new analysis of a lotsize model with patial acklogging. Naval Resea Logistics uately, 6, [4] KYUNG, S Inventoy model with patial ackodes. Intenational Jounal of Systems Science, 131, [5] Rainowitz, G., Mehez, A., Chu, C.-W., & Patuwo, B. E A patial ackode contol fo continuous eview, q inventoy system with poisson [9] [10] [11] [1] [13] dem constant lead time. Computes & opeations esea, 7, Zipkin, P Foundations of Inventoy Management McGaw-Hill. Fegany, H., & El-Wakeel, M Constained poailistic lost sales inventoy system with continuous distiutions vaying ode cost. Jounal of Association fo the Advancement of Modeling &Simulation Teniques in Entepises, 7, 3 4. Fegany, A., & El-Wakeel, M Constained poailistic lost sales inventoy system with nomal distiution vaying ode cost. Jounal of Mathematics Statistics, 1, El-Wakeel, M. F. 01. Constained ackodes inventoy system with vaying ode cost: Lead time dem unifomly distiuted. Jounal of King Saud Univesity-Science, 43, Hadley, G Analysis of Inventoy Systems: By G. Hadley TM Whitin. Pentice-Hall. Faycky, W. J., & Banks, J Pocuement inventoy systems: theoy analysis. Reinhold. Hillie, F. S., & Lieeman, G. J Intoduction to opeations esea, volume 6. McGaw-Hill New Yok. Nahmias, S., & Cheng, Y Poduction opeations analysis. S 13 Copyight Canadian Resea & Development Cente of Sciences Cultues