An Investigation of Vendor Managed Inventory Application in Supply Chain: The EOQ Model with Shortage

Transcription

1 An Investigation of Vendor Managed Inventory Application in upply Chain: The EOQ Model with hortage eyed Hamid Reza Pasandideh, Ph.D., Assistant Professor Railway Engineering Faculty, Iran University of cience and Technology, Tehran, Iran Phone: +98 (1) , Fax: +98 (1) , eyed Taghi Akhavan Niaki, Ph.D., Professor Department of Industrial Engineering, harif University of Technology, Tehran, Iran Phone: , Fax: , niaki@sharif.edu Ali Roozbeh Nia, M.c. tudent Department of Industrial Engineering, Islamic Azad University, Qazvin, Iran Phone: +98 (81) , Fax: +98 (81) , ali_roozbeh@yahoo.com Abstract Keywords: Recent researches have shown the importance of improving the supply chain competitiveness by means of strategic alliances. This study considers the retailer supplier partnership through a vendor managed inventory () system and develops an analytical model to explore the effect of important supply chain parameters on the cost savings realized from collaborative initiatives. This model is developed for a two-level supply chain consisting of a single supplier and a single retailer and examines the inventory management practices before and after implementation of. The results of analytical examination show that implementation in economic order quantity model when shortage is backlogged sometimes has the ability to reduce total costs of supply chains. Three numerical examples are also given to support this claim. Vendor managed inventory; upply chain management; Economic order quantity; hortage; acklog Corresponding Author 1

2 1. Introduction A supply chain is a dynamic system that includes all activities involved in delivering a product from the stage of raw material to the customer. These activities include manufacturing, inventory control, distribution, warehousing, and customer service. upply chain management coordinates and integrates all of these activities into a smooth process. The main objective of the supply chain management is to minimize system-wide costs while satisfying service level requirements. With increasing global competition and emergence of e- business, supply chain management is viewed as a major solution to cost reduction and profitability (Tyana and Wee 003). Inventory has been considered one of the major drivers of the supply chain and its management (Chopra and Meindl 001). There is nothing more important within the realm of supply chain management than the management of inventory (Levy and Dhruv 000). As such, decisions regarding inventory replenishment have a direct effect on supply chain performance. In a traditional supply chain each partner (company) is responsible for his own inventory control and production or distribution ordering activities and the interactions between partners are limited to the feed-forward physical flow of products and the feedback flow of information. One fundamental characteristic and problem that all partners of a traditional supply chain (such as retailers, distributors, manufacturers, raw material suppliers) must solve is just how much to order the production system to make (or the suppliers to supply) to enable a supply chain echelon to satisfy its customers demands. This is the classic production/inventory control problem (Disney and Towill 003). A simple schematic of a four-echelon supply chain, comprising retailer, distributor, warehouse, and factory is illustrated in Fig. 1 (Disney et al. 003). In a vendor managed inventory (), besides the traditional supply chain information and material flows, the supplier controls the retailer's inventory level so as to ensure that a desirable

3 customer service levels are maintained. In other words, has been described as an inventory and supply chain management tool in which the supplier has taken the responsibility for making decisions as to the timing and amounts of inventory replenishment (Disney et al. 003). Customer Retailer Distributor Warehouse Factory Flow of information up the supply chain Flow of materials down the supply chain Fig. 1: A traditional supply chain Evidence has shown that can improve supply chain performance by decreasing inventory levels and increasing fill rates; as a result, industry use of has grown over time (Emigh 1999). is a collaborative commerce initiative that integrates operations between suppliers and retailers through information sharing and business process reengineering. y the aid of information technologies, such as electronic data interchange (EDI) or internet-based XML protocols, retailers can share sales and inventory information with suppliers on a real time basis. A simple diagram of a supply chain is found in Fig. (Disney and Towill 003). Fig. : A supply chain 3

4 uppliers can then use this information to plan production runs, schedule deliveries, and manage order volumes and inventory levels at the retailer's stock-keeping facilities (Yao et al. 007). The potential benefits from are very compelling and can be summarized as reduced inventory costs for the supplier and retailer and improved customer service levels, such as reduced order cycle times and higher fill rates (Achabal et al. 000 and Waller et al. 1999). These benefits have been realized by successful retailers and suppliers, most notably Wal-Mart and key suppliers like Proctor and Gamble (Cetinkaya and Lee 000). In order to determine the impact of important supply chain parameters on the cost savings to be realized from collaborative initiatives such as, in this paper the previous research is extended and an analytical model is developed for an economic order quantity (EOQ) problem where shortage is backlogged. The model is considered for a two-level supply chain consisting of a single supplier (vendor or distributer) and a single buyer (retailer) to examine the inventory management practices before and after the -implementation. These results have managerial ramifications in that they can help demonstrate when and by how much is likely to produce benefits for supplier buyer dyads in practice. The rest of the paper is structured as follows: in the next section, a review of the literature on is presented. While section 3 presents the framework of the proposed model, section 4 contains the model along with its assumptions. ection 5 uses the framework to analyze cost savings due to. Three numerical examples are presented in section 6 to justify and demonstrate the application of the proposed method. Finally, conclusions and recommendations for future research are presented in section 7. 4

5 . Literature review An early conceptual framework of was described by Magee (1958) when discussing who should have authority over the control of inventories. However, interest in the concept has only really developed during the 1990s. Companies have looked to improve their supply chains as a way of generating a competitive advantage, with often advocated. This strategy has been particularly popular in the grocery sector but has also been implemented in sectors as diverse as steel, books and petrochemicals (Disney et al. 003). Examples of the companies that popularized in 1980s are Wal-Mart, K-Mart and Proctor & Gamble (latherwick 1998, Waller et al. 1999). The use and the benefits of were originally documented by a few authors with the primary research being conducted by latherwick (1998), Cachon & Fisher (1997), Clark & Hammond (1997), Fraza (1998), and Waller et al. (1999). In all documented cases of organizations that use, there was some connection between the members in order to facilitate the exchange of information on inventory levels, product usage, and re-supply issues. Generally this connection was provided with electronic data interchange (EDI) (Emigh 1999). Haavik (000) stated that EDI tools were needed to realize the full benefits of. Lawrence and Vokurka (1999) and Challener (000), however, described situations where the exchange was a manual process. In these cases, a representative from the upstream member physically monitored the inventory level. In order to illustrate the potential inventory cost savings from integrated inventory management, early researchers studied simple supply chains with a single supplier and a single retailer. Woo et al. (001) and Yu & Liang (004) extended the two-echelon inventory supply chains to three echelon ones where the supplier was a manufacturer and his raw materials inventory was involved. In order to streamline the supply chain, the supplier expected to synchronize his production cycles with his retailers orderings. The supplier and his retailers employed a common replenishment cycle policy to reduce supply chain inventory cost. Vigtil (007) described a set of 5

6 five case studies that indicated sales forecasts and inventory positions were the most valuable information provided to supplier by the retailers in a relationship. Generally speaking, the research can be classified into two groups: a) trict managerial and functional subjects along with simulation procedure (see for example Holmstrom 1998, Tyana & Wee 003, and Disney & Towill 003) b) Analytic and mathematical terms and presentations of the economic analysis (see for example Van Der Vlist et al. 007, Chang et al. 005, Dong & Xu 00, Yao et al. 007, Jasemi 006, and ofifard et al. 007.) Within the second group, Dong and Xu (00) presented an analytical model to evaluate the short-term and long-term impact of on supply chain profitability by analyzing the inventory systems of the parties involved. The impact of was also compared with that of full channel coordination and it was found that in the short-term can accomplish what full channel coordination is set to accomplish. They formulated appropriate mathematical models for a buyer supplier channel structure, examined the effects of a strategy on the various cost components of both parties, and then analyzed the role of in a supply chain initiative. In particular, the effects of an integrative program on total relevant costs and profits were investigated. everal common assumptions that also were used in their inventory-channel coordination research were: the inventory system of the buyer can be described by an economic order quantity (EOQ) policy, the demands are deterministic, there are no stock-outs, and the lead-times are also deterministic. Yao et al. (007) using the same assumptions as Dong and Xu s (00) along with an additional assumption (the order quantity for the supplier is likely to be an integer multiple of the buyer s replenishment quantity) presented an analytical model to determine how key logistics parameters, most notably ordering costs and inventory carrying charges, can affect the benefits to be derived from. They then determined how the benefits were likely to be distributed between a 6

7 buyer and supplier in a supply chain, given these logistics parameters. Finally, they presented a numerical example to illustrate their results. Van Der Vlist et al. (007) extended the Yao et al. (007) model along with the costs of shipments from the supplier to the buyer. ofifard et al. (007) presented an analytical model for a single-buyer-single-supplier model to explore the effects of collaborative supply-chain initiatives such as with the economic production quantity (EPQ) manner. Furthermore, Jasemi (006) developed a supply chain model with single supplier n-buyers and compared system with traditional types. He also made a pricing system for profit sharing between parties. It should be noted that in all of the previous researches within the second group the EOQ model with infinite production rate was considered. 3. Modeling framework In order to determine the impact of important supply chain parameters on the cost savings to be realized from collaborative initiatives such as, consider a two-level supply chain consisting of a single supplier and a single retailer for which we examine the inventory management practices before and after the implementation of. Although many of the results can be generalized to more complex supply chains, a simple supply chain is used for computational ease. We assume that the retailer faces external demand from consumers. In a supply chain without on the one hand, the supplier observes consumer demand only indirectly through the retailer's ordering policy. In fact, the retailer company appears to be the leader in this relationship and the supplier just takes the order quantity from the retailer and makes the necessary delivery. In other words, the supplier does not have any responsibility for the production holding. On the other hand, in a supply chain with, the retailer no longer manages 7

8 its inventory system and leaves it to the supplier to determine inventory levels, order quantities, lead times, etc. (Dong & Xu 00). In a supply chain with, the supplier s information system directly receives consumer demand data. As a result, the supplier has now the combined inventory with order setup and holding cost (Dong & Xu 00). The supplier with regard to his own inventory cost, which equals to the total cost of the supply chain, determines the timing and the quantity of production in every cycle. The major difference between not using and using is that the retailer's order quantity is determined by the supplier in a system (Yao et al. 007). Fig. 3 presents the modeling framework of and Non- supply chains, where the triangles show the stores. Fig. 3: The Modeling Frameworks 4. The economic order quantity policy when shortage is backlogged In this section, the assumptions and the notations are first defined. Then, the total cost of both the traditional (non-) and the integrated () supply chains are modeled. 8

9 4.1. Assumptions The mathematical model of this research will be developed on the basis of the following assumptions: (a) A single-supplier single-buyer supply chain with one item is considered (b) hortage is allowed and is completely backlogged ( 0 and 0) (c) Deliveries of orders are assumed to be instantaneous, that is, the lead time is zero (d) Costumer s demand is deterministic (e) The production rate is infinite (f) The product price is fixed in the planning period. 4. Notations The following notations will be used to develop the proposed model: TC no : Total costs of non- supply chain TC : Total costs of supply chain A : upplier s ordering cost per unit A : uyer's ordering cost per unit Q no : Order quantity of non- chain Q : Order quantity of chain D : uyer's demand rate h : Product s carrying cost per unit held in buyer's store in a period K 0 : uyer's inventory cost before K 1: uyer's inventory cost after K 0 : upplier's inventory cost before 9

10 K 1: upplier's inventory cost after b : Fixed amount of shortage in a cycle : Fixed cost of shortage per unit : Cost of shortage per unit per time 4.3 The case of non- (traditional) supply chain Prior to implementing, the inventory cost of the buyer and the supplier and hence the total inventory costs of the supply chain are calculated as follows: A D h b πbd 0 no Qno Qno Qno Qno K = + Q -b + + (1) K AD 0 () Qno A D h b πbd AD no 0 0 no Qno Qno Qno Qno Qno TC = K +K = + Q -b (3) The buyer's inventory cost in Eq. (1) is a function of both Q no and b for whom the optimal values can be obtained by partial derivatives. Differentiating Eq. (1) with respect to and setting it zero results in: Q no Q K 0 =0 no (4) h - DA + h (Qno -b) +πdb + b + Q no - b = 0 Q Q no no Q -b +Q (Q -b)= DA +πdb + b h no no no Q no = DA +πdb + b + b h (5) (6) (7) 10

11 At the same time, differentiating Eq. (1) with respect to b results in: b K 0 =0 (8) h b πd - Qno -b + + =0 (9) Q Q Q no no no hb b πd -h = 0 (10) Q Q Q no no no 1 hb+ b+πd = h (11) Q no b(h + ) πd Q no = + (1) h h πd Q no = +b1+ h h (13) Inserting Eq. (13) into Eq. (7), we can write 1 πd b1+ = DA +πdb + b + b h h h (14) + h b +π Db +(πd) - DA h = 0 (15) ince shortage is allowed and backlogged, i.e., 0 and 0, the optimal value of b can be obtained using Eq. (15) as b : + h b -DA h =0 (16) DA h b = + h (17) DAh b = (18) ( +h ) 11

12 Inserting Eq. (18) into Eq. (13) we can obtain the optimal order quantity of the non- supply chain as: DA h Q no = 1+ +h h DA h +h =. +h h (19) (0) DA h + Q no = h (1) Furthermore, inserting Eq. (18) and (1) into Eq. (3), the total inventory cost of the supply chain before implementing becomes: 1 h b TC no = DA + Qno -b + +A D Qno 1 h DA h DAh = DA A D DA h +h ( +h ) h + h 1 h DAh = DA + DA DA DA +h +h h + h 1 h h = DA DA DA +h +h h + h 1 h TC no = DA 1+ +DA DA +h h + h () (3) (4) (5) (6) 1

13 4.4 The case of (integrated) supply chain After the implementation of, the inventory costs of both the buyer and the supplier and hence the total inventory costs of the integrated supply chain are calculated as follows: K1 0 (7) K Q b (8) 1 AD AD h b bd - Q Q Q Q Q h TC K K A A Q b (9) D b bd Q Q Q Q Once again, the total inventory cost of the integrated supply chain in Eq. (9) is a function of both Q and b for whom the optimal values can be obtained by partial derivatives. Differentiating Eq. (9) with respect to Q and setting it zero results in: TC Q =0 (30) D h h b πbd - A +A - Q -b + Q -b- - =0 (31) Q Q Q Q Q 1 h b h - D A +A + Q -b + +πbd + Q - b = 0 Q Q (3) 1 h 1 b h - (Q -b) - D(A +A )+ +πbd + (Q - b) = 0 Q Q Q (33) 1 h h 1 b - Q -b + Q -b = D A +A + +πbd Q Q Q 1 1 b - Q -b +Q Q -b = D A +A + +πbd h 1 1 b Q -b - Q -b +Q = D A +A + +πbd h (34) (35) (36) 13

14 1 1 b Q -b Q +b = D A +A + +πbd h 1 1 b Q -b = D A +A + +πbd h 1 1 b 1 Q = D A +A + +πbd + b h (37) (38) (39) Differentiating Eq. (9) with respect to b and setting it zero results in: b TC 0 (40) h 1 πd (41) - Q -b + b + =0 Q Q Q hb 1 πd -h + + b + = 0 (4) Q Q Q 1 hb+ b+πd = h (43) Q b h + πd Q = + (44) h h πd Q = +b 1+ h h (45) Inserting Eq. (45) into Eq. (39) we can write: 1 πd 1 b 1 +b 1+ = D A +A + +πbd + b h h h (46) + h b +π Db + πd -D A +A h =0 (47) Considering 0 and 0 in Eq. (47) the optimal value of b is obtained as: + h b -Dh A +A =0 (48) 14

15 Dh A + A b = + h (49) Dh A + A b = ( +h ) (50) Therefore, by inserting Eq. (50) into Eq. (45) the optimal value of Q is obtained as: Dh A + A Q = 1+ +h h (51) = Dh A +A +h +h h (5) D A + A +h Q = h (53) Finally, by inserting Eq. (50) and (53) into Eq. (9), the total inventory cost of the supply chain under becomes: 1 h b TC = D A +A + Q -b + Q However, we need to obtain the term Q -b as follows: +h h +h D A +A Dh A +A Q -b = - +h h +h h D A + A Dh A + A = - D A + A +h h = - h +h (54) (55) (56) (57) = D A + A + h - h h +h (58) 15

16 = D A + A +h +h -h h ( +h ) (59) = + h h +h D A + A (60) D A + A +h h = + h +h +h D A + A h Q -b = 1+ h +h (61) (6) Now, by inserting Eq. (6) into Eq. (54) we can write: TC = = h h D A + A Dh A + A D A + A +h h D A +A h +h +h h h DA +A +DA +A 1+ +D A +A +h +h D A + A +h h (63) (64) = h h DA +A h +h D A + A +h h (65) TC = D A + A 1+ h +h D A + A +h h (66) 16

17 5. The analysis of inventory costs with and without In order to compare the total inventory costs of the supply chain in two cases of non- and implementation, lettc TCno. Then, in what follows a suitable requirement is derived on which this assumption holds. TC TC (67) no h h D A +A 1 + DA DA +h +h D A + A +h DA h + h h (68) h h A +A 1+ A 1+ +A +h +h (69) D A + A DA h h h h h 4A +A 4A A + 4A A h +h +h (70) D A +A DA h h h h h h h A +A 1 + h 4A A +4A A 1+ +h ( +h ) +h +h (71) D DA h 4A h 8A h 4A A h 4A A + 4A A + - DA +h + h + h h (A + A ) h h 1+ + o D +h +h 3 4Ah 4Ah 8Ah A h 4AAh DA DA DA + h DA DA + h 3 4AAh h (A + A ) h (A + A ) 4h (A + A ) o DA + h D D +h D + h (7) (73) 17

18 3 A h A h 4A h A h A h A h D D D DA D D +h +h +h 3 h (A + A ) h (A + A ) 4h (A + A ) - - o D D +h D +h A +A +h D +h 3 3 A h - h 4A h + A h - 4h A - 4h A + + D A h + A h - h A - h A D A h + o DA (74) (75) 3 -A h A h A h - + o (76) D +h DA +h D Ah -h h A - + o D +h +h A Ah -4A h -4A h +h +A +h D A + h o (77) (78) Ah A D + h Ah A D + h -4A h -4Ah-4Ah +A +h o -8A h -4Ah +A +h o (79) (80) The term A h A D + h is always positive, therefore: -8A h -4Ah +A +h o -Ah 8h A + h o (81) (8) A +h A h 8h +4 (83) A +h 4A h h + (84) 18

19 4h h + A A +h (85) Hence, the following conclusion can be made in a single buyer single supplier supply chain system where the EOQ model is used and shortages are completely backlogged: The integrated supply system is not always better than the traditional one. The implementation has the ability to reduce the total costs of the supply chain only when Eq. (85) holds The to Non- ratio of supplier s order quantity Let be the to Non- ratio of the supplier's order quantity, i.e. Q Q (86) no Then, D A + A +h h A +A A β = = =1+ =1+ d DA A A + h h (87) where d A A. ince the quantity of monotonically increases with respect to d, the following conclusions can then be made: a) The ratio of the order quantity to non- order quantity monotonically increases with respect to d A A. Fig. 4 depicts this conclusion. b) The larger the ratio of the supplier to the retailer order cost, the larger the ratio of to non- order quantity is. 19

20 c) ince small values of d cause the retailer' order cost to be larger than the supplier's order cost and since 1+ d, the implementation of reduces order quantities by a greater amount when the order cost ratio is small. This is a favor for the supplier, because its order cost decreases. 10 Value Fig. 4. The to non- ratio of the supplier's order quantity vs. d 6. Numerical examples In this section, three numerical examples are given to justify and demonstrate the application of the proposed method. In these examples, let =A inequality A supply chain. 4h h +. Hence, based on Eq. (85), if the +h holds, the total costs of the system will be smaller than the one of the Non- The initial data of the numerical examples are given in Table (1), in which the examples share equal demand, equal carrying cost, and equal shortage cost. However, different buyer and 0

21 supplier's ordering costs are used. The results of applying the proposed method on the data of the numerical examples are given in Tables (), (3), and (4). Table (1): The Initial Data of the Numerical Examples Numerical Examples The parameters Example 1 Example Example 3 A A h D Table (): The Results Obtained for Example 1 Non- -upply Does Eq. (85) hold? The elected Chain upply Chain Chain b Q Yes Total Cost Table (3): The Results Obtained for Example Non- -upply Does Eq. (85) hold? The elected Chain upply Chain Chain b Q No Non- Total Cost Table (4): The Results Obtained for Example 3 Non- -upply Does Eq. (85) hold? The elected Chain upply Chain Chain b Q Yes Total Cost The results of Tables (), (3), and (4) show that employing Eq. (85) is justified and the supply chain with less total cost is selected as the better one in all of the three examples. 1

22 7. Conclusions and future research This paper contributes to the literature an analytical model that helps to provide a better understanding of how important supply chain parameters affect the inventory cost savings to be realized from implementation. This model was considered for a two-level supply chain consisting of a single supplier and a single buyer by whom the inventory management practices before and after the implementation of was examined. Results from the model showed that implementation in economic order quantity model, when shortage is backlogged has the ability to reduce total costs of supply chain only in a special case (when Eq. 85 for A & A holds). In other words, we showed that cost savings earned by implementing collaborative initiatives such as vendor managed inventory that allows information sharing and integration among firms in the supply chain cannot be reached without limitation. ince the proposed model of this research is representative of a certain type supply chain, the results are only true when the assumptions are held. In particular, the demand was assumed to be known with certainty, the lead time was assumed to be zero and only one item assumed to be in the supply chain. The study of changing the above assumptions is a possible topic for further research. 8. References Achabal D, Mclntyre, mith, Kalyanam K. (000). A decision support system for vendor managed inventory. Journal of Retailing, 76: latherwick A. (1998). Vendor-managed Inventory: Fashion fad or important supply chain strategy? upply Chain Management, 3: Cachon G, Fisher M. (1997). Campbell oup's continuous replenishment program: Evaluation and enhanced decision rules. Production and Operations Management, 6:

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