Cardamom Planters Association College, Bodinayakanur, Tamil Nadu, India

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1 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07) Research Inda Publcatons OPPTIMIZATION OF TWO-ECHELON INVENTORY SYSTEM WITH TWO SUPPLIERS AND DIRECT DEMAND S. MOHAMED BASHEER R. KARTHIKEYAN K. KRISHNAN 3 Research Scholar, PG & Research Department of Mathematcs, Cardamom Planters Assocaton College, Bodnayakanur, Taml Nadu, Inda Lecturer n Mathematcs, Govt. Polytechnc College, Aundpatty 3 Assstant Professor, PG& Research Department of Mathematcs, Cardamom Planters Assocaton College, Bodnayakanur, Taml Nadu, Inda drkkmaths@gmal.com Abstract Inventores exst throughout the supply chan n varous form for varous reasons. Ths paper presents a contnuous revew two echelon nventory system. The operatng polcy at the lower echelon s (s, S) that s whenever the nventory level traps to s on order for = (S-s) tems s placed, the ordered tems are receved after a random tmewhch s dstrbuted as exponental. We assume that the demands accrung durng the stockout perod are lost. The retaler replenshes ther stock from the regular suppler whch adopts (0,M) polcy, M = n. When the regular suppler stock s empty the replacement of retaler stock made by the outsde suppler who adopts (0,N) polcy N = n. The jont probablty dsrupton of the nventory levels of retaler, regular suppler and the outsde suppler are obtaned n the steady state case.varous system performance measures are derved and the long run total expected nventory cost rate s calculated. Several nstances of a numercal examples, whch provde nsght nto the behavour of the system are presented. Key Words: Contnuous revew nventory system, two-echelon, postve lead tme.. Introducton Most manufacturng enterprses are organzed n to network of manufacturng and dstrbuted stes that procure Raw- materal, process them nto fnshed goods and dstrbuted the fnshed goods n to customers. The terms mult-echelon or mult-level producton dstrbuton network and also synonymous wth such networks (supply chan )when on tems move through more than one steps before reachng the fnal customer. Inventory exst throughout the supply chan n varous form for varous reasons. At any manufacturng pont they may exst as raw materals,work-n processorfnshed goods. The usual objectve for a mult-echelon nventory model s to coordnate the nventores at the varous echelons so as to mnmze the total cost assocated wth the entre mult-echelon nventory system. Ths s a natural objectve for a fully ntegrated corporaton that operates the entre system. It mght also be a sutable objectve when certan echelons are managed by ether the supplers or the retalers of the company. Mult-echelon nventory system has been studed by many researchers and ts applcatons n supply chan management has proved worthy n recent lterature.as supply chans ntegrates many operators n the network and optmze the total cost nvolved wthout compromsng as customer servce effcency. The frst quanttatve analyss n nventory studes Started wth the work of Harrs (95)[7].Clark and Scarf (960)[4]had put forward the mult-echelon nventory frst. They analyzed a N-echelon ppelnng system wthout consderng a lot sze, Recent developments n two-echelon models may be found n. One of the oldest papers n the feld of contnuous revew mult-echelon nventory system s a basc and semnal paper wrtten by Sherbrooke n 968.Hadley, G and Whtn, T. M., (963)[6],Naddor.E (966)[] Inventory System, John Wley and Sons, New York. Analyss of nventory systems, Prentce- Hall, Englewood Clff, New Jersey.HP's(Hawlett Packard) Strategc Plannng and Modelng(SPaM) group ntated ths knd of research n 977.Contnuous revew models of 89

2 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07) Research Inda Publcatons mult-echelon nventory system n 980s concentrated more on reparable tems n a Depot- Base system than as consumable tems (see Graves, Monzadeh and Lee).Kalpakam and Arvargnan(988) ntroduced multple reorder level polcy wth lost sales n nventory control system. All these papers deal wth reparable tems wth batch orderng. Jokar and Sefbarghy analyzed a two echelon nventory system wth one warehouse and multple retalers controlled by contnuous revew (R, ) polcy. A Complete revew was provded by Bento M. Beamon (998)[]. Sven Axsater (993)[] proposed an approxmate model of nventory structure n SC. He assumed (S-, S) polces n the Deport-Base systems for reparable tems n the Amercan Ar Force and could approxmate the average nventory and stock out level n bases. The supply chan concept grow largely out of two-stage mult-echelon nventory models, and t s mportant to note that consderable research n ths area s based on the classc work of Clark and Scarf (960).Contnuous revew pershable nventory models studed by Kalpakam. S and Arvargnan. G (998)[8] and a contnuous revew pershable nventory system at Servce Facltes was studed by Elango.C and Arvargnan.G,(00)[5].A contnuous revew (s, S) polcy wth postve lead tmes n two-echelon Supply Chan for both pershable and non pershable was consdered by Krshnan. K and Elango. C Krshnan.K And Elango.C.A contnuous revew (s, S) polcy wth postve lead tmes n two echelons Supply Chan was consdered by Krshnan. K. (007).Rameshpandy.M, et. al (04)[3] consder a Two-Echelon Pershable Inventory System wth drect and Retral demands and Satheeshkumar.R (04) [4] et. al consder a Partal Backloggng Inventory System n Two-echelon wth Retral and Drect Demands. The rest of the paper s organzed as follows. The model formulaton s descrbed n secton, along wth some mportant notatons used n the paper. In secton 3, steady state analyss are done: Secton 4 deals wth the dervaton of operatng characterstcs of the system. In secton 5, the cost analyss for the operaton.secton 6 provdes Numercal examples and senstvty analyss..the Problem Descrpton.Model The nventory control system n supply chan consdered n ths paper s defned as follows. A supply chan system consstng one Manufacturer(MF), two supplers (regular and outsde), sngle dsrupton centre(dc) and n dentcal retalers dealng wth a sngle fnshed product. These fnshed products moves from the manufacturer through the network consst of manufacture, suppler, DC, Retalers and the fnal customer. A fnshed product s suppled from MF to suppler (regular and outsde) whch adopts (0,M) and (0, N) replenshment polcy then the product s suppled to retaler who adopts(s,s) polcy. The demand at retalers node followsaposson dstrbutonwth rate>0. The replacement of tem n terms of product s made from regular suppler s admnstrated wth exponental dstrbuton havng parameter 0. The replenshment of tems of pocket s made from outsde suppler wth rate 0. Demands accrung durng the stock out perods are assumed to be lost. The maxmum nventory level at retaler node S s fxed, and the recorder pont s s and the orderng quantty s (=S-s) tems. The maxmum nventory at regular suppler nm(=n) and outsource suppler n N (=n) 90

3 . Notatons and varables We use the followng for the fourth comng analyss part of our theses C : The element of sub matrx at (,j)th poston of C j 0 : Zero matrx I: Identty matrx e: A column vector of s of approprate dmenson : Demand rate at Retaler : Mean reorder rate at retaler from regular suppler : Mean replacement rate at retaler from outsde suppler S : Maxmum nventory level at retaler s: Mnmum nventory level at retaler M: Maxmum nventory level at regular suppler N: Maxmum nventory level at outsources suppler H r : Holdng cost at regular H rd : Holdng cost at regular suppler H od : Holdng cost at outsources suppler O r : Orderng cost at retaler O rd : Orderng cost at regular suppler O od : Orderng cost at outsources suppler P r : penalty cost at retaler I r : Average nventory level at retaler I rd : Average nventory level at regular suppler I od : Average nventory level at outsource suppler R R : Mean reorder rate at retaler R Rd : Mean reorder rate at regular suppler R od : Mean reorder rate at outsource suppler S R : Shortage rate at retaler n : n Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07) Research Inda Publcatons 3. Analyss Let I (t), I (t) and I D (t) denote the on hand nventory levels of outsdesupplers, regular supplers and retaler respectvely at tme t +. We defne I (t) = { ( I (t), I (t), I D (t), ) : t 0 } as a Markov process wth state space E = { (, j,k) =,...n, j =,...n, k= 0,...S}. Snce E s fnte and all ts states are aperodc, recurrent non- null and also rreducble. That s all the states are ergodc. Hence the lmtng dstrbuton exsts and s ndependent of the ntal state The nfntesmal generator matrx of ths process C = ( a (, j, k, :l, m, n)) (, j, k )( l, m, n) E can be obtaned from the followng arguments. The arrval of a demand at retaler make a state transton n the Markov process from (, j, k ) to ( -, j, k ) wth the ntensty of transton > 0. The replacement of nventory atretaler from regular suppler makes a state transton from (, j, k ) to ( +, j-, k ) wth ntensty of transton > 0. The replacement of nventory at dstrbutor from outsde suppler makes a state transton from(, j, k ) to ( +, j, k- ) wth ntensty of transton > 0. 9

4 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07) Research Inda Publcatons The nfntesmal generator C s gven by A B A B C= A B B A Hence entres of C s gven by A p q; q n, ( n ),... B p q ; q ( n ),... C pq = B p q ( n ) q n 0 otherwse pq The sub matrces are gven by A p q; q n, ( n ),... A p q ; q ( n ),... A = A3 p q ( n ) q n A4 p q q 0 0 otherwse A p q; q 0 B = 0 otherwse pq The sub matrces of A and B are p q;q S,...s ( ) p q; q s,... A = p q; q 0 p q ; q S,...,0 0 otherwse p q ;q S,S... A = 0 otherwse pq;q S,...0 A 3 = 0 otherwse 9

5 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07) Research Inda Publcatons ( ) p q;q S,...s ( ) p q; q s,... A = ( ) p q; q 0 p q ; q S,...,0 0 otherwse 4 3.Steady StateAnalyss The structure of the nfntesmal matrx C, reveals that the state space E of the Markov process { I (t) : t 0 } s fnte and rreducble. Let the lmtng probablty dstrbuton of the nventory level process be lmpr{( I( t), I( t), I3( t) (, j, k)} where Let n n n,,... j, k j, k j, k ( (, j, k ), normalzng condton t s the steady state probablty that the system be n state (, j, k). denote the steady state probablty dstrbuton. For each can be obtaned by solvng the matrx equaton C = 0 together wth (, j, k ) E 4. Operatng characterstc In ths secton we derve some mportant system performance measure. 4.Average nventory Level The event I R,I Rd,IOd denote the average nventory level at Retaler, Regular suppler, and Outsde suppler respectvely, () R n n S I k j 0 () () Rd n S n I j Od k 0 j n S n I k j 0 k 0 4.Mean Reorder Rate Let RR, RRd,ROd be the mean reorder rate at retaler, regular suppler, outsource suppler respectvely, () () () R R R Rd R Od n n k j * n s s j,k k, k 0 * n s j, j 0 93

6 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07) Research Inda Publcatons 4.3Shortage rate Let S R, be the shortage rate at retaler and t s gven by () S R n n k j j,0 5 Cost analyss In ths secton we mpose a cost structure for the proposed model and analyze t by the crtera of mnmzaton of long run total expected cost per unt tme. The long run expected cost rate C(S, ) s gven by c( s, ) ( Hr *I r ) ( Hrd *I rd ) ( Hod *I od ) (O r*r r) (O rd*r rd ) (O od*r od ) (P r*s r) Although we have a not proved analytcally the convexty of the cost functon C(S,) our experence wth consderable number of numercal examples ndcate that C(s,) for fxed appears to be convex s. In some cases t turned out to be ncreasng functon of s. For large number case of C(s,) revealed a locally convex structure. Hence we adopted the numercal search procedure to determne the optmal value of s. 6. Numercal Example 6 Numercal Example and Senstvty Analyss In ths secton we dscuss the problem of mnmzng the structure. We assumehr Hrd Hod the holdng cost at dstrbuton node s less than that of regular dstrbutor node and an outsde dstrbutor node. Holdng cost at the regular dstrbutor node s less than outsource dstrbutor node as the rental charge may be hgh at outsource dstrbutor. AlsoOr Ord Ood the orderng cost at retaler node s less than that of regular dstrbutor node and an outsource dstrbutor node. Orderng cost at the regular dstrbutor s less than outsource dstrbutor node. The results we obtaned n the steady state case may be llustrated through the followng numercal example, S =6, M = 80, N =60, = 4, 3, Hr., Hrd., Hod.3 O., O., O.3 P 3., P 3., r rd od The cost for dfferent reorder level are gven by r rd s 3 4 * C(s, ) Table: Total expected cost rate as a functon s and For the nventory capacty S, the optmal reorder level s* and optmal cost C(s,) are ndcated by the symbol *. The Convexty of the cost functon s gven n the graph. 94

7 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07) Research Inda Publcatons s s C(s, ) C(s, ) Concluson Ths paper deals wth an Inventory problem wth two suppler, namely a regular suppler and outsde suppler. The demand at retaler node follows Posson wth rate λ. The structure of the chan allows vertcal movement of goods from to regular suppler to Retalers. If there s no stock n regular suppler, then the retaler wll get products from outsde suppler. The model s analyzed wthn the framework of Markov processes. Jont probablty dstrbuton of nventory levels at retaler, Regular and Outsde supplers n the steady state are computed. Varous system performance measures are derved and the long-run expected cost rate s calculated. By assumng a sutable cost structure on the nventory system, we have presented extensve numercal llustratons to show the effect of change of values on the total expected cost rate.it would be nterestng to analyze the problem dscussed n ths paper by relaxng the assumpton of exponentally dstrbuted lead-tmes to a class of arbtrarly dstrbuted leadtmes usng technques from renewal theory and sem-regeneratve processes. Once ths s done, the general model can be used to generate varous specal eases. References [] Axsater, S. (993). Exact and approxmate evaluaton of batch orderng polces for two level nventory systems. Oper. Res [] Benta M. Beamon. (998). Supply Chan Desgn and Analyss: Models and Methods. Internatonal Journal of Producton Economcs.Vol.55, No.3, pp [3] Cnlar.E, Introducton to Stochastc 'Processes, Prentce Hall, Engle-wood Clffs, NJ, 975. [4] Clark, A. J. and H. Scarf, (960). Optmal Polces for a Mult- Echelon Inventory Problem.Management Scence, 6(4): [5] Elango.C and Arvargnan.G, (00). Atleastst sales nventory system wth multple reorder levels (In Russa t ). ElectronnoeModelrovane, 3, 74,8. [6] Hadley, G and Whtn, T. M., (963), Analyss of nventory systems, Prentce- Hall, Englewood Clff, [7] Harrs, F., 95, Operatons and costs, Factory management seres, A.W. Shah Co., Chcago,48-5. [8] Kalpakam, S and Arvargnan, G. (998). A Contnuous revew Pershable Inventory Model, Statstcs 9, 3,

8 Internatonal Journal of Computatonal and Appled Mathematcs. ISSN Volume, Number (07) Research Inda Publcatons [9] Karln.S and Taylor.-I. M, (998), An Introducton to Stochastc Modelng, Thrd edton, Academc press, New York. [0] Krshnan. K, (007), Stochastc Modelng In Supply Chan Management System,unpublshed Ph.D. Thess, Madura Kamaraj Unversty, Madura. [] Medh.J,(009) Stochastc processes, Thrd edton, New Age Internatonal Publshers, New Delh. [] Naddor.E (966), Inventory System, John Wley and Sons, New York. [3] Rameshpandy.M, Peryasamy.C, Krshnan.K(04). Analyss of Two- EchelonPershable Inventory System wth drect and Retral demands IOSR, Journal of Mathematcs, Vol. 0, Issue 5, Ver., pp [4] Satheeshkumar.R, Rameshpandy.M, Krshnan.K, (04). Partal BackloggngInventory System n Two-echelon wth Retral and Drect Demands InternatonalJournal Of Mathematcal Scences Vol.7, No.., pp