Incorporating Transportation Cost into Joint Economic Lot Size For Single Vendor-Buyer

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1 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY Incorporating Transportation Cost into Joint Econoic Lot Size For Single Vendor-Buyer Raden Achad Chairdino Leuveano a,c,fairul Azni Bin Jafar a, Chairul Saleh b, Mohd Razali Bin Muhaad a, Mohd Niza Abd Rahan c a Faculty of Manufacturing Engineering, Universiti Teknikal Malaysia Melaka, Melaka, Malaysia Eail: herdi.leuveano05@gail.co; {fairul, ohdrazali }@ute.edu.y b Faculty of Industrial Technology, Islaic University of Indonesia, Yogyakarta, Indonesia Eail: pfchs@yahoo.co c Faculty of Engineering & Built Environent, National University of Malaysia, Selangor, Malaysia Eail: niza@eng.uk.y Abstract Considerable attention to coordinate the syste between buyer and vendor has becoe an interesting issue to efficiently increase the perforance of supply chain activities. Joint econoic lot size odel (JELS) has been introduced by any researchers as the spirit of coordinating the flow of aterial fro the vendor to its downstrea. As an inventory replenishent technique, JELS odel is centered on reducing joint total cost of vendor and buyer by siultaneously deciding optial delivery lot size, nuber of deliveries, and batch production lot. It is appropriate to take into account transportation costs as the function shipping weight and distance since delivery lot size has interrelated with shipping weight. Hence, this study constitutes an effort to develop the odel of JELS by incorporating transportation cost. The solution procedure of the odel is developed for solving two probles which are incapacitated and capacitated odel. In addition, nuerical exaples were provided to illustrate the feasibility of the solution procedure in deriving optial solution. The result presents central decision aking which is useful for coordination and collaboration between vendor and buyer. Index Ters Supply chain, Joint econoic lot size, Transportation. I. INTRODUCTION Supply chain anageent has been succeeded to divert the copanies' attention, not to focus on internal organization only, but also should consider coordinating with their vendor. It is due to global arket copetitions which ephasize the copanies to provide low cost strategy while still increasing custoer service level []. The ain eleent to cost reduction is an integration of supply chain stages such as procureent, production, and distribution that should be straightforward deterined optial solutions [3]. In the practice, ost of procureent scenarios prove that vendor and buyer are treated independently for giving policy to order and production [4, 5, 6, 9]. It is Manuscript received 3 May 03 ; revised 0 june 03; accepted October 0, 03. Corresponding authors: Telp: E-ail addreses: herdi.leuveano05@gail.co iportant to acknowledge that if one party leads to an optial solution to the decision, the other ay get disadvantage. Unfortunately, if both parties still regulate separate decision, costly possibility will be occurred in inventory and distribution. Therefore, it is good to have a cooperative strategy in coordinating the flow of aterial fro vendor to buyer. One of the ost well-known inventory techniques that represent coordination and collaboration between vendor and buyer is joint econoic lot size (JELS) odel. It was started by Goyal []. The objective of this odel is to reduce the joint total cost between vendor and buyer. Rather than independent policy, joint policy resulting good agreeent for inventory replenishent and considerable saving can be achieved. As planning tools, JELS odels provide useful decisions in deterining optial policies which can be econoically benefit for all parties involved in the supply chain. The iportant decision is usually pointing at delivery lot size, nuber of deliveries, and batch production lot. Paying attention to delivery lot size, would be better to consider transportation as well. It is due to ore than 50% of total annual logistic costs can be pointed toward transportation that known as freight cost [7]. Freight costs here are the function of shipping weight and distance [8]. Therefore, delivery lot size has a direct ipact to the total freight cost of each shipent. In order to increase the degree of practical relevance as well, incorporating transportation cost into JELS odel becoe very attracting to be developed and investigated, especially its effect in the deterination of the optial solution. In addition, it is iportant to note that the objective of the odel is to reduce joint total cost between vendor and buyer. As the reainder of this paper, the next section provides the literature related on joint econoic lot size and transportation as a support for a new insight. Section III forulates the developent of atheatical odeling of JELS by incorporating transportation cost and its solution procedures. Section IV presents nuerical exaples to illustrate and test the feasibility of the solution procedure of the odel. The last section doi:0.4304/jsw

2 34 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 04 suarizes the results and recoendation for future research are given. II. LITERATURE REVIEW Traditionally, inventory replenishent decisions are treated independently fro the viewpoints of vendor and buyer based on econoic production quantity (EPQ) and econoic order quantity (EOQ) respectively [4, 5, 6, 9]. In instance procureent cases, the optial EOQ solution for the buyer was not acceptable to the vendor and otherwise as well. Instead of independent decision, the research studies about collaboration and coordination between vendor and buyer have been presented by useful inventory technique well known as joint econoic lot size odel. The first odel was initiated by Goyal []. The objective is to iniize joint total cost for both vendor and buyer proble by deterining optial lot size. Banerjee [9] generalized Goyal s odel by assuing shipent policy based on the lot for lot basis under deterinistic conditions. It eans that deliveries can be carried out after vendor copleting the production period. Unlike lot for lot policy, Banerjee and Ki [0] proposed better shipent policy by splitting the batch production lot size into sub-lot size so that the ultiple-shipent can be executed during production period. It will be ore econoical for vendor by incurring a single setup cost for ultiple shipent rather than lot for lot policy. This ultiple-shipent policy is also ore preferable in a anufacturing environent. Thereby, beside deterination of lot sizes, the optial nuber of shipents becoes iportant decisions as well to see how uch batch production lot will be executed. Alost all the research regarding joint econoic lot size has been extensively reviewed by [0]. As entioned earlier that any deterination of delivery lot size should then consider freight rate costs which are usually coputed based on shipping weight and distance. Bauol and Vinod [] research work was the first introduced inventory theoretic odels as the integration of transportation and inventory costs. Whereas, the first incorporated freight rates function into lot sizing decision was Langley []. Actual shipping decision fall into three categories: ) delivery which resulted in true truckload (TL) shipping quantities, ) delivery which supposed to be over declare as TL, and 3) delivery are possibly shipped at less-than-truckload (LTL) [7]. Swenseth and Godfrey [3] have proposed freight rate function that eulates reality and better to represent actual freight rates such as freight rate functions such as proportional, adjusted inverse, constant, exponential, and inverse functions. Swenseth and Godfrey [7] hence studied the effect of joining freight function into an EOQ odel for inventory replenishent decision. The study analyzed the cobination of EOQ with inverse and adjusted inverse function. The result shows that inverse function is able to eulate freight rate for true TL and adjusted inverse function eulates LTL freight rates. Mendoza and Ventura [4] also have added quantity discount for EOQ odel and transportation in which considering TL and LTL shipent. Mendoza and Ventura [5] studied estiation of freight rates in inventory replenishent decisions for vendor selection decision. Up to this explanation, the research concerned with incorporating transportation costs into inventory replenishent decision only focused on single stage odel. Now, consider incorporating transportation costs into JELS odel. Nie, Xu, and Zhan [6] developed Ki and Ha [4] s odel by joining freight costs function of Swenseth and Godfrey [7] either inverse or adjusted inverse into JELS odel. Chen and Sarker [7] considered a coplex ulti-stage of supply chain include ulti- vendor and a single buyer. The study assued that the ilk run syste was used to consolidate all the ites fro ulti-vendor using a single truck under just-in-tie (JIT) environents. Unlike previous freight cost function in [7], Chen and Sarker [7] revised the freight costs becoes the function of shipping weights and distances. It is due to the prior freight rates proposed by Swenseth and Godfrey [7] only focused upon the shipping weight, whereas distance assued to be fixed. In the practical exaple, JIT syste has characteristic on regulating frequent deliveries in sall lot size, shorter lead tie, and close vendor ties [8]. Thereby, it is better to take advantage on shipping distance coputed in the freight rate as well since vendor should be closer to the ain buyer. Since it is ore appropriate to take into account of transportation as the function of shipping weight, and distance into inventory replenishent decision, this research proposes a developent of JELS odel. The objective is to iniize joint total cost between vendor and buyer by deciding optial delivery lot size, nuber of deliveries, and batch production. III. PROBLEM DEFINITIONS Suppose that a single vendor and single buyer have regulated long ter JIT relationship. Taking the exaple in JIT practice, frequent delivery in saller lot size becoes useful strategy to iprove the syste s perforance of internal copany where lower inventory cost constitutes its purpose. Nonetheless, the first proble is the ibalance lot sizing decision occurs between buyer and vendor. Since buyer leads to an optial solution, the vendor should produce on required basis and certainly incurs high setup costs for each lot produced. On the other hand, high delivery cost will be incurred by the buyer because of conducting frequent delivery while reducing inventory cost. Coordination and collaboration between vendor and buyer should exist to overcoe this proble. This study then proposed JELS odel to reduce joint total cost of both parties by siultaneously optiizing delivery lot size, nuber of shipents, and batch production lots. IV. MODEL FORMULATION In this section, the atheatical odels are elaborated to represent the coordination between buyer and vendor. This odel consists of buyer and vendor cost functions.

3 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY Firstly, it will be described independently. Furtherore, JELS is presented as the coposite of those cost functions. In addition, the solution procedures are also proposed to find the optial solutions. There are two solution procedures to solve regarding transportation probles such as incapacitated and capacitated. Incapacitated proble discusses to actual shipping weight does not exceed capacity truckload. Otherwise, capacitated proble occurs if actual shipping weight exceeds the capacity of a truckload. Hence, the following assuptions and notations are introduced as: A. Assuptions and Notations To boundary the research, these assuptions are adopted fro Ki and Ha [4] and its additional described as follows: ) Deand, production, and delivery lead tie assued to be deterinistic and constant. ) No shortages and backordered allowed. 3) No quantity discount. 4) Decision variables are delivery lot size, nuber of shipents, and batch production lot. 5) The shipent policy can be carried out during production period. 6) The perforance of the supply chain indicator is joint total cost. Notations of the odel are presented as below: D Deand rate of the buyer. P Production rate of the vendor P > D. S Setup cost per setup of the vendor. A Ordering cost per order of the buyer. H b Holding costs per unit inventory per period of the buyer. H v Holding costs per unit inventory per period of the vendor, Hv < Hb Fo Transportation costs per shipent. w Weight of a unit part (lbs). d Transportation distance (iles). α Discount factor for LTL shipents Fx The freight rate in dollar per pound for a given per ile for full truckload (FTL) product. Fy The freight rate in dollar per pound for a given per ile for the partial load. Wx Wy Q q F ( Dqwd,,, ) TC b Full truckload (FTL) shipping weight (lbs). Actual shipping weight (lbs). Production lot size of parts of vendor (unit) Q = q. Delivery lot size of buyer (unit), q = Q. Nuber of shipents. Total freight cost as the function of shipping weight and distance Total cost of the buyer. TC v JTC Total cost of the vendor. Joint total cost of the systes. B. Model Forulation Previously, it is stated that JELS odel coposites of vendor and buyer cost functions. Let firstly discusses on vendor cost function. Vendor cost function consists of setup cost, fix transportation for preparing and receiving the aterial, and inventory cost respectively. In addition, the inventory cost is derived fro Jogklekar [8]. The expression of vendor cost function is given as D D q D D TCv( q, ) = S + F0 + + H q q P P Buyer cost function includes ordering cost, holding cost, and variable transportation cost or freight rates as the function of weight and distance. The first two ters are basically obtained fro EOQ odel. The third ter, recall to JIT practice in which regulating frequent delivery in saller lot size, ephasizing the buyer to use less-than truckload (LTL) shipent. One of freight rates function that able to eulate LTL shipent is adjusted inverse function [7, 3]. Nonetheless, the adjusted inverse function in previous research constitutes the function of shipping weight, then need to redefine it by adding the shipping distance paraeter. Firstly needs to deterine the freight rate for partial load F y based on adjusted inverse function given as [9]. Wx Wy Fy = Fx + α Fx W y Where α has interval fro 0 to that indicated as a discount factor for LTL shipents to increase the freight rate per pound over a given distance as W y increase. Moreover, estiation of total freight cost per period as the function of shipping weight and distance for adjusted inverse yields v () () D F ( Dqwd,,, ) = FWd x x Ddw( ) Fx q α + α (3) Hence, the buyer cost function can be forulated as: D q TCb( q) = A + Hb + F ( D, q, w, d ) (4) q So far, both cost functions have been clearly introduced. Actually, either vendor or buyer cost functions have own optial solution that can iniize his independent total cost. It is indicated that one party will get an advantage when one of the leads an optial solution. Before oving toward JELS odel, it is better to know the vendor and the buyer inventory status that shown in Fig. which is adopted fro [4]. It can be seen that the inventory status of vendor regulates ultiple shipent that can be executed during production period. By cobining Eqs. () and (4) then JELS odel can be obtained.

4 36 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 04 D D JTC( q, ) = S + ( A + F0 ) q q q D D + Hv H + + b P P + F( D, q, w, d) (5) In order to test the convexity of joint total cost JTC q, respect to q for fixed value of, hence taking second derivatives of Eq. (6). JTC q, D S = A F α FxWxd> 0 q (9) Quantity Quantity Q Prod. Period Q/P q Cycle tie (Q/D) Non-Prod.Period (Q/D-Q/P) Vendor Inventory Status Tie Substitute Eq. (8) into Eq. (6) then obtained optial joint total cost as the function of nuber deliveries is as follows: S JTC ( ) = D A + + F0 + α FxWxd D D + + Hv + Hb P P (0) By taking first partial derivatives of Eq. (0) with respect to equal to zero, then optial nuber deliveries can be written as q/d Buyer inventory status Figure. Vendor and buyer inventory status Tie Where F ( Dqwd,,, ) could be coputed by querying actual freight rates to obtain total actual transportation cost. Meanwhile, if joint total cost is considered based on adjusted inverse function, Eq. (5) can be revised as D D JTC( q, ) = S + ( A + F0 ) q q q D D + Hv H + + b P P D + FW x x d Ddw ( ) F x q α + α Subject to ( q ) > 0, ( ) > 0, and integer value. In order to obtain an optial solution, take the first derivatives of Eq. (6) with respect q equal to zero. (, ) JTC q D S = A+ + F + α FW d 0 x x q D D + + Hv + Hb= 0 P P Then resulting optial q as, q S D A+ + F0 + α FW x xd = D D + Hv + H P P b (6) (7) (8) = D S Hv + Hb P D x x P ( A F α FW d) () The nuber of deliveries constitutes positive and integer value [6]. Therefore the value of ( ) should eet the requireents likewise D S Hv + Hb P =,forjtc( ) JTC( + ) D ( A+ F0 + α FxWxd) P () Or D S Hv + Hb P = +, for JTC ( ) JTC ( ) D ( A+ F0 + α FW x xd) P (3) In order to test the convexity of joint total cost JTC q, respect for fixed value of q, then taking second derivatives of Eq. (6). JTC q, DS = > 0 3 q (3) Refer to practical relevance; it ight be eerged two probles regarding shipent policy. The probles are incapacitated odel and capacitated odel that has already discussed above. Recall to incapacitated proble, it is condition where actual shipping weight is not larger than capacity truckload W y W x, otherwise capacitated proble occurs when actual shipping weight larger than

5 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY capacity truckload W y > W x. Incapacitated are siply proble if the buyer provided larger truck. On the other hand, the Lagrangian relaxation approach is required to solve capacitated proble that has included in the solution procedures. Hence, in order to deterine the optial solution, the following solution procedures are introduced to solve two probles of shipent condition are described as: Step 0: Deterine the nuber of deliveries using Eq. (), and set it as an integer value. Step : Set value of using Eqs.() and (3), then substituting and into Eq. (0). Choose optial nuber of deliveries get iniu JTC ( ). Step : Deterine optial order quantity q using Eq. (8) For fixed, and set it as an integer value. Step 3: With q and are optial solutions. Then find iniu joint total cost without actual freight rates using Eq. (6) Step 4: Evaluate actual shipping weight W y Step 5: Step 6: = qw. If truckload restriction W y W x is satisfied, then copute joint total cost using Eq. (5), where total transportation cost is coputed by querying actual freight rates. Otherwise, do next step if truckload constraint is not satisfied Wy > Wx. Revised nuber of deliveries. Δ W = q w W x ΔW Δ = and = +Δ Wx Revised delivery lo size. q q = Wx and Wy = q w qw The value of q is rounded to integer Step 7: Recoputed joint total cost JTC ( q, ) using Eq. (5) where transportation cost is also coputed using querying actual freight rates. V. NUMERICAL EXAMPLE In order to illustrate the effectiveness of the solution procedure, a nuerical exaple is presented. The nuerical exaple here includes actual freight rate schedule and paraeter data. The actual freight rate schedule is adopted fro Swenseth and Godfrey [7]. Since the previous freight rates are only the function of shipping weight and because of under study consider distance paraeter, then it needs to redefine by considering the shipping distance. Table I presents the new freight rate schedule by considering shipping weight and distance. The essence of the prior data still exists such as a constant charge per pound and constant charge per shipent that signed by. The data are redefined fro the freight rate per pound to freight rate per pound per ile. For instance, assued that distance can be delivered up to 600 iles and on the ranges 4,750-9,999 lb considered as a constant charge per shipent. So that freight rate per pound per ile is obtained by dividing freight rate per shipent with the highest break point and distance. Unlike the ranges of 0,000-8,000 lb are considered as a constant charge per pound. To change it into in freight rate per pound per ile can be coputed by dividing freight rate per pound with distance. In addition, Swenseth and Godfrey [7] entioned that soe weights are exist and when ultiplied by its corresponding freight rate will provide the sae total cost as that for the next weight break. These concepts provide a choice to over-declared as TL or well-known by indifference points that used by shipper to obtain a lower total transportation cost. This is accoplished by artificially inflating the weight to a higher weight breakpoint resulting in a lower total cost [7]. TABLE I. FREIGHT RATE SCHEDULE AS THE FUNCTION OF SHIPPING WEIGHT AND DISTANCE Weight break F x /pound F x /pound/ile -7 lb $40 $ lb $0.76/lb $ lb $74 $ lb $0.48/lb $ lb $38 $ ,000-,855 lb $0.38/lb $ ,999 lb $56 $ ,000-4,749 lb $0.8/lb $ ,750-9,999 lb $608 $ ,000-8,56 lb $0.0608/lb $ ,57-46,000 lb $0 $ Paraeter TABLE II PARAMETER DATA Value D 0,000 units P 40,000 units A $30/order S $3,600/setup H b $45/unit/year H v $38/unit/year F 0 $50/unit/year α 0.46 w lb d 600 iles F x $ /lb/iles W x 46,000 lb

6 38 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 04 In order to calculate total actual freight rate by considering shipping weight and distance for constant charge per pound, actual shipping weight is then just ultiplied by freight rate per pound per ile and distance. Conversely, to calculate total actual freight cost for constant charge per shipent, then the highest weight breakpoints is ultiplied by freight rate per pound per ile and distance. Furtherore, Table II also presents inforation data of vendor and buyer. The data were obtained fro Swenseth and Godfrey [7], Ki and Ha [4], and Nie, Xu, and Zhan [6]. Let consider testing the proposed odel using data provided. By taking the first exaple of incapacitated proble and applying the solution procedure, so obtained the result obtained is as follows: Step 0: Nuber of deliveries is 4 Step : JTC ( = 4) =$55,69.44 JTC ( = 5) =$55,87.8. JTC ( = 5) - ( 4) JTC = =$ Step : Optial delivery lot size q =397 units. Step 3: Miniu joint total cost without actual freight JTC q, = $60, rates Step 4: Actual shipping decision W y W y W x = qw= 8,734 lb,. Miniu joint total cost by considering actual freight rates (, ) JTC q = $67,790. It is feasible solution and procedure stops. Elaborating the result above shows that, optial delivery lot size q = 397 units, the nuber of deliveries = 4, batch production lot Q =, 588 units. Actual shipping weights of 397 units were 8,734 lbs and fall into a corresponding freight rate of $ which yielding variable transportation cost per shipent $ ,999 lb 600 iles= $608. This actual shipping quantity can be over-declared as truckload (0,000 lb). Shipping weight of 8,734 lbs resulting the sae freight rate as the next weight break-point. The annual nuber of shipents was 0,000 units/ 397units= 5.88 and total actual variable transportation cost 5.88 $608= $5, Based on optial solutions, obtain iniu join total cost without considering actual freight rates is $60,33.08 while the iniu joint cost by considering actual freight rates is $67,790. A coparison between the joint total cost with and without actual freight rates are deviated around %. Now try to solve the second proble of shipping condition that is capacitated proble. Having restriction of capacity truckload ight be resulting actual shipping weight exceed the capacity of a truckload. To overcoe this proble, the Lagrangian relaxation ethod is then required. This approach also aied to try relaxing the solution in order to adjust it to the capacity of truckload while still reducing joint total cost. Assued that truckload capacity W x is revised as 5,000 lb, freight rate F x changes to be $ and the other paraeter data reain unchanged. Hence, the result of illustrating the solution procedures to find optial solution can be seen as follows: Step 0: Nuber of deliveries is 4 JTC = =$53,0. Step : 5 Step : Step 3: JTC ( = 6) =$53,046. JTC ( = 6) - ( 5) JTC = =$5.. Optial delivery lot size q =35 units. Miniu joint total cost without actual freight JTC q, = $64, rates Step 4: Actual shipping decision W y = qw =6,930 lb, Wy > Wx. It is not feasible solution and do the next step. Step 5: The revised nuber of deliveries =7 Step 6: Revised the delivery lot size q 7 units and shipping weight W y =4,994, Wy Wx. The solution is feasible and approxiate to the optial solution. JTC q, = Step 7: Recalculate joint total cost $78, Procedures stop. It can be observed that delivery lot size decrease as the nuber of deliveries increase and the solution becoes feasible. Therefore, the Lagrangian relaxation ethod provides a good solution where truckload liitation can be solved. The joint total cost for capacitated proble increase 5.88%, copared to incapacitated proble. A sensitivity analysis is carried out for this odel to study how the delivery lot size, nuber of shipents and total cost of the syste are affected due to the change paraeters: S, H b, F x, α. The values of S, A, H b, F 0, F x, α will be varied fro 5% to 00% to test the ipact of paraeters. And the other paraeters reain unchanged. The syste perforance to be analyzed here use joint total cost without considering actual freight because it used to see the behavior of true odel before connected to actual freight rate. The result of sensitivity analysis is presented in Table III. The optial order quantity, nuber of deliveries, and joint total cost are dependent on the various cost paraeters. Also, as reinder, batch production lot has not analyzed its effect due to the change of paraeter because it is just the function of delivery lot size and nuber of deliveries. How sensitive one paraeter to the joint total cost is can be deterined by calculating Δ JTC = JTC JTC JTC 00% where JTC is the initial joint total cost. There are three categories to deterine about how sensitive the joint total cost of the change of paraeters such as slightly sensitive, oderately sensitive, and highly sensitive [9]. If the deviation ΔJTC on average less than %, between -3%,

7 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY and above 3% so it is considered as slightly sensitive, oderately sensitive, and highly sensitive respectively. In addition, the rate of changes of paraeters can be seen and observed whether the solution is directly related or inversely related to the paraeter over feasible range. TABLE III SENSITIVITY ANALYSIS Paraeter Changed Value q JTC ($) Δ JTC (%) S = A = H b = F 0 = F x = α = The rate of changing a paraeter of the optial solution can be observed by taking the partial derivatives of with respect to S, A, H b, H v, F 0, F x, ( q ) and q α by atheatical expression are,,, S A Hb q,, and,,,,, F0 Fx α S A Hb F0 Fx. All the rate of direction of change of solution can α be seen in the figure placed on Appendix B. The result of Table III can be described as follows: ) Effect of setup cost S to the solutions and joint total cost. It can be observed that the above feasible range fro 5% to 00% of setup cost S, the deviation Δ JTC of the joint total cost is highly sensitive to setup cost. Moreover, the solution (, ) q and joint total cost JTC are directly related to setup cost. It eans that the value of optial solution and joint total cost increase as setup cost increase. The rate of direction of change of solution due to paraeter q S can be seen by atheatical expression, S. S ) Effect of ordering cost A to the solution and joint total cost. It can be noted that the deviation of Δ JTC of the joint total cost is slightly sensitive and directly related to ordering cost A. Based on the rate of direction of change of solution due to paraeter A q by atheatical expression,, it can be A A inferred that delivery lot size q is directly related to paraeters A and nuber of deliveries is inversely related to the paraeter A. Inversely relationship eans that nuber of deliveries decrease as ordering cost increase. 3) Effect of holding cost H b to the solution and joint total cost. Based on the result in Table III shows that the deviation Δ JTC of the joint total cost is highly sensitive and directly related to ordering cost H b. However, it can be suarized fro the rate of direction of change of solution, due to Hb Hb the paraeter Hb that delivery lot size is inversely related to paraeters A and nuber of deliveries is directly related to the paraeter A. 4) Effect of fix transportation F 0 to the solution and joint total cost. Above feasible range fro 5% to 00% of fix transportation shows that the deviation Δ JTC of the q

8 30 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 04 joint total cost is slightly sensitive and directly related to fix transportation F 0. Also, delivery lot size is directly related to fix transportation F 0. Otherwise the nuber of deliveries is inversely related to fix transportation F 0. The rate of direction of change of paraeters can be seen by q atheatical expression,. F0 F0 5) Effect of freight rate cost F x to the solution and joint total cost. It can be observed that above a feasible range fro 5% to 00% of freight rate shows that the deviation Δ JTC of the joint total cost is highly sensitive and directly related to freight cost. The rate of direction of change of solution can be seen by q atheatical expression, and inferred Fx Fx that delivery lot size is directly related to freight rate. Otherwise the nuber of deliveries is inversely related freight rate. 6) Effect of discount factor α to the solution and joint total cost. The deviation Δ JTC of the joint total cost is oderately sensitive and directly related to discount factor α. Based on the rate of direction of change q of solution,, can be observed that α α delivery lot size is directly related to α. Otherwise the nuber of deliveries is inversely related freight rate α. VI. CONCLUSION One of the iportant points in this study is synchronization of the production flow fro vendor to buyer. Joint econoic lot size (JELS) odel has proved its ability to represent the coordination and collaboration for both parties as well as anaging the flow of inventory replenishent by siultaneously deterining optial delivery lot size and nuber of deliveries. It has been ephasized refer to the iportance of considering the transportation cost into inventory replenishent decisions in the supply chain syste. Transportation cost has been included to deterine optial delivery lot size and nuber of deliveries. Thus, by incorporating transportation cost into JELS odel, lower joint total cost as a coposite of inventory cost, setup cost, ordering cost, fix transportation cost and freight rate cost can be achieved. The advantages fro the ipact of the total cost reduction, provide the econoic benefit that can be shared by both parties through bargaining and negotiation process which significantly iprove the relationship of vendor and buyer. It is also contributing a useful insight for iproving the perforance of the supply chain. For further research, firstly, since the nuber of deand and production are assued to be deterinistic; the research ay be extended by changing the deand and production to be dynaic. It requires soe ethods to face uncertainty in deand and production like artificial intelligence. Secondly, in the practice, consignent can be transported using TL or LTL shipent. In order to provide a coplete range of shipping decision, this research also needs to consider TL shipent. Therefore, a copany can decide which ode of transportations that efficiently eet the requireent of the copany. Third, quality is always becoe iportant issues to increase custoer service level. In the ost practical issues in a anufacturing environent, ites that shipped by a vendor do not qualified 00% accepted. So that, incorporating quality issues into inventory replenishent decision becoes ore attracting to be developed. Lastly, to be ore practical, then the research also can be extended by considering ultiple actors, ultiple ites, and shortage ites. APPENDIX A CONVEXITY TEST OF JOINT TOTAL COST USING THE HESSIAN MATRIX Proof. Taking the second derivatives of JTC with respect to q and. JTC q, D S = A F α FW x xd> 0 q JTC q, DS = > 0 3 q JTC q, DS D = + Hv,With P q P Thus, (, ) (, ) (, ) JTC q JTC q JTC q 4D AS 4DFS 0 4Dα FWdS x x 4D S = q q q q DS DS D D H v + q q P 4 P D For condition li =, then obtained D P P 4DFS 0 4Dα FWdS x x D S H A. A. > D A.3 v A.4 4D AS 3 = > 0 A.5 q q q q Since D, A,, F 0, F x, JTC is a convex function at point (, ) satisfied. W x, d > 0, therefore q. The proof is APPENDIX B THE RATE OF DIRECTION OF CHANGE OF SOLUTION TO PARAMETER.

9 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 04 3 B. The rate of direction of the solution q and the paraeter setup cost S B3. The rate of direction of the solution q and the holding cost H b Figure. The rate of direction of the solution q to the S Figure 6. The rate of direction of the solution q to the H b Figure 3. The rate of direction of the solution the S B. The rate of direction of the solution q and the ordering cost A Figure 7. The rate of direction of the solution the holding cost B4. The rate of direction of the solution q and the fix transportation cost F 0 Figure 4. The rate of direction of the solution q to the A Figure 8. The rate of direction of the solution q to the F 0 Figure 5. The rate of direction of the solution the A Figure 9. The rate of direction of the solution to the F 0

10 3 JOURNAL OF SOFTWARE, VOL. 9, NO. 5, MAY 04 B.5 The rate of direction of the solution q and the freight rate F x Figure 3. The rate of direction of the solution to the α Figure 0. The rate of direction of the solution to q the F x ACKNOWLEDGMENT The authors are indebted to the anonyous reviewers for giving valuable suggestions and coents. REFERENCES Figure. The rate of direction of the solution to the B.6 The rate of direction of the solution q and the discount factor α Figure. The rate of direction of the solution to F x q the α [] S. K. Goyal, An integrated inventory odel for a single vendor and single custoer proble, International Journal Production Research, Vol. 5, pp. 07, 977. [] D. J. Thoas and P. M. Griffin, Coordinated supply chain anageent, European Journal of Operational Research, Vol. 94, pp. 5, 996. [3] H. Stadler, Supply chain anageent and advanced planning-basics overview and challenges, European Journal of Operational Research, Vol. 63, pp , 005. [4] S. L. Ki and D. Ha, A JIT lot splitting odel for supply chain anageent: Enhancing buyer-vendor linkage. International Journal Production Econoics, Vol. 86, pp. -0, 003. [5] Yang, P.C., Wee, H.M., and Yang, H.J. Global optial policy for vendor-buyer integrated inventory syste within just in tie environent. Journal Global Optial, Vol. 37, pp , 007. [6] Sucky, E. Inventory anageent in supply chains: A bargaining proble. International Journal Production Econoics, Vol. 93, pp. 53-6, 005. [7] S. R. Swenseth and M. R. Godfrey, Incorporating transportation costs into inventory replenishent decisions, International Journal Production Econoics, Vol. 77, pp.3 30, 00. [8] S. R. Swenseth, and F. P. Buffa, Just-in-Tie: Soe effects on the logistic function, The International Journal of Logistic Manageent, Vol., pp. 5-34, 990. [9] A. Banerjee, A joint econoic lot size odel for buyer and vendor, Decision sciences, Vol. 7, pp. 9 3, 986. [0] A. Banerjee and S. L. Ki, An integrated JIT inventory odel, International Journal of Operation & Production Manageent. Vol, 5, pp , 995. [] W. J. Bouul and H. D. Vinod, An inventory theoretic odel of freight transport deand. Manageent Science, Vol. 6, pp. 43-4, 970. [] J. C. Langley, The inclusion of transportation costs in inventory odels: soe consideration. Journal of Business Logistic, Vol., pp. 06-5, 980. [3] S. R. Swenseth and M. R. Godfrey, The estiating freight rates for logistic decisions. Journal of Business Logistic, Vol. 7, pp. 3-3, 996.

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