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1 Journal of Physcs: Conference Seres PAPER OPEN ACCESS An ntegrated producton-nventory model for the snglevendor two-buyer problem wth partal backorder, stochastc demand, and servce level constrants To cte ths artcle: Nughthoh Arfaw Kurdh et al 0 J. Phys.: Conf. Ser Vew the artcle onlne for updates and enhancements. Related content - An ntegrated producton-nventory model for food products adoptng a general raw materal procurement polcy G Fauza, H Prasetyo and B S Amanto - Analyss of producton-nventory decsons n a decentralzed supply chan wth prcedependent demand N.A. Kurdh, S.T. Irsananto and Sutanto - A collaboratve vendor-buyer productonnventory systems wth mperfect qualty tems, nspecton errors, and stochastc demand under budget capacty constrant: a Karush-Kuhn-Tucker condtons approach N.A. Kurdh, R.A. Nurhayat, S.B. Wyono et al. Ths content was downloaded from IP address on /0/08 at :

2 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 An ntegrated producton-nventory model for the snglevendor two-buyer problem wth partal backorder, stochastc demand, and servce level constrants Nughthoh Arfaw Kurdh, Toray Ad Dwryo, Sutanto Department of Mathematcs, Faculty of Mathematcs and Natural Scence, Sebelas Maret Unversty, Surakarta, 57, Indonesa Emal: arfa@mpa.uns.ac.d, torayad@yahoo.com, sutantompa@gmal.com Abstract. Ths paper presents an ntegrated sngle-vendor two-buyer producton-nventory model wth stochastc demand and servce level constrants. Shortage s permtted n the model, and partal backordered partal lost sale. The lead tme demand s assumed follows a normal dstrbuton and the lead tme can be reduced by addng crashng cost. The lead tme and orderng cost reductons are nterdependent wth logartmc functon relatonshp. A servce level constrant polcy correspondng to each buyer s consdered n the model n order to lmt the level of nventory shortages. The purpose of ths research s to mnmze jont total cost nventory model by fndng the optmal order quantty, safety stock, lead tme, and the number of lots delvered n one producton run. The optmal producton-nventory polcy ganed by the Lagrange method s shaped to account for the servce level restrctons. Fnally, a numercal example and effects of the key parameters are performed to llustrate the results of the proposed model. Keywords: Integrated model, partal backorder, servce level, lead tme. Introducton In the past few decades, frms have realzed that an effcent management across the dfferent partes n a supply chan s crtcal to reducng nventory costs. Ths effcent management can be acheved through greater cooperaton and better coordnaton among the dfferent partes. Ths means that the vendor and the buyers should work cooperatvely towards mnmzng ther costs. Integrated nventory management has receved a great deal of attenton. Probably, Goyal [5] has been one of the frst poneers n studyng the jont optmzaton problem consstng of a sngle vendor and a sngle buyer where the rate of producton s nfnte for the vendor. The Goyal s [5] model has been generalzed by Banerjee [] by consderng the fnte producton rate. Later, Goyal and Nebebe [] proposed new soluton approaches for the sngle-vendor sngle-buyer model under dfferent shpment strateges. Pan and Yang [] also nvestgated the sngle-vendor sngle-buyer ntegrated model under assumpton that the lead tme demand follows normal dstrbuton functon. Further, Ouyang et al. [0] mprove the Pan and Yang s [] model by assumng that the shortage s permtted durng the lead tme. Recently, the sngle- Content from ths work may be used under the terms of the Creatve Commons Attrbuton.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Publshed under lcence by Ltd

3 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 vendor sngle-buyer ntegrated models have been developed by many scholars n dfferent assumptons, for example n Rad et al. [], An and Lee [], and Gr and Bardhan [4]. Gr and Bardhan [4] proposed a sngle-vendor sngle-buyer model by consderng the space lmtaton at the buyer s end. Further, only few researchers have addressed the ntegrated sngle-vendor mult-buyer system. In general, n the system the vendor has a problem wth determnng optmum producton quantty and shppng schedule, and the buyer has a problem wth determnng the order quantty, whch mnmze the jon total operatng cost. Durng the last three decades, researchers have been developng the model and the soluton to these problems. Lu [9] proposed the sngle-vendor mult-buyer nventory model wth the vendor manufacturng at a fnte rate. Yao and Chou [5] then ntroduced a heurstc for the snglevendor mult-buyer problem. Other related papers have been developed by Abdul-Jalbar et al. [], Rad et al. [], and Jha and Shanker [7]. Abdul-Jalbar et al. [] proposed an ntegrated nventory model for the sngle-vendor two-buyer problem. In the model, t s assumed that the vendor manufactures the tem at a fnte rate, each buyer faces a constant determnstc demand, and the shortages are not allowed. Rad et al. [] consdered a two-echelon supply chan model wth a sngle vendor and two buyers. The mathematcal model s developed for the ntegrated vendor-managed nventory (VMI) polcy. Later, n partcular, Jha and Shanker [7] analyzed an ntegrated producton-nventory model wth controllable lead tme and stochastc demand durng the lead tme. They assumed that the unsatsfed demand at the buyers s completely backordered and a servce level constrant correspondng to each buyer s ncluded n the model, whch lmts that the stock-out level per cycle of each buyer s bounded. In ths paper, we address an ntegrated producton-nventory model for the sngle-vendor two-buyer problem wth controllable lead tme and servce level constrants. Unlke Jha and Shanker [7], we assume that the partal backorder stuaton s consdered. Ths means that the shortages are partal backordered and partal lost sale wth a certan backorder rate. Further, the lead tme and orderng cost reductons are nterdependent. Accordng to Kurdh et al. [8], the mplementaton of electronc data nterchange (EDI) may reduce the lead tme and orderng cost smultaneously. Hence, t s more close to the real stuaton that orderng cost reductons vary accordng to dfferent lead tmes. It s also assumed that both buyers order the same tem to the vendor and the demand from customers to each buyer s stochastc. When the stochastc demand s consdered, lead tme becomes an mportant ssue. The controllable lead tme leads to many benefts such as mproves customer servce level and ncreases the compettve advantage of busness. In many practcal stuatons, lead tme can be controlled at the expense of extra cost whch s known as lead tme crashng cost. In the present study, we also assume that the lead tme of each buyer has several components n whch all components can be shortened at an added crashng cost. We have focused on the sngle-vendor two-buyer system because t s the smplest case wthn the sngle-vendor mult-buyer problems. In ths system, we can show a guarantee that the optmal soluton obtaned satsfes the second order suffcent condton for the mnmzng problem wth two servce level constrants. We also beleve that the results of ths study wll offer the possble strateges whch can be analyzed for the mult-buyer case. The rest of ths paper s organzed as follows: The notatons and assumptons used n ths paper are ntroduced n Secton. In Secton, we formulate the ntegrated producton-nventory model contanng sngle vendor and two buyers wth controllable lead tme, stochastc demand, and a servce level constrant on each buyer n the partal backorder case. The Lagrangan multpler technque and a detaled soluton procedure to solve the proposed model are presented n Secton 4. In Secton 5, a numercal example and dscusson of the results are provded. Fnally, some conclusons and suggestons for some future research are gven n Secton.. Notatons and Assumptons The followng notatons and assumptons are used n developng mathematcal model... Notatons th buyer parameter ( =,) D Average demand per unt tme

4 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 C b A b S k h b L Q r X Unt purchase cost Orderng cost per order Safety stock Safety factor (decson varable) Holdng cost rate (per monetary unt nvested n nventory) per unt tme Length of lead tme (decson varable) Order quantty (decson varable) Reorder pont Lead tme demand, whch s normally dstrbuted wth fnte mean D L and standard devaton σ L, where σ denotes the standard devaton of demand per unt tme, X ~N(D L, σ L ) β Fracton of the shortage that wll be backordered ε Proporton of demands that cannot be met by stock, ( ε ) s the servce level Vendor parameter P Producton rate, P > D (D = = D ) C v Unt producton cost (C v < C b, ) h v Holdng cost rate (per monetary unt nvested n nventory) per unt m Number of lots delvered from the vendor to each buyer n a producton cycle (same for all the buyers), a postve nteger (decson varable) A v Setup cost per setup Q Shpment lot sze n each delvery to meet the demand of all the buyers, Q = Q =... Assumptons. The system conssts of two buyers who are suppled wth a sngle-tem by a sngle-vendor.. Buyer orders a lot of sze Q (Q = = Q ) and the vendor manufactures mq unts wth a fnte producton rate P (P > D) n one setup but shps n quantty Q over m tmes to meet the demands of all the buyers such that Q = D Q/D.. The tems are delvered by the vendor at the same tme to buyer and buyer. 4. The lead tme demand X has fnte mean (D L ) and t follows a normal dstrbuton wth standard devaton σ L. 5. Each buyer revews nventory usng contnuous revew polcy and places an order whenever nventory level falls to the reorder pont.. The reorder pont r = expected demand durng lead tme (D L ) + safety stock (S ), and S = k σ L,r = D L + k σ L, where k s the safety factor. 7. The lead tme L of buyer has n mutually ndependent components. The rth component of lead tme of buyer has a mnmum a,r, normal duraton b,r and a crash cost per unt tme c,r, where c, c, c,n. Let L,r be the length of the lead tme wth components,,...,r crashed to ther mnmum duraton, then L,r can be expressed as n L,r = r j=r+ b,j j= a,j, r =,,, n. The lead tme crashng cost C (L ) per cycle for a gven L [L,r, L,r ] s r C (L ) = c,r (L,r L ) + j= (b,j a,j ). 8. The lead tme and orderng cost reductons have the followng logartmc functonal relatonshp as: A,0 A = τ ln ( L ), A L,0 where τ < 0 s constant scalng parameter for the logartmc relatonshp between percentages of reductons n lead tme and orderng cost.

5 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008. Model Formulaton In ths research, there are two buyers who ordered sngle-tem to the same vendor. buyer have a number of orders (Q ) and lead tme demand (X ). However, both the buyers and vendor work together = Q to determne the optmal Q, where Q =. Snce the setup cost for producton s consdered qute expensve, the vendor producng mq unts, but sendng a number of Q unts any buyer to order. Sngletem that have been produced wll be sent to each buyer wth lead tme demand L weeks, where the lead tme L can be reduced by crashng cost. A mathematcal model of the sngle-vendor two-buyer system wll be formulated to mnmze the jont total annual expected cost whle satsfyng the servce level constrant on each buyer by determnng the optmal order quantty, lead tme, safety factor of the buyers and the number of shpments n a producton cycle between the vendor and the buyer. The nventory pattern of the system can be seen n Fgure. Fgure. The nventory pattern for the sngle-vendor two-buyer system.. Vendor s expected total annual cost 4

6 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 When the buyer orderng sngle-tem to vendor amounted Q unts, the vendor manufactures mq at one setup wth a fnte producton rate P. The length of system cycle s mq, m lots of Q sze are D delvered from the vendor to all the buyers. To obtan a mnmum vendor total cost, vendor wll determne the optmal values of the frequency of delvery (m). The resultng expected total annual cost for the vendor, whch s composed of setup cost and holdng cost, s expressed as TEC v (Q, m) = A vd + Q h mq vc v [m ( D ) + D ]. () P P.. Buyer s expected total annual cost The lead tme demand from customers to each buyer follows a normal dstrbuton wth mean D L and standard devaton σ L. The expected demand because of the occorrence of stockout s gven by E(X r ) = σ L φ(k ), where φ(k ) = φ(k ) k [ Φ(k )], and φ, Φ are the standard normal probablty densty functon and cumulatve dstrbuton functon. Thus, the expected number of backorders per cycle s β σ L φ(k ) and the expected loss n sales per orderng cycle s ( β )σ L φ(k ), where β s the fracton of the shortage that wll be backordered. The resultng expected total annual cost for each buyer, whch s composed of orderng cost, holdng cost, and crashng cost, s expressed as TEC b (Q, k, L ) = A(L )D Q Substtutng Q = D Q D TEC b (Q, k, L ) = A(L )D Q + h b C b ( Q + k σ L + ( β )σ L φ(k )) + D C Q (L ). () nto (), the expected annual total cost for the th buyer can be wrtten as + h b C b ( Q D D + k σ L + ( β )σ L φ(k )) + D Q C (L ). () Further, each buyer use servce level constrant polcy to lmt the proporton of demands not met from stock, whch should not exceed a certan value. The servce level constrant for buyer can be expressed as Expecteddemand shortages at the end of cycle of buyer Quantty avalable at buyer for satsfyng the demand per cycle ε,.e., σ L φ(k ) Q ε, so ε s the servce level. Substtuton of Q n the nequalty gves Dσ L φ(k ) D Q ε. (4).. Jont expected total annual cost The jont expected total annual cost s the sum of the expected total annual cost for the vendor n () and the expected total annual cost for the buyer and n (). The servce level constant n (4) ndcates that the shortage per cycle for buyer I s lmted. Hence, the problem that must be resolved s Mnmze JTEC(Q, k, k, L, L, m) = TEC b (Q, k, L ) + TEC v (Q, m) = = D Q [A v + A m = (L ) + C (L )] + h b C b ( Q D D + k σ L + ( β )σ L φ(k )) + Q h vc v [m ( D P ) + D P ] subject to Dσ L φ(k ) D Q ε,. = (5) 4. Soluton Technque The problem (5) can be solved frst by addng slack varables H 0 and H 0 to convert the constrants σ L φ(k ) ε Q and σ L φ(k ) ε Q, respectvely, to equalty. Then, the Lagrange functon of (5) s JTEC(Q, k, k, L, L, λ, λ, H, H, m) 5

7 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 = D Q [A v + A m = (L ) + C (L )] + h b C b ( Q = D D + k σ L + ( β )σ L φ(k )) () + Q h vc v [m ( D ) + D ] + λ P P = (σ L φ(k ) + H Qε ), where λ and λ are Lagrange multpler. Takng the second partal dervatves of () wth L [L,r, L,r ], =,, we get JTEC(Q,k,k,L,L,λ,λ,H,H,m) = h L b C b k σ h 4L L b C b ( β )σ φ(k 4L L ) λ σ φ(k 4L L ) < 0,. Consequently, f Q, k, k, λ, λ, H, H, and m fxed, JTEC(Q, k, k, L, L, λ, λ, H, H, m) s concave n L [L,r, L,r ],. Hence, for fxed (Q, k, k, λ, λ, H, H, m), the mnmum jont total expected cost wll occur at the end ponts of the nterval [L,r, L,r ],. Takng partal dervatves of JTEC(Q, k, k, L, L, λ, λ, H, H, m) n () wth respect to Q, k, k, λ, λ, H, and H, respectvely, and equalzng the results to zero, we have JTEC(Q,k,k,L,L,λ,λ,H,H,m) = D Q Q [A v + A D m = (L ) + C (L )] + h b C = b D + h vc v [m ( D ) + D ] λ P P = ε = 0, (7) JTEC(Q,k,k,L,L,λ,λ,H,H,m) = h k b C b σ L h b C b ( β )σ L ( Φ(k )) λ σ L ( Φ(k )) = 0, (8) JTEC(Q,k,k,L,L,λ,λ,H,H,m) = σ λ L φ(k ) + H ε Q = 0, (9) JTEC(Q,k,k,L,L,λ,λ,H,H,m) = λ H H = 0,. (0) From Equaton (0), we obtan λ = 0 or H = 0,. If H 0 and λ = 0, one has Φ(k ) = β,. Snce Φ(k β ) cannot be negatve, then H = 0, λ 0,. Hence, the servce level constrants are actve when the optmal soluton s obtaned. Furthermore, solvng Equatons (8) and (9), we have the followng results: λ = h bc b h bc b ( β ) D( Φ(k )) D, φ(k ) = ε D Q,. () Dσ L Thus, subttutng Equatons () and () to Equaton (7), we get Q = D[ A v m + = (A (L )+C (L )) ] h v C v [m( D P ) +D P ] D [h b C b D ( h b C b D( Φ(k )) h b C b ( β ) )ε D ] = D { } The followng proposton shows that pont (Q, k, k ) s the local optmal soluton, whch mnmzes the jont expected total annual cost JTEC(Q, k, k, L, L, m) and satsfes the servce level constrants Dσ L φ(k ) ε D Q,.. () Proposton. For gven m and L [L,r, L,r ],, the pont (Q, k, k ) satsfes the second order suffcent condton (SOCS) for the mnmzng problem wth two constrants. Proof. For gven m and L [L,r, L,r ],, we frst obtan the Bordered Hessan matrx H B s as follows:

8 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 H B = JTEC(.) Q λ JTEC(.) k λ JTEC(.) JTEC(.) Q λ JTEC(.) k λ JTEC(.) [ k λ where JTEC( ) = JTEC(Q, k, k, L, L, λ, λ, m), JTEC( ) = D Q Q (A v JTEC( ) = JTEC( ) Q λ λ Q JTEC( ) = JTEC( ) k λ λ k JTEC( ) k JTEC( ) = JTEC( ) k Q Q k m + A (L ) + C (L ) k λ = ), = ε D,, = Dσ L ( Φ(k )),, JTEC(.) λ Q JTEC(.) λ Q JTEC(.) Q JTEC(.) k Q JTEC(.) k Q = h b C b ( β )σ L Φ(k ) + λ Dσ L Φ(k ),, = JTEC( ) λ k = JTEC( ) Q k = JTEC( ) k λ = JTEC( ) k Q = JTEC( ) λ λ = JTEC( ) k k = JTEC( ) λ λ JTEC(.) λ k JTEC(.) λ k JTEC(.) Q k JTEC(.) k JTEC(.) k k = JTEC( ) k k = JTEC( ) λ JTEC(.) λ k JTEC(.) λ k JTEC(.) Q k JTEC(.) k k δ JTEC(.) k ] = JTEC( ) k λ = JTEC( ) λ = 0.. = JTEC( ) λ k For gven value of m and L [L,r, L,r ],, snce there are three varables (Q, k, k ) and two constrants, therefore we need to check the sgn of the last one prncpal mnor determnant of H B at pont (Q, k, k ). We have H 55 = D5 σ L ( ϕ(k )) σ L ( ϕ(k )) [ A v m + A (L ) + C (L ) Q = ] + ε D Dσ L ( Φ(k )) h b C b ( β )σ L Φ(k ) > 0. ( Φ(k )) = Snce the sgn of H 55 s postve, hence, t can be concluded that (Q, k, k ) satsfes the SOCS for the mnmzng problem wth two constrants. Next, we take the second partal dervatve of JTEC(Q, k, k, L, L, m) wth respect to m. We obtan JTEC(.) = D A v > 0. Hence, JTEC(Q, k m Q m, k, L, L, m) s convex n m. Snce, the value of m must be a postve nteger, then the optmal number of shpments to each buyer, m, s reduced to fnd a local mnmum and can be determned by satsfyng the followng condton. JTEC(Q, k, k, L, L, m ) JTEC(Q, k, k, L, L, m ) JTEC(Q, k, k, L, L, m + ). The followng algorthm s developed to descrbe the soluton procedure for the ntegrated snglevendor two-buyer model to obtan the optmal soluton of the buyer s order quantty, safety factor and lead tme and number of shpments to each buyer n each producton cycle of the vendor. Algorthm Step 0: Set m =. Step : For each buyer =,, perform Step. Step : For each L,r, r = 0,,, n, perform () to (v). () Start wth k 0,r = 0, and obtan φ(k 0,r ) = and Φ(k 0,r ) = 0.5. () Substtutng Φ(k 0,r ) nto Equaton () to obtan Q 0 r. 7

9 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9// () Substtutng Q r nto Equaton () to evaluate φ(k,r ). Check the value of φ(k,r ) n the standard normal table n Slver and Peterson [4] to fnd k,r, φ(k,r ), and Φ(k,r ). (v) Repeat () and () untl no change occurs n the values of Q r and k,r. Then denote the value of Q r and k,r by Q r and k,r. (v) Compute JTEC(Q r, k,r, k,r, L,r, L,r, m) usng Equaton (5). Step : Set C(Q m, k,m, k,m, L,m, L,m, m) = mn {JTEC(Q r, k,r, k,r, L,r, L,r, m)}, r=0,,,n then (Q, k, k, L, L, m) s the optmal soluton for m value. Step 4: Set m = m + and repeat Step () to () to get JTEC(Q m, k,m, k,m, L,m, L,m, m). Step 5: If JTEC(Q m, k,m, k,m, L,m, L,m, m) JTEC(Q m, k,m, k,m, L,m, L,m, m ) then go to step 4, otherwse go to step. Step : Set JTEC(Q, k, k, L, L, m ) = JTEC(Q m, k,m, k,m, L,m, L,m, m ), then (Q, k, k, L, L, m ) s the optmal soluton. Step 7: Determne the optmal order quantty of each buyer usng the relatonshp Q = D Q. D 5. A numercal example In ths secton, we present an nstance of a sngle-vendor two-buyer system to llustrate the soluton procedure developed n Secton 4. Consder a system consstng of two buyers and a vendor wth the followng parameter values of the vendor: P = 5000 unts per year, A v = $400 per setup, C v = $5 per unt, and h v = 0.. The parameter values for the buyers are gven n Table, whereas the lead tme of each buyer has three components wth the data shown n Table. Table. Data of the buyers Buyer D (unts per year) A b ($) C b ($) h b σ (unts per weak) 0 50 ε (%) 9 97 Table. The buyers lead tme data Buyer Lead tme component r Normal duraton b,r (days) Mnmum duraton a,r (days) Unt crashng cost c,r ($/days) We solve the case when τ = 0.5. By applyng the algorthm procedure, the results can be seen n Table. From the table, t can be observed that JTEC has the mnmum value of $57. for the optmal number of shpments to each buyer as m =, the optmal shpment lot sze as Q = unts, the lead tme of the buyers as L =, L = 4 weeks, and the safety factor of the buyers as k = 0.99, k = Usng the relatonshp mentoned n Step 7 of the algorthm, the optmal order quantty for the buyers can be calculated as Q =.5, Q = unts. Table. Results of the soluton procedure for τ = 0.5 (lead tme n weeks) r L,r Q k k m JTEC( ) 8

10 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 L,0 = 7 L, = 5 L, = 4 L, = * Mnmum jont expected total annual cost * Further, Table 4 shows the comparson of non-ntegrated model, partal ntegrated model, and ntegrated model. On the partal-ntegrated model, one of the two buyers coordnate wth the vendor to make decsons cooperatvely, whereas another buyer has ndependent decson n determnng the optmal soluton. From Table 4, we observe that the jont expected total cost n the ntegrated model generally s less than that n the partal-ntegrated model and the non-ntegrated model. Moreover, t s noted that when the number of buyer on ntegrated system ncreases, the total reorder pont of buyers decreases. Therefore, the ntegrated system can also help an organzaton reduce ts level of safety stock. Next, a far and acceptable proft sharng mechansm s the key to the success of an ntegrated model. The jont expected total annual cost should be allocated to the vendor and to each buyer as follows: Cost to the buyer : ω b JTEC(Q, k, k, L, L, m ), Cost to the buyer : ω b JTEC(Q, k, k, L, L, m ), Cost to the vendor : ( ω b ω b )JTEC(Q, k, k, L, L, m ), where ω b = ω b = TEC b (Q,k,L ), TEC b (Q,k,L )+TEC b (Q,k,L )+TEC v (Q,m ) TEC b (Q,k,L ). TEC b (Q,k,L )+TEC b (Q,k,L )+TEC v (Q,m ) For example, for τ = 0,5, we obtan ω b = 0, and ω b = 0,05. Then the allocaton of the total annual cost to the buyer, buyer, and the vendor are $9.9, $598.55, and $947.5, respectvely. Table 4. Cost comparson for non-ntegrated model, partal-ntegrated model, and ntegrated model for τ = 0.5 Non-ntegrated model Partal-ntegrated model Integrated model Order quantty of buyer Order quantty of buyer Lead tme of buyer Lead tme of buyer Reorder pont of buyer Reorder pont of buyer Vendor s producton quantty Total annual cost of buyer Allocated annual cost of buyer

11 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 Total annual cost of buyer Allocated annual cost of buyer Total annual cost of vendor Allocated annual cost of vendor Jon cost Conclusons In ths paper, an ntegrated producton-nventory model for sngle-vendor two-buyer problem has been studed wth partal backorder and controllable lead tme under ndependent normally dstrbuted demand on the buyers. Addtonally, a servce level constrant correspondng to each buyer s ncluded n the model to lmt the shortages. Mnmzng the jont expected total annual cost functon and satsfyng the servce level constrant on each buyer, a Lagrangan multpler technque and an algorthm procedure are proposed to determne the optmal order quantty, safety factor, lead tme of the buyers and number of lots delvered from the vendor to the buyers n a producton cycle. Numercal results show that the optmal soluton obtaned satsfes the servce level constrant on all the buyers. Moreover, we show that by vewng the vendor and the two buyers as a system rather than as separate ndvduals, total system cost can be reduced sgnfcantly. Regardng some future research, we can extend the present model by consderng the mperfect producton process on the vendor and the nspecton errors on the buyers. In order to show the uncertanty and vagueness, the lead tme demand can be consdered as a fuzzy random varable. References [] Abdul-Jalbar, B., Guterrez, J. M., and Scla, J., An ntegrated nventory model for the snglevendor two-buyer problem, Internatonal Journal of Producton Economcs, 08 (007), [] An, H. and Lee, H., An ntegrated sngle vendor-sngle buyer producton nventory system ncorporatng warehouse szng decsons, Journal of the Korean Insttute of Industral Engneers, 40 (04), [] Benerjee, A., A jont economc-lot-sze model for purchaser and vendor, Decson Scences, 7 (98), 9-. [4] Gr, B. C. and Bardhan, S., A vendor-buyer JELS model wth stock-dependent demand and consgned nventory under buyer s space constrant, Operatonal Research, 5 (05), [5] Goyal, S. K., An ntegrated nventory model for a sngle suppler-sngle customer problem, Internatonal Journal of Producton Research, 5 (97), 07-. [] Goyal, S. K. and Nebebe, F., Determnaton of economc producton-shpment polcy for a snglevendor-sngle-buyer system, European Journal of Operatonal Research, (), [7] Jha, J. K. and Shanker, K., Sngle-vendor mult-buyer ntegrated producton-nventory model wth controllable lead tme and servce level constrants, Appled Mathematcal Modellng, 7 (0), [8] Kurdh, N. A., Lestar, S. M. P., and Susant, Y., A fuzzy collaboratve supply chan nventory model wth controllable setup cost and servce level constrant for mperfect tems, Internatonal Journal of Appled Management Scence, 7 (05), 9-. [9] Lu, L., A one-vendor mult-buyer ntegrated nventory model, European Journal of Operatonal Research, 8 (995), -. [0] Ouyang, L. Y., Wu, K. S., and Ho, C. H., Integrated vendor-buyer cooperatve models wth stochastc demand n controllable lead tme, Internatonal Journal of Producton Economcs, 9 (004), 55-. [] Pan, J. C. H. and Yang, J. S., A study of an ntegrated nventory wth controllable lead tme, Internatonal Journal of Producton Research, 40 (00), -7. [] Rad, M. A., Khoshalhan, F., and Setak, M., Supply chan sngle vendor-sngle buyer nventory model wth prce-dependent demand, Journal of Industral Engneerng and Management 7 (04),

12 ICMAME 05 Journal of Physcs: Conference Seres 9 (0) 0008 do:0.088/74-59/9//0008 [] Rad, R. H., Razm, J., Sangar, M. S., and Ebrahm, Z. F., Optmzng an ntegrated vendormanaged nventory system for a sngle-vendor two-buyer supply chan wth determnng weghtng factor for vendor s orderng cost, Internatonal Journal of Producton Economcs, 5 (04), [4] Slver, E. A. and Peterson, R., Decson Systems for Inventory Management and Producton Plannng, John Wley, New York, 985. [5] Yao, M. and Chou, C., On a replenshment coordnaton model n an ntegrated supply chan wth one vendor and multple buyers, European Journal of Operatonal Research, 59 (004),