International Journal of Industrial Engineering Computations

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1 International Journal of Industrial Engineering Computations ( 44 Contents lists available at Groingcience International Journal of Industrial Engineering Computations homepage: Groingciencecom/iec Optimal pricing and lot sizing in VMI model M Ziaee a and J L ouquard b a Department of Industrial Engineering University of ojnord ojnord Iran b University Franςois-abelias Tours France T I C L E I N F O rticle history: eceived January eceived in revised form March ccepted pril vailable online pril eyords: Vendor Managed Inventory upply Chain Management Optimal Pricing Economic Order uantity Geometric Programming One-buyer One- supplier VMI To-buyer To-supplier VMI T C T Vendor Managed Inventory (VMI is one of the effective techniques for managing the inventory in supply chain VMI models have been proven to reduce the cost of inventory compared ith traditional economic order quantity method under some conditions such as constant demand and production expenditure oever the modeling of the VMI problem has never been studied under some realistic assumptions such as price dependent demand In this paper three problem formulations are proposed In the first problem formulation e study a VMI problem ith one buyer and one supplier hen demand is considered to be a function of price and price elasticity to demand and production cost is also a function of demand The proposed model is formulated and solved in a form of geometric programming For the second and the third models e consider VMI problem ith to buyers and to suppliers assuming that each buyer centre is relatively close to the other buyer centre Each supplier has only one product hich is different from the product of the other supplier To suppliers cooperate in customer relationship management and to buyers cooperate in supplier relationship management as ell so the suppliers send the orders of to buyers by one vehicle simultaneously For the third model an additional assumption hich is practically applicable and reasonable is considered For all the proposed models the optimal solution is compared ith the traditional one We demonstrate the implementation of our proposed models using some numerical examples Groing cience Ltd ll rights reserved Introduction During the past to decades Vendor Managed Inventory (VMI has become one of the most effective and popular approaches in supply chain management (VMI (Lee & Chu 5; De Toni & Zamolo 5; usdiansyah & Tsao 5;Yao & Evers 7 VMI can shift the retailers responsibility for planning and replenishment activities to the manufacturers based on demand information sent by the customers or retailers to the manufacturers There are the folloing advantages for VMI model: There is an improvement on performance of the supply chain by collaborative partnerships beteen buyer(s and supplier(s The customer service levels increase The inventory levels decrease and fill rates increase 4 Total costs and lead-time are minimized This approach has been idely used in the literature (Cachon and Fisher 997; Cachon ; Dong & Xu Fry et al ( study the effects of logistics in VMI for problems ith one buyer and one supplier Luca ertazzi et al (5 compare the VMI policies ith more traditional retailermanaged inventory (MI policy and sho that the VMI policies significantly reduce the average cost compared ith MI policy Yan Dong et al ( study VMI effects on total costs in long-term and shortterm planning horizons and sho that in certain conditions VMI may decrease the supplier s profit The VMI problem ith one buyer and one supplier has been ell studied in the context of some assumptions such as constant demand and production expenditure oever these assumptions are often unrealistic for real orld Corresponding author addresses: ziaee@iubacir (M Ziaee Groing cience Ltd ll rights reserved doi: 567/jiec

2 4 problems We first consider the one buyer-one supplier VMI problem hen demand is a function of price and price elasticity to demand This assumption has already been used for different problems (Lee & im 99; Lee & im 99; oon & Cerry 5 Lee and im (99 are believed to be the first people ho model and solve some traditional problems under this ind of assumption adjadi et al (5 extend the modelling of the problem under some different conditions adjadi and Ziaee (6 study the effects of price for a price discrimination model The reminder of this paper is organized as follos The first proposed model of this paper studies a one-buyer one-supplier VMI model in hich e assume hen demand for a product increases the production is less costly We propose a solution procedure for the proposed model hich is based on Geometric Programming (GP In sections and 4 e consider the to-buyer to-supplier VMI problem For both models e assume that each buyer centre is relatively close to other buyer centre but there is a relatively a big distance beteen the buyer and all other suppliers To suppliers collaborate ith each other in a form of a customer relationship management and to buyers also cooperate in a form of a supplier relationship management as ell so the suppliers send the orders of to buyers by one vehicle simultaneously Note that this ind of conditions holds for many real-orld problems For all three proposed models the optimal solution is compared ith the traditional one and a numerical example is presented in order to illustrate the implementation of the proposed method Finally the conclusion remars are given to summarize the contribution of this paper Model Consider a single product here demand is affected by the selling price Let P be the selling price per unit and elasticity of price ith respect to demand in the maret respectively We assume that the demand or production (D is a function of price per unit (P that is D P > ( The scaling parameter represents the effect of other related factors and > implies that D increases as P decreases We also assume that the unit production cost (C can be discounted ith β Therefore e have C ud β β > ( here D is the amount of demand or production in the planning period and u is a scaling parameter The exponent β represents total production elasticity of production unit cost ith a small value for β (say β We consider a traditional inventory model (EO here no shortage is permitted The folloing assumptions hold in this paper The consumption rate is constant The amount of order is constant Each ne order arrives as soon as the inventory level is equal to zero Transportation times are negligible and each order is delivered at once The setup and holding costs in VMI model are paid by the supplier The other notations of our model are as follos : Total inventory costs for the supply chain hich includes the costs of the supplier and the buyer in traditional model : Total inventory costs for the supply chain hich includes the costs of the supplier and the buyer in VMI model : The optimal values of respectively : Total costs of the supplier in traditional and VMI models respectively : Total costs of the buyer in traditional and VMI models respectively : The size of each order : The optimal value of in traditional and VMI models respectively : The cost of each order for the supplier - Economic Order uantity

3 M Ziaee/ International Journal of Industrial Engineering Computations ( 5 : The cost of each order for the buyer D: The demand of buyer or the amount of production by supplier in each period : The inventory holding cost per unit of product and per period In a traditional inventory model the costs of inventory and procurement are computed as follos ( here D PD (4 and D C D (5 Using ( and ( the problem of minimizing the total costs of inventory for the buyer in traditional model can be ritten as follos min ( P P (6 Problem (6 is a posynomial form of geometric programming and its degree of difficulty (eightler & Phillips 976; Dembo 98; Freedland 98 is -( uppose e assume δ ( P δ P (7 δ lso let i i be the dual variables associated ith δ i i respectively Therefore the dual problem formulation for the problem (6 is as follos d( max f( (8 subject to ( i i (9 Thus and ( Note that in order to have i for i e must limit to < < and the optimal value of f ( can be determined using i i as follos f f( ( ( ( No e can use the folloing equations to determine the optimal value of the primal variables P

4 6 ( f P δ f δ ( Thus ( P ( ( ( The total inventory cost for the supply chain in traditional model hich is the sum of the costs of buyer and supplier is computed as follos PD CD D (4 In (4 the parameter is the sum of and (ie Using equations ( ( and ( yields ( ( ( (5 ( ( ( ( ( β β u imilarly the total cost of inventory in VMI model is calculated as follos PD CD D (6 Using ( and ( the problem of minimizing the total costs of inventory in VMI model is computed as follos ( ( min β β P P P u (7 Problem (7 is a posynomial form of geometric programming and its degree of difficulty is 4-( Let i for i 4 be the dual variables Therefore the dual problem formulation for the problem (7 is as follos ( 4 4 max g( d( β u (8 subject to 4 ( β ( 4 i i 4 (9 We rerite the equation (8 in terms of only one dual variable Therefore e have ( ( ( ( ( ( ( 4 β β β β

5 M Ziaee/ International Journal of Industrial Engineering Computations ( 7 ccording to the assumptions concerning the values of and β hich ere mentioned earlier e can no assume β < and the constraints i for i 4 require that < < and < β < 5 and therefore < 5 ( (- ( β ( β ( ( ( ( ( ( ( d( u β β β β β ( (- ( ( β β Therefore e can use a simple linear search to find the optimal solution (d ( Then the values of and 4 can be easily calculated using ( ith the values of and corresponding (that is P and are also obtained as follos P 4 Obviously the VMI model is preferred to the traditional one if ( ( Numerical example We no illustrate the proposed model using a numerical example The parameter values for this example are as follos β u The results obtained from solving the test problem are as follos For traditional model: P For VMI model: P ( 4 The above results sho that the total costs of VMI model are much loer than those of traditional model (about one-third Model In this model e assume that there are to buyers and to suppliers Each buyer centre is relatively close to the other buyer centre and so are the to suppliers but any buyer is far aay from any supplier Other assumptions of this model are as follos Each supplier procures only one product hich is different from the product of other supplier Therefore e have to products in our model For each product to buyers have the same holding cost To suppliers cooperate in customer relationship management and to buyers cooperate in supplier relationship management therefore they send the orders of to buyers by only one vehicle simultaneously and therefore for each buyer the to suppliers have the same fixed ordering cost; and for each supplier the to buyers have also the same fixed ordering cost (see Fig The demands of suppliers and buyers are constant and given Transportation times are negligible and each order is delivered at once hortage is not alloed Each ne order arrives as soon as the inventory level is equal to zero We assume that each supplier has a constant selling price for his product (as opposed to the first model and there is no discounting

6 8 The consumption rate is constant The etup and holding costs in VMI model are paid by supplier uyer upplier uyer upplier Fig To-buyer to-supplier system dditional notations used for the second model are as follos : The cost of each order (fixed ordering cost for the supplier j j : i The cost of each order (fixed ordering cost for the buyer i : j The inventory holding cost per unit of product and per period for supplier j o : Total costs of supplier j for procuring all demands of buyer i in traditional model o : Total costs of buyer i for procuring all his demands from supplier j in traditional model : The sum of o and o o : o Total costs of supply chain including the costs of all suppliers and buyers in traditional model : Total costs of supply chain including the costs of all suppliers and buyers in VMI model v v v : Total costs of to suppliers in VMI model : Total costs of to buyers in VMI model : The demand of buyer i for the product of supplier j in each period : o The order quantity of the product of supplier j for buyer i in traditional model : v The order quantity of the product of supplier j for buyer i in VMI model and o are decision variables; v o and traditional model e have i o i ; j j are the optimal values of v o and respectively For v ( i j o o o o j o (4 Therefore

7 M Ziaee/ International Journal of Industrial Engineering Computations ( 9 j j o o o ( j i j i j i i (5 j o o j i ( i j i j i (6 In VMI model the number of orders for the supplier j is equal to the folloing expression i / i (7 ased on the assumptions mentioned earlier the to suppliers have the same number of orders that is (8 We assume that for each product and for each buyer the proportion of the buyer order quantity ith respect to the total order quantity is equal to the proportion of the buyer demand ith respect to the total demand that is or (9 imilarly e have Therefore the number of orders in VMI model is equal to v or Therefore ( ( ( j v i j Thus ( v v v ( ( ( For the sae of the convenience e assume that and x (4 Therefore

8 4 v ( ( (5 ecall that e have Therefore e have v x ( ( x x (6 The optimal value of x is obtained by taing the derivative of v ith respect to x and equating it to zero v x ( ( x (7 ( ( x (8 nd therefore the optimal value of v is obtained as follos: [ v ( ( ( ] (9 The difference beteen the minimum total costs of traditional model and those of VMI model is calculated as follos j (9 o v i j ( i j i ( [ ( ( ] Numerical example The general notion behind the problem formulation is best illustrated by a numerical example The parameter values for the test problem are as follos: The optimal results are as follos For traditional model: For VMI model: The above results sho that the total costs of the VMI model are less than those of the traditional one

9 M Ziaee/ International Journal of Industrial Engineering Computations ( 4 4 Model This model is similar to the second model but e no consider an additional assumption hich is practically applicable and profitable ssume that the number of orders for the supplier j (j is equal to n j and one of them (say n is greater than the other one that is n > n In such a situation it is better to send n orders by the method of model (ie the orders of to suppliers are sent by one vehicle simultaneously and n n orders (hich belongs to the supplier are sent separately by supplier uppose that the cost of each order sent by the second model method is equal to c s previously mentioned the folloing relations hold for the number of orders for suppliers and : If and then the total costs of to suppliers in VMI model is calculated as follos v nd if v ( c min ( c ( < then the total costs of to suppliers in VMI model is obtained as follos ( c ( c (4 (4 Let v c min ( nd the total costs of to buyers in VMI model is obtained as follos (4 j v ( max ( i j (4 Therefore the total costs in VMI model is calculated as follos: v v v c min ( ( max ( ( ( (44 issumption : min ( (45 ased on the above assumption the folloing relationship holds

10 4 v ( c ( ( ( (46 denotes the value of v( v under the assumption Therefore e have v ( ( c ( ( To obtain the optimal value of ( e use the first order necessary conditions v( v ( ( c ( v( ( ( Therefore e have (48 (47 ( c ( ( ( (49 and v ( c ( ( ( ( (5 ii ssumption : min ( (5 ased on the above assumption the folloing relationship holds c v ( ( ( ( v ( denotes the value of v under the assumption Therefore e have: v ( ( ( ( ( nd the optimal value of is calculated as follos: v( c (5 (5

11 M Ziaee/ International Journal of Industrial Engineering Computations ( 4 v ( ( ( v ( c ( ( ( ( ( c ( ( ( ( c ( (56 v( To solve the third model e first obtain the values of and using (49 and (55 and then select the one hich satisfies the corresponding assumption nother ay is to solve the folloing mathematical programming: v ( subject to v : y Numerical example v y ( y c ( ( y We no use the numerical example of model (ith c 8 to illustrate the approach Under the assumptions and v ( v ( and are obtained as follos: Under assumption : 748 v ( Under assumption : v ( Therefore the optimal solution of the problem is the data for assumptions and the total costs for the VMI model is equal to (Note that if C then the value of total costs of VMI in this model is equal to hich is less than the corresponding value in the model ( 5 Conclusions In this paper e have presented three problem formulations for Vendor Managed Inventory The first proposed model of this paper considers the demand as a function of price and price elasticity to demand and the production cost is also considered to be a function of demand The optimal solutions of both VMI and (54 (55

12 44 traditional models are derived and compared We have also explained through a numerical example ho VMI could reduce the total inventory cost compared ith traditional method In the second and third models e have studied VMI problem ith to buyers and to suppliers and discussed the optimal solutions cnoledgment The authors ould lie to than the anonymous referees for their comments on the earlier version of this or eferences ertazzi L Paletta G & peranza M G (5 Minimizing the total cost in an integrated vendor managed inventory system Journal of euristics 9 49 Cachon G ( toc ars: inventory competition in a to echelon supply chain Operations esearch 49( Cachon G & Fisher M (997 Campbell soup s continuous product replenishment program: evaluation and enhanced decision rules Production and Operations Management 6( De Toni F & Zamolo E (5 From a traditional replenishment system to vendor-managed inventory: case study from the household electrical appliances sector International Journal of Production Economics Dembo (98 ensitivity analysis in geometric programming Journal of Optimization Theory and pplications 7 Dong Y & Xu ( supply chain model of vendor managed inventory Transportation esearch: Part E Freeland J (98 Coordination strategies for production and mareting in a functionally decentralized firm IIE Transactions 6 Fry M apuscinsi & Olsen T ( Coordinating production and delivery under a (z Z-type vendormanaged inventory contract Manufacturing & ervice Operations Management ( 5-7 oon J & Cerry M (5 Optimal inventory policies for an economic order quantity model ith decreasing cost functions European Journal of Operational esearch im D & Lee W J (998 Optimal joint pricing and lot sizing ith fixed and variable capacity European Journal of Operational esearch 9 7 Lee C C & Chu W J (5 Who should control inventory in a supply chain? European Journal of Operational esearch Lee W J & im D (99 Determining selling price and order quantity by geometric programming optimal solution bounds and sensitivity Decision ciences Lee W J & im D (99 Optimal and heuristic decision strategies for integrated production and mareting planning Decision ciences 4 adjadi J & Ziaee M (6 price discrimination modeling using geometric programming Lecture Notes in Computer cience adjadi J Orougee M & ryanezhad M (5 Optimal production and mareting planning Computational Optimization and pplications ( 95 usdiansyah & Tsao D (5 n integrated model of the periodic delivery problems for vendingmachine supply chains Journal of Food Engineering Yao Y Evers P T & Dresner M E (7 upply chain integration in vendor-managed inventory Decision upport ystems 4(

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