A Two-Echelon Inventory Model for Single-Vender and Multi-Buyer System Through Common Replenishment Epochs

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1 A Two-Echelon Inventory Model for Sngle-Vender and Mult-Buyer System Through Common Replenshment Epochs Wen-Jen Chang and Chh-Hung Tsa Instructor Assocate Professor Department of Industral Engneerng and Management Ta-Hwa Insttute of Technology Hsn-Chu, Tawan, ROC Abstract To cope wth the compettveness of global market, the enterprse s decson has now been based on seekng a global optmum of supply chan. The nventory polcy of enterprse can be made by the all members of supply chan n order to reach the wn-wn for both venders and buyers. Ths study consders the problem of one-vender supplyng a product to mult-buyer (customers). The objectve s to mnmze the vender s total costs of order processng and transportaton subject to the maxmum costs whch buyers are prepared to ncur. In the proposed model, the vender offers a prce dscount to entce the buyers to accept the polcy of common replenshment epochs (CRE). Fnally, ths study evaluates the beneft of the proposed coordnated strategy wth a numercal study. Keywords: Supply Chan, Common Replenshment Epochs, Replenshment Polcy 1. Introducton Loosely speakng, a supply chan management s a management tool that nvolves three man factors such as procurement, producton, and dstrbuton. In the past, the enterprse experenced a stage marked by the producer-orented manufacturng envronment. In that stage, a sngle ndvdual generally lmts the enterprse decsons. Snce the responsbltes of the three factors are all ndependent, the enterprse must adopt an approprate nventory control to pursue ts normal operatons. Accordng to a statstcal data from U.S. manufacturers, the average nventory cost shares about thrty to thrty-fve percent of ther annual ncomes. The nventory s regarded as a very mportant asset at that tme. However, the nventory s no longer to be a good management strategy due to the rapd product development. Snce the new nnovatve technology has made the product lfe cycle become shorter and shorter, the excess nventory wll block the cash flow and ndeed gves an adversely effect on the enterprse. It s a common belef that an optmum nventory polcy s the only way to lower the mpact to a lowest level. Therefore, 48

2 the supply chan management has become a focus ssue n the enterprse. There are many scholars and owners of enterprses have devoted ther tme and money to mprove the supply chan management technques. As the popularty of the Internet system, the dstance between the suppler and the customer s greatly shortened. As a result, the globalzaton has changed the trend of the commercal behavor between two organzatons (Thomas and Grffn, 1996). To cope wth the compettveness of global market, the enterprse s decsons have now been based on seekng a global optmum of supply chan nstead of the tradtonal local optmum. The enterprse owner also recognzes the fact that the ntegratng and coordnatng wth all members of upstream and downstream of supply chan wll result n more effectve management, and reach the wn-wn for both venders and buyers. Therefore, ths research wll study the problem of one-vender supplyng a product to mult-buyer (customers). Through the mutual agreement between buyers and vender, the vender offers a prce dscount to entce the buyers to accept the polcy of common replenshment epochs (CRE). The objectve s to determne the most optmum orderng and replenshment epochs by mnmzng the vender s total costs of order processng and transportaton subjected to the maxmum costs whch buyers are prepared to ncur. 2. Lterature Revew Supply chan management arses n many physcal stuatons. It has become a hot ssue because of the rapd development of the Internet. The so-called supply chan provdes a vocabulary to descrbe the actvtes that may be encountered n the enterprse durng the perod between order recevng and product delvery. These actvtes nclude obtanng materals, product desgn, manufacturng and dstrbuton. If dfferent unts perform these actvtes smultaneously, the system formed by these unts s called supply chan. The man factors that supply chan plannng must consder s to conduct ntegraton. Banerjee (1986) frst proposed a two-echelon nventory model for vender and buyer. These after, there are several researchers who have explored many problems on the ntegraton of vender and buyer. Hll (1997), Vswanathan (1999), and Hll (1999) nvestgated the nventory polcy for sngle-vender and sngle-buyer respectvely. Lu (1995), Vswanathan and Pplan (2001) evolved a model, whch consdered sngle-vender and mult-buyers. In Lu s study (1995), prevous purchasng nformaton and hghest expected acceptable cost were utlzed to determne the mnmum value of the sum of the nventory cost and setup cost. Ths paper manly proposed a soluton method to seek the best soluton of nventory for the sngle-vender and sngle-buyer. In addton, Lu (1995) also developed a heurstc model to dscuss the coordnatng mechansm among sngle-vender and mult-buyer. Hll [4] constructed a producton and nventory model wth ntegraton as a whole. In dealng wth the model, he assumed that the venders produced the product at ther greatest rate. The objectve was to mnmze the total nventory costs per unt tme. Ths model was expected to provde the venders wth the optmum producton and delvery schedulng. Vswanathan and Pplan (2001) further proposed a model, whch allowed the venders to offer a prce dscount so as to entce the buyers to accept the common replenshment epochs. Furthermore, the problem consdered that there was only one delvery when buyers placed an order each tme. The most optmum common replenshment epochs were then obtaned by Internatonal Journal of The Computer, The Internet and Management, Vol. 10, No.3, 2002, pp

3 mnmzng the prce dscounts and the order processng cost. In real stuatons, the vender pursue batch delvery wth one orderng by buyers s very common. Ths paper wll take account ths stuaton, and wll utlze the Vswanathan and Pplan s model (2001) to nvestgate the most optmum common replenshment epochs for sngle-vender and mult-buyer under the condton of batch delvery. In ths study, the objectve s to mnmze the total costs of order processng, prce dscount, and transportaton so as to determne the most optmum replenshment epochs, prce dscount, and replenshment. 3. Mathematcal Model 3.1 Notatons A s : vender s order setup cost when vender recevng an order at each tme. A : vender s order processng cost for processng a specfc order from buyer. C : vender s delvery cost to buyer (/each tme). D : demand quantty for buyer. G 0 b : total nventory costs for buyer before reachng agreement wth the members of supply chan (ncludng the cost of order processng, nventory holdng, and replenshment). G 0 v : vender s total costs for order processng and delvery before reachng agreement wth members of supply chan. G c b : total nventory costs for buyer after reachng agreement wth members of supply chan (ncludng the cost of order processng, nventory holdng, and replenshment). G c v : vender s total costs for order processng and delvery after reachng agreement wth members of supply chan. h : unt nventory holdng cost for buyer. K : orderng cost for buyer. n : a postve nteger, where the replenshment perod for buyer s n T o. N : a postve nteger, where the order perod for buyer s N T o. Q : order quantty for buyer before reachng agreement wth members of supply chan. R : replenshment processng cost for buyer. S: vender s compensaton rate for buyers. T 0 : vender s replenshment perod before reachng agreement wth members of supply chan. T : replenshment perod for buyer before reachng agreement wth members of supply chan. T : orderng perod for buyer after reachng agreement wth members of supply chan. T : replenshment perod for buyer after reachng agreement wth members of supply chan. TC : total nventory costs for buyer. x : an postve nteger, where x = ( N / n ). Z : lowest prce dscount accepted by buyer. Z: vender s fnal proposed prce dscount, where Z = max{z }. 3.2 Assumptons Ths study wll develop a two-echelon nventory model for sngle product, sngle-vender and mult-buyer for supply chan management problems. The assumptons are lsted as followng: 1. The market demand s known and fxed for buyers. 2. Buyer doesn t allow any materal shortage. 50

4 3. Vender s replenshment epochs must satsfy the condton: T? X,?? X = 1, a, where a s a postve? ? nteger. 4. The vender s replenshment perod (T 0 ) remans fxed. Buyers orderng must be at the perod of vender s replenshment. 5. There must be batch delvery for each buyer orderng, and the condton should be met: n T 0 = (N / x )T Mathematcal Modelng The total nventory costs for buyer to be consdered n ths study nclude the cost of orderng, nventory holdng, and the processng of replenshment. Hence, the total nventory costs (TC ) for buyer are gven by: D Q D TC = K + h + R Q 2 Q (1) Mnmzng the total costs and lettng 1 H = Dh, the most optmum economcal 2 orderng quantty (Q * ) and replenshment epoch (T ) for buyer can be expressed by Eq. (2) and (3). Q T D = 2 ( K + R ) * (2) h Q ( K + R ) = = (3) D H Substtutng Eq. (2) and (3) nto (1) and notng that each party can make ther own decsons; the total nventory costs (G 0 b ) for buyer can be smplfed as: ( K + R ) 0 ( K R ) b G 0 = + H T = 2 H + T (4) When vender accepts buyers orderng, the costs of the order processng and transportaton for delvery would be ncurred. These costs for buyer are (A + A s + C ). In order to satsfy the demand for buyers, the total costs (G 0 v ) for vender ncludng the cost of order processng and delvery are represented by Eq. (5): v n? = 1 ( A + A + C ) s G0 = (5) T However, from prevous orderng nformaton the vender could know the buyer s acceptable least prce dscount (Z ) under both agreement condton. Based on that, vender then proposes a Common Replenshment Epochs or Perods (CRE) or (T 0 ), and determnes the fnal prce dscount (Z) and cost compensaton rate (S) that could offer to buyers. Ths would entce the buyer to accept the strategy of the fxed replenshment epochs. At ths study, vender wll adopt batch delvery n order to reduce the nventory holdng and ncrease the flexblty for changng the specfcaton. Under the strategy of CRE, the buyer can pursue replenshment at the tme of suppler s delvery. The orderng (T ) and replenshment perod (T ) for buyer, and the total nventory costs (G c b ) before dscount can be expressed by Eq. (6) and (7), respectvely. G T = N T 0 T = n T 0, where N, n 1 (6) ( n T ) K D R K R 0 2 nt0 (7) b 0 c = + h + = + H nt0 + N T nt0 xnt0 Vender proposes a fxed replenshment perod based on prevous orderng nformaton before buyers and vender Internatonal Journal of The Computer, The Internet and Management, Vol. 10, No.3, 2002, pp

5 reachng an agreement. The lowest acceptable prce dscount (Z ) can be estmated by Eq. (8): Z 1? K R? + H nt0 + D? xnt0 nt0 2 1 (8) ( S) H ( K + R ) = Let the fnal prce dscount Z = Max{ Z }. If the total prce dscount s hgher than the ncrease amount of total nventory costs, buyers can then accept the CRE strategy proposed by the vender. Therefore, the total prce dscount ( D Z ) ganed by buyer must satsfy Eq. (9): K ( S) H ( K R ) D Z? + H nt xnt0 nt0 (9) Moreover, the orderng and replenshment perod based on assumpton 4 and 5 must be a multple of a postve nteger. Snce the total prce of the buyer possesses the characterstc of convex functon, a varaton method can be used to solve Eq. (7). Hence the correspondng N, n to have a lowest total prce must satsfy Eq. (10): K + x R n ( n + 1)?? n ( 1 2 n ) (10) x H T 0 Therefore, the total costs of order processng and delvery (G v c ) and the total nventory costs (G b c ) after prce dscount for buyer can be expressed by Eq. (11) and (12), respectvely: v As? A C??? Gc = + D Z + + (11) T0? N T0 nt0 b K R Gc = + H nt0 + DZ (12) xnt0 nt0 Ths research ntends to pursue mnmzaton of vender s total cost of orderng processng and delvery so as to obtan the most optmum replenshment R perod ( T 0 ), prce dscount (Z), and the buyer s orderng and replenshment perod. All equatons nvolved n the above dervaton are summarzed by Eq. (13):?? v A? Mn.???? s A C G = + c D Z + + T0? N T0 nt0 (13) s.t. N nteger. K,? n 1 ( S) H ( K R ) D Z? + H nt xnt0 nt0 n K + x R ( n )?? n ( n 1) R x H T0, where both are postve?? T 0? X, X = 1, a, where a s a? ? postve nteger. 4. Model Verfcaton and Comparson Ths secton wll justfy the applcablty of proposed method and make a comparson for each result. In ths model, a two-echelon nventory model for a sngle-vender and fve buyers wll be used. The annual demand quantty (D ), cost of order processng (K ), and replenshment processng (R ) are shown n the Table 1. The delvery cost (C ) for vender s assumed to be 1,000 dollars and the unt nventory holdng cost (h ) for buyers s fxed. When a cost compensaton rate (S =10) offered by vender, the vender must absorb the amount of the ncreased nventory cost. In other word, vender offers an approprate prce dscounts (Z) to buyers. 52

6 Table 1: Numercal data for Relevant Cost Buyer No. Annual Demand D Order Processng Cost K Replensh-m ent Processng Cost R 1 200, , , , , , Snce the vender s order processng cost s dfferent from the cost of buyers unt nventory holdng cost. The vender processng cost for A s are 0, 100, 500 and 1000, A are 100, 500 and 1000, and buyers unt nventory holdng cost for h are 1, 2 and 3, wll be utlzed n ths study. A comparson wth Vswanathan and Pplan s model (2001) s shown n Table 2. A numercal example for A s = 100, A = 500 and h = 1, the resultng total costs for vender and system wth makng decson ndependently are 247,716 and 362,309 respectvely. If the buyers and vender reach an agreement, and buyers are wllng to accept vender s proposed replenshment perod, the resultng optmum replenshment perod wll be fve weeks. The total cost savngs for vender and system are 151,303 and 175,132 respectvely. The buyers wll have 14.1 cost savngs. A numercal data s shown n Table 2. When vender takes buyers orderng, and decde to batch delvery upon both sde agreements, the favorable prce dscounts offered by vender wll entce the buyers to accept fxed replenshment perod. The vender s replenshment perod (T 0 ) wll be decreased wth the ncreasng of unt nventory holdng cost. Also from Table 2, makng decson based on ths model can have a tremendous savngs for both members. But n Vswanathan and Pplan s model (2001), only the buyers can have benefts. In other condtons, the vender should take more costs. In the case of A s = 0, A = 100, and h = 1, the buyer would have 14.1 cost savngs, but the cost of vender would ncrease to Fgure 1 and 2 show the correlaton among vender order processng cost, vender savngs, system savngs and buyers unt nventory holdng cost (h = 1) wth the condton of vender delvery cost, buyers unt nventory holdng cost, and replenshment processng cost remaned constant. It s shown by the fgure when A s s kept constant and A s ncreased, ths wll result n more cost savngs for vender and system. The cost savngs for vender and system wll gradually be ncreased f A s kept at a specfc value and A s s ncreased. Fgure 3 and 4 show the correlaton among vender order processng cost, vender savngs percentage, system savngs percentage and buyers unt nventory holdng cost (h = 1) wth the condton of vender delvery cost, buyer unt nventory holdng cost, and replenshment processng cost remaned constant. From the fgure, t s obvous that ths model wll ncrease the cost savngs percentage for vender and system when A s s fxed and A s ncreased. Internatonal Journal of The Computer, The Internet and Management, Vol. 10, No.3, 2002, pp

7 Table 2: Detaled results of numercal study A s A Unt holdng cost, h Dscount Z () CRE T 0 (weeks) System wthout coordnaton Savngs wth CRE strategy Vender cost System cost Buyers savngs Vender savngs System savngs Buyers Vender System savngs savngs savngs Vswanathan and Pplan s model Buyers savngs Vender System savngs savngs , ,898 16,695 83, , , ,827 25, , , , ,238 25, , , , ,380 28,742 90, , , ,309 23, , , , ,720 25, , , , ,309 21, , , , ,238 25, , , , ,650 25, , , , ,720 21, , , , ,650 25, , , , ,061 25, , , , ,907 31, , , , ,487 31, , , , ,963 31, , , , ,802 31, , , , ,383 31, , , , ,859 31, , , , ,383 31, , ,

8 , ,963 31, , , , ,440 38, , , Internatonal Journal of The Computer, The Internet and Management, Vol. 10, No.3, 2002, pp

9 Table 2: Detaled results of numercal study (contnued) A s A Unt holdng cost, h Dscount Z () CRE T 0 (weeks) System wthout coordnaton Vender cost System cost Buyers savngs Savngs wth CRE strategy Vender savngs System savngs Buyers savngs Vender savngs System savngs Vswanathan and Pplan s model Buyers savngs Vender System savngs savngs , ,859 31, , , , ,440 31, , , , ,916 38, , , , ,458 30, , , , ,722 45, , , , ,802 45, , , , ,274 30, , , , ,538 45, , , , ,618 45, , , , ,538 45, , , , ,802 45, , , , ,882 45, , , , ,618 45, , , , ,882 45, , , ,481 1,002,963 45, , ,

10 Vender savngs (, h = 1) (As = 0) (As = 500) (As = 1000) (As = 2000) (As = 5000) Vender's order processng cost (A ) Fgure 1: Vender s cost savngs System savngs (, h = 1) (As = 0) (As = 500) (As = 1000) (As = 2000) (As = 5000) Vender's order processng cost (A ) Fgure 2: System Cost Savngs Internatonal Journal of The Computer, The Internet and Management, Vol. 10, No.3, 2002, pp

11 Vender savngs (, h = 1) (As = 0) (As = 500) (As = 1000) (As = 2000) (As = 5000) Vender's order processng cost (A ) Fgure 3: The percentage of cost savngs for vender System Savngs (, h =0.1) (As = 0) (As = 500) (As = 1,000) (As = 2,000) (As = 5,000) Vender's Order Processng Cost (A ) Fgure 4: The percentage of cost savngs for System 60

12 5. Conclusons Ths research explores a two-echelon nventory model for a sngle-vender and mult-buyer. By way of mutual agreement, vender can propose the most optmum fxed replenshment perod, a reasonable prce dscount, and cost compensaton rate n order to entce the buyers to accept the proposed strategy. From the model and assocated data, we obtan the followng conclusons: 1. Under the CRE strategy, the vender s replenshment perod (T 0 ) wll decrease wth ncreasng buyers unt nventory holdng cost (h ) when both sdes reach agreement. 2. There would be a sgnfcant cost savng for each member f the upstream and downstream members of supply chan can communcate and ntegrate well. 3. If the vender order processng cost ncreases, the cost savngs for both sdes wll also ncrease. 4. The man advantage of batch delvery to the buyers s cost savngs. In addton, t s possble to have a specfcaton modfcaton n order to respond customer requrement n a short tme. 5. Ths model can save a sgnfcant cost for vender and buyers at any tme, but Vswanathan and Pplan s model (2001) s only benefcal to buyers. There are only a few specfc cases that are benefcal to both buyers and vender. References 1. Banerjee, A., 1986, A jont economc lot sze model for purchaser and vender, Decson Scences 17, Hll, R.M., 1997, The sngle-vender sngle-buyer ntegrated producton-nventory model wth a generalzed polcy, European Journal of Operatonal Research 97, Hll, R.M., 1999, The optmal producton and shpment polcy for the sngle-vender sngle-buyer ntegrated producton-nventory problem, Internatonal Journal of Producton Research 37, Lu, L., 1995, A one-vender mult-buyer ntegrated nventory model, European Journal of Operatonal Research 81, Thomas, D.J. and Grffn, P.M., 1996, Coordnated supply chan management, European Journal of Operatonal Research 94, Vswanathan, S., 1998, Optmal strategy for the ntegrated vender-buyer nventory model, European Journal of Operatonal Research 105, Vswanathan, S., and Pplan, R., 2001, Coordnatng supply chan nventores through common replenshment epochs, European Journal of Operatonal Research 129, Internatonal Journal of The Computer, The Internet and Management, Vol. 10, No.3, 2002, pp