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Coordination and profit sharing between a manufacturer and a buyer with target profit under credit option
European Journal of Operational Research 182 (2007) 1469–1478 www.elsevier.com/locate/ejor
O.R. Applications
Coordination and profit sharing between a manufacturer and a buyer with target profit under credit option S.P. Sarmah
a,*
, D. Acharya a, S.K. Goyal
b
a
b
Department of Industrial Engineering and Management, IIT Kharagpur, Kharagpur 721302, India Department of Decision Sciences and MIS, John Molson School of Business Concordia University, Montreal, Canada H3G1M8 Received 19 September 2005; accepted 12 September 2006 Available online 22 February 2007
Abstract Several studies have focused on buyer vendor coordination through quantity discount/credit option mechanism but few quantitative models and investigations are available that have explored the mechanism for transfer of surplus generated due to coordination. In this paper, we develop a coordination mechanism through credit option such that both the parties can divide the surplus equitably after satisfying their own profit targets. Two situations are explored here; in the first situation; both the parties have no individual profit target from the business whereas in the second situation, there are individual profit target for both the parties. The proposed mechanism for division of surplus is studied through a numerical study and the impact of different parameter values on the results are examined. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Supply chain; Coordination; Target profit; Credit; Discount
1. Introduction In the current business environment, relations between buyer and manufacturer have undergone significant changes with increasing emphasis on cooperation and information sharing. A buyer and a manufacturer together constitute a simple twostage supply chain. Woo et al. (2001) suggested that the cooperation between vendor and buyer for improving the performance of a supply chain has received a great deal of attention from researchers.
* Corresponding author. Tel.: +91 3222 283734; fax: +91 3222 282272. E-mail address: [email protected] (S.P. Sarmah).
Recently, supply chains with multiple decision makers have begun to receive considerable interest due to the fact that independent entities in the supply chain acting in their own self-interest often make decisions that are sub-optimal. It is well established that the total supply chain profit under decentralized decision scenario is less than the profit achievable by a single central decision maker with complete information. Therefore decentralized control is believed to be inefficient as compared to centralized decision making. Since different entities are involved in the supply chain and centralized decision-making is difficult, much of the existing research focuses on how this gap can be reduced by implementing different classes of coordination mechanisms (e.g. contracts).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.09.047
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Tsay et al. (1999) refer to this as system wide performance improvement objective. The most commonly used term in the literature for system wide performance improvement is the channel coordination. Thomas and Grifin (1996) have given a comprehensive review of supply chain coordination. There is no universal coordination strategy that will be efficient and effective for all supply chains because the performance of a coordination strategy is supply chain characteristics dependent. Various types of coordination mechanisms have been used in supply chain coordination such as quantity discount, credit option, buy back/return policies, quantity flexibility, and commitment of purchase quantity, etc. It is natural that when doing business, each member of a supply chain wants certain fixed amount of profit from the business and it is termed here as target profit of the member. Therefore, parties will not be interested in coordinating if their target profit is not achieved. In this paper, a manufacturer and a buyer’s coordination problem is considered where target profits of both the parties are known to each other. Considering credit policy as coordination mechanism between the two parties, the problem objective is to divide the surplus equitably between the two parties. 2. Review of literature Study of integrated inventory models can be viewed as one of the origin of supply chain coordination study. Corbette and Groote (2000) mentioned that the most well established framework for studying coordination in supply chain is perhaps that of choosing lot sizes in a tightly coupled system with lot for lot production. The integrated inventory models mainly examine the benefits accrued in the system due to coordination in order quantity between the two parties. Goyal and Gupta (1989), Munson and Rosenblatt (1998), Sherafali and Co (2000) and recently Sarmah et al. (2006) have reviewed the literature on buyer vendor coordination models. In an integrated inventory model, under EOQ type environment, both the buyer and the manufacturer can be financially better off by coordinating their lot-sizing decisions. When the setup cost of the manufacturer is higher than the ordering cost of the buyer, then either party determining lot size independently would lead to inefficient outcome. In such a situation, the manufacturer induces the
buyer to order larger quantity than the economic order quantity (EOQ) of the buyer. This reduces the setup cost of the manufacturer but at the same time increases the inventory related costs of the buyer. As an inducement, the manufacturer may offer quantity discount or credit to compensate the increase in cost of the buyer. The agreed order quantity between the parties of the supply chain is referred to as the joint economic lot size (JELS). Joint economic lot size reduces the total inventory related costs of the system and improves the supply chain performance. The idea of joint optimization for buyer and vendor was initiated by Goyal (1977) and later reinforced by Bannerjee (1986). The assumption that the supplier has the larger setup cost compared to the buyer is the starting point for the joint economic lot-sizing literature. Monahan (1984) showed that a seller can increase profits by offering an all units quantity discount with a single price break point to encourage the buyer to increase the order quantity. Relaxing the assumptions of lot-for-lot policy, Lee and Rosenblatt (1986) simultaneously determined the desired buyer’s order quantity and seller’s lot size. Rosenblatt and Lee (1985) have studied a linear discount schedule in the same setting as Lee and Rosenblatt (1986). Goyal (1987) provided an improved algorithm based on Lee and Rosenblatt (1986). All these models assume that demand rate is constant and independent of the selling price. Lal and Staelin (1984), Dada and Srikanth (1987), Weng (1995a,b), etc. also established the fact that both the buyer and the supplier can be better off by coordinating lot size. Weng (1999) showed that when both parties coordinate, the order quantity and joint profit will increase and the selling price will decrease. Recently, Munson and Rosenblatt (2001) and Khouja (2003) have extended two level supply chain coordination problems to three level supply chain considering quantity discount as coordination mechanism. All these papers have attempted to develop coordination mechanisms that can achieve either an improved solution or the joint optimal solution for the vendor and the buyer(s). Further, these papers have used quantity discount as coordination mechanism between the two members/three members of the supply chain but most of these papers have ignored the mechanism to divide the surpluses generated due to coordination between the parties. The literature on inventory theory generally assumes that the buyer immediately on receiving
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the consignment makes payment to the vendor. However, it is not unusual for a vendor to provide a credit period to the buyer for settling the accounts. Shinn and Hwang (2003) have mentioned that for the sake of better production and inventory control and lower average production cost per unit, some pharmaceutical companies and agricultural machinery manufacturers in Korea prefer infrequent orders in larger lot sizes to frequent orders in smaller lots. In such a situation, rather than giving on some discount on unit selling price for larger amount of purchase, Korean manufacturers offer a larger credit period. In India also some distributors of consumer durable goods offer items on credit to the retailer. In developing countries, seller providing credit to their customers is an important form of financing for business and particularly role of trade credit is immense where growth of financial institutions is less compared to developed nations. Trade credit can be defined as the purchase of goods or services that involves delivery of goods or services at a certain date with payment at a latter date (Issakson, 2002). The advantages of trade credit are discussed in literature by authors like Wilson et al. (2000), Issakson (2002) and Shinn and Hwang (2003) and some of the advantages associated with trade credit are mentioned here (i) It can be used as a tool to compete in the market for generating sales. Through credit, supplier can gain a competitive edge over the competitor. The length of the credit period offered by the supplier can be considered as a strategy against another supplier for winning over the buyer’s order. (ii) Credit extension helps in developing good long-term relationship with customers, which ultimately helps in generating future income for the firm. Thus, credit option can generate repeat purchase orders from the customer. (iii) A manufacturer may differentiate their offering to the market by extending trade credit to their customers as a commitment to the quality of their products. Due to the credit option, the customer is getting time to inspect the quality of product before paying for the goods. (iv) Finally, use of trade credit provides information on future cash needs by allowing buyers to accumulate invoices for payment. The information enables firms to predict their cash needs better. With the advantages associated with the credit option, some authors have used delay in payment as a coordination mechanism in their models. Goyal (1985) developed a mathematical model for obtaining the economic order quantity for an item
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for which the supplier permits a fixed delay in settling the amount owed to him. Kim et al. (1995) have determined the optimum length of credit period for the product supplier sells to retailers in order to maximize profit. Delay in payment condition as incentive mechanism is found mostly adopted for inventory replenishment of deteriorating items. Aggarwall and Jaggi (1995), Hwang and Shinn (1997) and Chang et al. (2001) have adopted the credit period/delay in payment option in their models for deteriorating products. Chu et al. (1998), Chu and Chung (1998) and Jamal et al. (2000) consider the lot-sizing problem for a deteriorating item under the condition of fixed demand and permissible delay in payment. Abad and Jaggi (2003) have studied seller and buyer problem in which the end demand is price sensitive and the seller may offer trade credit to the buyer. In this paper, the credit period is used as an incentive to coordinate the activities of the two members of the supply chain where both the members have certain amount of target profit from the business and when there is no individual predecided target profit. A procedure is developed for division of profit equitably generated due to coordination between the parties. A comparison of the two coordination policies namely credit and quantity discount for two-stage supply chain coordination is also given to enable the members of the supply chain to choose the most appropriate coordination mechanism.
3. Model formulation 3.1. Assumptions and notations The following assumptions and notations are used to develop the proposed coordination model. (i) All the parameters of the model are deterministic. (ii) The manufacturer follows lot-for-lot manufacturing policy. (iii) Replenishment is instantaneous. (iv) Shortages are not allowed. Notations D annual demand of the item Q lot size or order quantity d discount rate on unit purchase price t credit period in year
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C P R
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unit manufacturing cost unit selling price unit retail price of the buyer
In the following notations, subscript m and b represent manufacturer and buyer respectively, Si hi1 hi2 hi YNPi TCb
setup/ordering cost of the firm i, i ¼ fm; bg, ($ per setup) capital cost of the firm i, i ¼ fm; bg, ($ per unit per time period) other inventory cost the firm i, i ¼ fm; bg, ($ per unit per time period) holding cost of the firm i, =hi1 + hi2 where i ¼ fm; bg, ($ per unit per time period) yearly net profit of the firm i, i ¼ fm; bg total cost of the buyer
3.2. Case I: No pre-decided individual target profit for the two members In this case, both the parties of the supply chain do not have any pre-decided target profit. The model assumes that the two parties are willing to adopt a common order quantity Q that is larger than the economic order quantity of the buyer. Due to the adoption of larger order quantity Q, the manufacturer will generate additional profit. This additional profit generated by the manufacturer should be shared equitably through credit option mechanism. Initially, the buyer orders the EOQ and the total annual inventory related cost to the buyer is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCb ¼ 2DS b hb ; ð1Þ when buyer orders at EOQ, the manufacturer’s profit consists of gross revenue minus setup cost is YNPm ¼ ðP CÞD
DS m : Qb
ð2Þ
Since the manufacturer’s setup cost is higher than that of buyer’s ordering cost, the manufacturer will induce the buyer to order in quantities larger than buyer’s EOQ, and any increase in cost at buyer’s side is compensated through credit option. The buyer’s total inventory related cost is the sum of ordering cost and holding cost. If Q > Qb, the ordering quantity of the buyer, then the buyer’s total annual inventory related cost is given by TCb1 ¼
DS b Q þ hb : 2 Q
ð3Þ
Increase in buyer’s cost is the difference of the cost at buyer’s new ordering quantity Q and the cost at the EOQ and is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DS b Q þ hb 2DS b hb : ð4Þ Z¼ 2 Q This increase in cost of the buyer is to be compensated by the manufacturer. Manufacturer’s yearly net profit (YNPm) due to changed ordering policy is YNPm ¼ Gross revenue Setup cost Amount of compensation to the buyer equal to the increase in cost of the buyer pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DS m DS b Q YNPm ¼ ðP CÞD hb þ 2DS b hb : 2 Q Q ð5Þ Differentiating Eq. (5) with respect to Q and equating to zero provides the optimum value of order quantity Q and is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðS b þ S m Þ Q ¼ : ð6Þ hb The manufacturer compensates any increase in buyer’s cost due to changed ordering policy through credit. The buyer will be interested to order a lot size Q larger than the buyer’s EOQ, provided buyer’s cost does not increase from the initial cost, which is the optimum cost for the buyer. Minimum credit time required by the buyer that compensates the increase in cost due to coordinated order quantity from Eq. (4) is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DS b Q hb 2DS b hb ¼ Dtmin hb : þ Q 2
ð7Þ
Eq. (7) results the minimum credit time required by the buyer to compensate buyer’s increase in cost and is given by sffiffiffiffiffiffiffiffi Sb Q 2S b tmin ¼ þ : ð8Þ Q hb 2D Dhb When manufacturer offers minimum credit time to the buyer, manufacturer’s maximum yearly net profit is YNPm max ¼ ðP CÞD
DS m Dtmin hm : Q
ð9Þ
The manufacturer can give credit to the buyer as long as manufacturer’s cost with credit due to coor-
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dinated order quantity Q* does not exceed the cost without credit, i.e. cost incurred when buyer is ordering at EOQ. From the difference of Eqs. (9) and (2), we get the following condition DS m DS m þ Dtmax hm ¼ : Q Qb The maximum credit time that manufacturer can offer to the buyer is given by Sm 1 1 tmax ¼ : ð10Þ hm Q b Q The two values tmin and tmax are the two bounds of the credit period over which negotiation between the two parties can take place to transfer the benefit of profit due to coordination. The additional profit available at manufacturer’s side is the difference of profit at the new order quantity of the buyer to the profit when the buyer orders at EOQ. The difference of Eqs. (9) and (2) gives the maximum extra profit manufacturer obtains due to the change of the ordering policy of the buyer. With the assumption that the surplus is to be divided equitably between the two parties by giving extra credit period to the buyer and let this extra credit period be equal to Dt. For equal distribution, the expression can be written as 1 1 2DtDhb ¼ S m D Dtmin hm Qb Q Sm 1 1 tmin hm ) Dt ¼ : ð11Þ 2hb Qb Q 2hb Through the credit giving mechanism, manufacturer is not only compensating the buyer’s extra cost but also at the same time shares equitably the surplus generated due to coordination by giving credit for an extra period of time Dt and now total credit time t becomes t ¼ tmin þ Dt:
ð12Þ
After sharing the surplus equitably, buyer’s profit is YNPb ¼ ðR P ÞD
DS b Q hb þ Dðtmin þ DtÞhb : Q 2 ð13Þ
Similarly, manufacturer’s profit after sharing the surplus equitably is given by DS m YNPm ¼ ðP CÞD Dðhm tmin þ hb DtÞ: Q ð14Þ
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3.3. Case II: With pre-decided target profit for the two members The buyer and the manufacturer of a supply chain depending on their power structure in the channel may have some target profit from the business and their target profits may be different. In the development of the model, it is presumed that both the parties are rational in deciding their target profits. Also it is assumed that both the parties know each other’s target profit and the parties are willing to coordinate as long as they achieve their target profits. The buyer’s target profit, Tb can be expressed as T b ¼ ð1 þ xÞT b min ; T b min
where DS b Qb hb and x P 0: ¼ ðR P ÞD Qb 2
ð15Þ
Similarly, target profit of the manufacturer is expressed as T m ¼ ð1 þ yÞT m min ; T m min
where DS m ¼ ðP CÞD and y P 0: Qb
ð16Þ
The profit obtained in the channel is maximum for centralized coordinated solution and it can be expressed as CP max ¼ ðR CÞD þ Dtðhb hm Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðS b þ S m Þhb : ð17Þ
It is observed from Eq. (17) that centralized profit is an increasing function of credit time and it will be maximum when t ¼ tmax . When the sum of target profit of buyer and manufacturer is higher than that of maximum profit achievable through centralized solution i.e. T b þ T m > CP max , then no solution is possible that satisfies the target profit of both the members. In such cases, the target profits of both the members should be lowered proportionately. The revised target profit of the buyer and the manufacturer under such situation can be written as Tb CP max ; Tb þ Tm Tm ¼ CP max : Tb þ Tm
T 0b ¼ T 0m
ð18Þ ð19Þ
However, when T b þ T m 6 CP max , then there exists a solution. When sum of the target profit of two
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members is less than profit obtained due to centralized solution, extra profit left over after achieving each one’s individual target profit is divided equally between the two parties. Thus, new profit of the manufacturer with initial target profit Tm can be written as 1 T m1 ¼ T m þ ½CP max ðT b þ T m Þ: 2
ð20Þ
Similarly, new profit of the buyer with initial target profit Tb of buyer becomes 1 T b1 ¼ T b þ ½CP max ðT b þ T m Þ: 2
ð21Þ
Therefore, it can be said that when T b1 ¼ kCP max , where k is a fractional value between 0 and 1, credit time required for the buyer to achieve the profit is t¼
kCP max þ
DS b Q
þ
Q 2
hb ðR P ÞD
Dhb
age cost, cost of handling, insurance, deterioration, etc. Further, cost of capital considered here refers to cost of borrowed capital to finance the inventory. 4.1. All unit quantity discount policy Many authors have used quantity discount as coordination mechanisms for supply chain coordination where the manufacturer provides discount to the buyer for ordering quantity larger than buyer’s EOQ. When the buyer’s orders quantity Q is larger than EOQ, manufacturer provides quantity discount d per unit to compensate the increase in cost of the buyer. Buyer’s profit with order quantity of Q and discount d is YNPb1 ¼ ðR P ÞD
:
ð22Þ
Correspondingly, the manufacturer’s profit is given by ð1 kÞCP max . 4. A comparison of credit period and quantity discount policies for two-stage supply chain coordination The purpose of this section is to make a comparison of the two incentive mechanisms namely credit giving policy and quantity discount policy as coordination mechanism for two-stage supply chain coordination. Arcelus and Srinivasan (1990) considered delay of payment and price discount for the case of extra ordinary purchases by the buyer and derived the relationship between credit period and discount from buyer’s perspective and mentioned that sometimes it is not possible to go for discount policy for fear of future retaliation from other manufactures. On the other hand, the buyer with less cash on hand may not be able to accept a discount offer from the manufacturer, even if it is beneficial. In such a situation, credit may be an acceptable alternative option. The equivalence of the two policies is studied here for supply chain coordination when both the parties belong to two different organizations. The two components of the holding costs are considered here separately and one component is the cost of capital and the other one is the cost of holding the item physically in the store. This includes stor-
DS b Q hb þ dPD: 2 Q
ð23Þ
Manufacturer’s total profit corresponding to the above discount is given by YNPm1 ¼ ðP CÞD
DS m dPD: Q
ð24Þ
4.2. Credit option policy For the benefit of the reader and continuity, the profit equations of buyer and manufacturer are rewritten here again. Manufacturer provides credit period t to the buyer for order quantity of Q larger than buyer’s EOQ and buyer’s profit is given by YNPb2 ¼ ðR P ÞD
DS b Q hb þ Dthb : 2 Q
ð25Þ
Manufacturer’s profit corresponding to the given credit period t is YNPm2 ¼ ðP CÞD
DS m Dthm : Q
ð26Þ
4.3. Comparison of the two coordination policies Manufacturer will be indifferent to offer credit for t time period or discount d per unit provided YNPm1 ¼ YNPm2 thm : d¼ P
which implies that ð27Þ
Similarly, the buyer will be indifferent to get credit for t time period or discount d per unit provided
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YNPb1 ¼ YNPb2 thb : d¼ P
which implies that ð28Þ
From (27) and (28), it is found that for both the policies to be equivalent it must be hm ¼ hb :
ð29Þ
Eq. (29) can be written as hm1 þ hm2 ¼ hb1 þ hb2 :
ð30Þ
Normally, there is no appreciable difference in the physical holding cost component of the two parties and the difference is mainly due to the capital cost component. Therefore, one gets hm1 ¼ hb1 :
ð31Þ
It is also well known that the cost of borrowed capital of a financially weak organization is higher than that of a financially strong organization. Therefore, the parties will be indifferent to two policies provided both the parties are either financially strong or financially weak, i.e. when hm1 ¼ hb1 . Two cases are considered here where hm1 6¼ hb1 , i.e hm 6¼ hb Case 1. When cost of borrowed capital of manufacturer is higher than that of buyer i.e. hm > hb . Proposition 1. For higher cost of borrowed capital for the manufacturer compared to the buyer, the manufacturer prefers discount policy to credit option. Proof. For a common coordinated order quantity Q(>EQOb) of the two parties, for an acceptable amount of benefit of coordination that the manufacturer is willing to give to the buyer through discount mechanism is dPD. For the same amount of benefit, the credit period t1 that manufacturer offers satisfies the equation dPD ¼ Dt1 hm :
ð32Þ
Similarly, to avail the same benefit of dPD from the manufacturer, the buyer expects a credit period t2 such that dPD ¼ Dt2 hb :
ð33Þ
From Eqs. (32) and (33); it is seen that t 1 hm ¼ t 2 h b :
ð34Þ
Since, hm > hb , hence, t1 < t2 . Therefore, manufacturer will prefer to give quantity discount to the buyer as this will increase
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manufacturer’s profit by an amount equal to (t2 t1)Dhm. h Case 2. When cost of borrowed capital of manufacturer is lower than that of buyer, i.e. hm1 < hb1 . Proposition 2. For the lower cost of borrowed capital of the manufacturer compared to the buyer, the manufacturer prefers credit policy to discount option. Proof. Following the similar line of argument as in Case 1, it can be easily established that in such situation, t1 > t2 . Thus, when the manufacturer is financially strong compared to the buyer (i.e hm < hb ) manufacturer may prefer to offer credit option to the buyer, as this will increase manufacturer’s profit by an amount = (t1 t2)Dhm. h In this situation, the buyer should also prefer this option, as it will bring down his capital needs.
5. Numerical study First, we have studied a base case considering the following data to examine the proposed procedure for division of surplus generated due to coordination and then an extensive numerical experiment is carried out. Annual demand D ¼ 1000 units, Setup cost of manufacturer S m ¼ $400 per setup, Ordering cost of the buyer S b ¼ $25 per order, Holding cost of buyer per unit per year hb ¼ $5, Holding cost of manufacturer per unit per year hm ¼ $4. Contract price between the buyer and the manufacturer P ¼ $120 per unit. Cost of acquiring the item by the manufacturer C ¼ $100 per unit, sell price of the item by the buyer R ¼ $140 per unit. Economic order quantity of the buyer Qb ¼ ffiffiffiffiffiffiffiffi q 2DS b ¼ 100 units. hb Channel optimum order quantity from Eq. (6) is Q ¼ 412:3. Minimum credit time offered by manufacturer tmin ¼ 0:0327 year. Maximum credit time manufacturer can offer tmax ¼ :757 year. Total channel profit is obtained from Eq. (17). When manufacturer offers minimum credit period to the buyer CP min ¼ $17; 971:25.
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When manufacturer offers maximum credit period to the buyer CP max ¼ $18; 695:45. Before coordination between the parties b Profit of the buyer = ðR P ÞD DS Q2b hb ¼ Qb $19; 500. m Profit of the manufacturer = ðP CÞD DS ¼ Qb $16; 000. We have considered here the case of equitable distribution of profit between the members of supply chain. When the manufacturer offers minimum credit time and there is no pre decided target profit between the partners, using Eqs. (13) and (14) respectively, one gets Profit of the buyer YNPb ¼ $20; 521:87, i.e. an increase of 5% from without coordinated situation. Profit of the manufacturer YNPm ¼ $17; 449:12, i.e. an increase of 8.3% from without coordinated situation. Further, to test the practical performance of the proposed procedure for division of surplus after coordination and the impact of the various parameters value on the results, we conducted a computational study including 243 problem instances. These problems were generated using three different settings for each parameter and the numerical experiments were carried out using Matlab 7. Experimental factors considered in the analysis are given in Table 1. Since most of buyer vendor coordination models assume that hb > hm (because of the added value to items as they are carried through the supply chain) we have also followed here the same logic in considering the value of hm and hb respectively. Further, various prices considered in the model are constant and since demand is deterministic, we have set prices at the following level for the numerical experiment C ¼ 100; P ¼ 120; R ¼ 140. From Eq. (17) and the results of the numerical experiments performed; some interesting findings can be summarized as follows: Table 1 Sm Sb hm hb D
175 20 0.5 4 200
350 100 1 8 650
Total number of problem instances
700 150 2 16 1200
3 3 3 3 3
values values values values values
243
1. There is an increase in total channel profit from minimum credit time offered by the manufacturer to the maximum credit time offered for channel coordination. 2. When inventory carrying cost (hm) of the manufacturer increases with the values of D, Sm, Sb and hb fixed at a particular level, it is found that in all the cases, total channel profit decreases. It means that when cost of borrowed capital of manufacturer increases to maintain inventory then with credit option channel profit starts decreasing. 3. With the increase of inventory carrying cost of buyer (hb), the size of the buyer’s EOQ and coordinated order quantity of the channel decreases. As the value of hb increases for the fixed value of D, Sm, Sb and hm at a particular level, it is found that in 48 cases out of the total 243 cases buyer’s individual profit increases and in rest of the cases, buyer’s individual profit decreases. In all the 48 cases where individual profit of buyer increases, it is seen that setup cost of manufacturer is substantially higher than the ordering cost of the buyer. Other way round, one can say that when the ratio of ordering cost of buyer and setup cost of manufacturer is significantly low then as holding cost of buyer (hb) increases, value of individual profit of buyer (YNPb) also increases. However the total channel profit and individual profit of the manufacturer decreases in all the cases. Due to decrease in the lot size, there is a negative impact on the profit of the manufacturer as well as on total channel profit, which is largely dependent on the lot size. 6. Summary and conclusions In this paper, we present credit scheme for coordination between two parties of supply chain and develop a simple procedure for equitable distribution of profit generated in the channel through credit mechanism when both the parties have certain amount of target profit from the business and when there is no target profit. The range of credit time over which coordination between the two members can take place and improves the channel profit has been derived here. We have also compared two incentive mechanisms, i.e. credit and discount policy and derive the conditions under which both discount and credit policies, as incentive mechanisms for coordination are equivalent. The result shows that when cost of borrowed capital
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for inventory for both the buyer and the manufacturer are same then both the policies are equivalent. When manufacturer is financially strong, manufacturer prefers credit policy to discount option and when financially weak, the reverse is true. The numerical study conducted on 243 problem shows that the coordination mechanism enhances the individual profit of both the members. We can summarize the following conclusions: First, there is an increase in channel profit when the credit time offered by the manufacturer increases from the lower bound to the upper bound. Second, there is a decrease in the total channel profit with increase in the holding cost of the manufacturer. As the cost of borrowed capital increases, the benefits accrued due to coordination through credit option decreases. Thirdly, as the inventory holding cost of buyer increases, it is observed that when the ratio of ordering cost of buyer and setup cost of manufacturer is low, individual profit of buyer increases. On the other hand, as the ratio increases, individual profit of the buyer decreases. However, in all the cases individual profit of manufacturer and total channel profit decreases. There is a negative impact on profit of the manufacturer as well in the channel due to decrease in the lot sizes. In this paper, it is assumed that both the parties share all the information. However, in practice, the members of supply chain may not be interested to disclose all information and in future, models should be developed to consider the case of division of profit under imperfect information sharing between the parties of a supply chain. Acknowledgement The authors are grateful to the anonymous reviewers for their helpful comments and suggestions. References Abad, P.L., Jaggi, C.K., 2003. A joint approach for setting unit price and the length of the credit period for a seller when end demand is price sensitive. International Journal of Production Economics 83, 115–122. Aggarwall, S.P., Jaggi, C.K., 1995. Ordering policies of deteriorating items under permissible delay in payment. Journal of Operational Research Society 46 (6), 658–662. Arcelus, F.J., Srinivasan, G., 1990. Delay of payment vs price discount for extra ordinary purchases: The buyers perspective. Engineering cost and Production Economics 19 (1–3), 273– 279.
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Bannerjee, A., 1986. A joint economic lot size model for purchaser and vendor. Decision Science 17 (3), 292–311. Chang, J.H., Hung, H.C., Dye, Y.C., 2001. An inventory model for deteriorating items with linear trend demand under the condition of permissible delay in payments. Production Planning and Control 12 (3), 274–282. Chu, P., Chung, K.H., 1998. A theorem on the determination of economic order quantity under conditions of permissible delay in payments. Computer and Operations Research 25 (1), 49–52. Chu, P., Chung, K.H., Lan, S.P., 1998. Economic order quantity of deteriorating items under permissible delay in payments. Computer and Operations Research 25 (10), 817– 824. Corbette, C.J., Groote, X.De., 2000. A supplier’s optimal quantity discount policy under asymmetric information. Management Science 46 (3), 444–450. Dada, M., Srikanth, K.N., 1987. Pricing policies for quantity discounts. Management Science 33 (10), 1247–1253. Goyal, S.K., 1977. An integrated inventory model for a single supplier single customer problem. International Journal of Production Research 15 (1), 107–111. Goyal, S.K., 1985. Economic order quantity under condition of permissible delay in payments. Journal of Operation Research Society 36 (4), 335–338. Goyal, S.K., 1987. Determination of a supplier’s economic ordering policy. Journal of the Operational Research Society 38 (9), 853–857. Goyal, S.K., Gupta, Y.P., 1989. Integrated inventory model: The buyer vendor co-ordination. European Journal of Operational Research 41 (3), 261–269. Hwang, H., Shinn, S.W., 1997. Retailer’s pricing and lot sizing policy for exponentially deteriorating products under the conditions of permissible delay in payments. Computer and Operations Research 24 (6), 539–547. Issakson, A., 2002. Trade credit in Kenyan manufacturing: Evidence from plant level data. Working paper, source:. Jamal, A.M.M., Sarkar, B., Wang, S., 2000. Optimal payment time for a retailer under permitted delay of payment to the wholesaler. International Journal of Production Economics 66, 59–66. Khouja, Moutaz, 2003. Optimizing inventory decisions in a multistage multi customer supply chain. Transportation Research Part E 39 (3), 193–208. Kim, S.J., Hwang, H., Shinn, W.S., 1995. An optimal credit policy to increase wholesaler’s profit with price dependent demand function. Production Planning and Control 6 (1), 45–50. Lal, R., Staelin, R., 1984. An approach for developing an optimal discount pricing policy. Management Science 30 (12), 1524– 1539. Lee, H.L., Rosenblatt, M.J., 1986. A generalized quantity discount-pricing model to increase supplier’s profits. Management Science 32 (9), 1177–1185. Monahan, J.P., 1984. A quantity discount pricing model to increase vendor profits. Management Science 30 (6), 720–726. Munson, L.C., Rosenblatt, J.M., 1998. Theories and realities of quantity discounts: An exploratory study. Production and Operations Management 7 (4), 352–369. Munson, L.C., Rosenblatt, J.M., 2001. Coordinating a three level supply chain with quantity discounts. IIE Transactions 33 (4), 371–384.
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Rosenblatt, M.J., Lee, H.L., 1985. Improving profitability with quantity discounts under fixed demand. IIE Transaction 17 (4), 388–395. Sarmah, S.P., Acharya, D., Goyal, S.K., 2006. Buyer vendor coordination models in supply chain management. European Journal of Operational Research 175 (1), 1–15. Sherafali, M., Co, HC., 2000. Some models for understanding the cooperation between the supplier and the buyer. International Journal of Production Research 38, 3425–3449. Shinn, W.S., Hwang, H., 2003. Optimal pricing and ordering policies for retailers under order size dependent delay in payments. Computers and Operations Research 30 (1), 35– 50. Thomas, D.J., Grifin, P.J., 1996. Coordinated supply chain management. European Journal of Operational Research 94 (1), 1–15. Tsay, A., Nahmias, S., Agarwal, N., 1999. Modeling supply chain contracts: A review. In: Tayur, S., Magazine, M., Ganeshan, R. (Eds.), Quantitative Models for Supply Chain
Management. Kluwer Academic Publishers, 1999, pp. 301– 336. Weng, Z.K., 1995a. Channel coordination and quantity discounts. Management Science 41 (9), 1509–1522. Weng, Z.K., 1995b. Modeling quantity discounts under general price sensitive demand functions: Optimal policies and relationship. European Journal of Operational Research 86 (2), 300–314. Weng, Z.K., 1999. The power of coordination decisions for short life cycles products in a manufacturing and distribution supply chain. IIE Transactions 31, 1047–1049. Wilson, N., Wetherhill, P., Summers, B., 2000. Business to Business credit: A strategic tool for the ‘New Economy’. Source:, Credit Management Research Centre. Woo, Y.Y., Hsu, S.L., Wu, S., 2001. An integrated inventory model for a single vendor and multiple buyers with ordering cost reduction. International Journal of Production Economics 73 (3), 203–221.
O.R. Applications
Coordination and profit sharing between a manufacturer and a buyer with target profit under credit option S.P. Sarmah
a,*
, D. Acharya a, S.K. Goyal
b
a
b
Department of Industrial Engineering and Management, IIT Kharagpur, Kharagpur 721302, India Department of Decision Sciences and MIS, John Molson School of Business Concordia University, Montreal, Canada H3G1M8 Received 19 September 2005; accepted 12 September 2006 Available online 22 February 2007
Abstract Several studies have focused on buyer vendor coordination through quantity discount/credit option mechanism but few quantitative models and investigations are available that have explored the mechanism for transfer of surplus generated due to coordination. In this paper, we develop a coordination mechanism through credit option such that both the parties can divide the surplus equitably after satisfying their own profit targets. Two situations are explored here; in the first situation; both the parties have no individual profit target from the business whereas in the second situation, there are individual profit target for both the parties. The proposed mechanism for division of surplus is studied through a numerical study and the impact of different parameter values on the results are examined. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Supply chain; Coordination; Target profit; Credit; Discount
1. Introduction In the current business environment, relations between buyer and manufacturer have undergone significant changes with increasing emphasis on cooperation and information sharing. A buyer and a manufacturer together constitute a simple twostage supply chain. Woo et al. (2001) suggested that the cooperation between vendor and buyer for improving the performance of a supply chain has received a great deal of attention from researchers.
* Corresponding author. Tel.: +91 3222 283734; fax: +91 3222 282272. E-mail address: [email protected] (S.P. Sarmah).
Recently, supply chains with multiple decision makers have begun to receive considerable interest due to the fact that independent entities in the supply chain acting in their own self-interest often make decisions that are sub-optimal. It is well established that the total supply chain profit under decentralized decision scenario is less than the profit achievable by a single central decision maker with complete information. Therefore decentralized control is believed to be inefficient as compared to centralized decision making. Since different entities are involved in the supply chain and centralized decision-making is difficult, much of the existing research focuses on how this gap can be reduced by implementing different classes of coordination mechanisms (e.g. contracts).
0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2006.09.047
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Tsay et al. (1999) refer to this as system wide performance improvement objective. The most commonly used term in the literature for system wide performance improvement is the channel coordination. Thomas and Grifin (1996) have given a comprehensive review of supply chain coordination. There is no universal coordination strategy that will be efficient and effective for all supply chains because the performance of a coordination strategy is supply chain characteristics dependent. Various types of coordination mechanisms have been used in supply chain coordination such as quantity discount, credit option, buy back/return policies, quantity flexibility, and commitment of purchase quantity, etc. It is natural that when doing business, each member of a supply chain wants certain fixed amount of profit from the business and it is termed here as target profit of the member. Therefore, parties will not be interested in coordinating if their target profit is not achieved. In this paper, a manufacturer and a buyer’s coordination problem is considered where target profits of both the parties are known to each other. Considering credit policy as coordination mechanism between the two parties, the problem objective is to divide the surplus equitably between the two parties. 2. Review of literature Study of integrated inventory models can be viewed as one of the origin of supply chain coordination study. Corbette and Groote (2000) mentioned that the most well established framework for studying coordination in supply chain is perhaps that of choosing lot sizes in a tightly coupled system with lot for lot production. The integrated inventory models mainly examine the benefits accrued in the system due to coordination in order quantity between the two parties. Goyal and Gupta (1989), Munson and Rosenblatt (1998), Sherafali and Co (2000) and recently Sarmah et al. (2006) have reviewed the literature on buyer vendor coordination models. In an integrated inventory model, under EOQ type environment, both the buyer and the manufacturer can be financially better off by coordinating their lot-sizing decisions. When the setup cost of the manufacturer is higher than the ordering cost of the buyer, then either party determining lot size independently would lead to inefficient outcome. In such a situation, the manufacturer induces the
buyer to order larger quantity than the economic order quantity (EOQ) of the buyer. This reduces the setup cost of the manufacturer but at the same time increases the inventory related costs of the buyer. As an inducement, the manufacturer may offer quantity discount or credit to compensate the increase in cost of the buyer. The agreed order quantity between the parties of the supply chain is referred to as the joint economic lot size (JELS). Joint economic lot size reduces the total inventory related costs of the system and improves the supply chain performance. The idea of joint optimization for buyer and vendor was initiated by Goyal (1977) and later reinforced by Bannerjee (1986). The assumption that the supplier has the larger setup cost compared to the buyer is the starting point for the joint economic lot-sizing literature. Monahan (1984) showed that a seller can increase profits by offering an all units quantity discount with a single price break point to encourage the buyer to increase the order quantity. Relaxing the assumptions of lot-for-lot policy, Lee and Rosenblatt (1986) simultaneously determined the desired buyer’s order quantity and seller’s lot size. Rosenblatt and Lee (1985) have studied a linear discount schedule in the same setting as Lee and Rosenblatt (1986). Goyal (1987) provided an improved algorithm based on Lee and Rosenblatt (1986). All these models assume that demand rate is constant and independent of the selling price. Lal and Staelin (1984), Dada and Srikanth (1987), Weng (1995a,b), etc. also established the fact that both the buyer and the supplier can be better off by coordinating lot size. Weng (1999) showed that when both parties coordinate, the order quantity and joint profit will increase and the selling price will decrease. Recently, Munson and Rosenblatt (2001) and Khouja (2003) have extended two level supply chain coordination problems to three level supply chain considering quantity discount as coordination mechanism. All these papers have attempted to develop coordination mechanisms that can achieve either an improved solution or the joint optimal solution for the vendor and the buyer(s). Further, these papers have used quantity discount as coordination mechanism between the two members/three members of the supply chain but most of these papers have ignored the mechanism to divide the surpluses generated due to coordination between the parties. The literature on inventory theory generally assumes that the buyer immediately on receiving
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the consignment makes payment to the vendor. However, it is not unusual for a vendor to provide a credit period to the buyer for settling the accounts. Shinn and Hwang (2003) have mentioned that for the sake of better production and inventory control and lower average production cost per unit, some pharmaceutical companies and agricultural machinery manufacturers in Korea prefer infrequent orders in larger lot sizes to frequent orders in smaller lots. In such a situation, rather than giving on some discount on unit selling price for larger amount of purchase, Korean manufacturers offer a larger credit period. In India also some distributors of consumer durable goods offer items on credit to the retailer. In developing countries, seller providing credit to their customers is an important form of financing for business and particularly role of trade credit is immense where growth of financial institutions is less compared to developed nations. Trade credit can be defined as the purchase of goods or services that involves delivery of goods or services at a certain date with payment at a latter date (Issakson, 2002). The advantages of trade credit are discussed in literature by authors like Wilson et al. (2000), Issakson (2002) and Shinn and Hwang (2003) and some of the advantages associated with trade credit are mentioned here (i) It can be used as a tool to compete in the market for generating sales. Through credit, supplier can gain a competitive edge over the competitor. The length of the credit period offered by the supplier can be considered as a strategy against another supplier for winning over the buyer’s order. (ii) Credit extension helps in developing good long-term relationship with customers, which ultimately helps in generating future income for the firm. Thus, credit option can generate repeat purchase orders from the customer. (iii) A manufacturer may differentiate their offering to the market by extending trade credit to their customers as a commitment to the quality of their products. Due to the credit option, the customer is getting time to inspect the quality of product before paying for the goods. (iv) Finally, use of trade credit provides information on future cash needs by allowing buyers to accumulate invoices for payment. The information enables firms to predict their cash needs better. With the advantages associated with the credit option, some authors have used delay in payment as a coordination mechanism in their models. Goyal (1985) developed a mathematical model for obtaining the economic order quantity for an item
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for which the supplier permits a fixed delay in settling the amount owed to him. Kim et al. (1995) have determined the optimum length of credit period for the product supplier sells to retailers in order to maximize profit. Delay in payment condition as incentive mechanism is found mostly adopted for inventory replenishment of deteriorating items. Aggarwall and Jaggi (1995), Hwang and Shinn (1997) and Chang et al. (2001) have adopted the credit period/delay in payment option in their models for deteriorating products. Chu et al. (1998), Chu and Chung (1998) and Jamal et al. (2000) consider the lot-sizing problem for a deteriorating item under the condition of fixed demand and permissible delay in payment. Abad and Jaggi (2003) have studied seller and buyer problem in which the end demand is price sensitive and the seller may offer trade credit to the buyer. In this paper, the credit period is used as an incentive to coordinate the activities of the two members of the supply chain where both the members have certain amount of target profit from the business and when there is no individual predecided target profit. A procedure is developed for division of profit equitably generated due to coordination between the parties. A comparison of the two coordination policies namely credit and quantity discount for two-stage supply chain coordination is also given to enable the members of the supply chain to choose the most appropriate coordination mechanism.
3. Model formulation 3.1. Assumptions and notations The following assumptions and notations are used to develop the proposed coordination model. (i) All the parameters of the model are deterministic. (ii) The manufacturer follows lot-for-lot manufacturing policy. (iii) Replenishment is instantaneous. (iv) Shortages are not allowed. Notations D annual demand of the item Q lot size or order quantity d discount rate on unit purchase price t credit period in year
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C P R
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unit manufacturing cost unit selling price unit retail price of the buyer
In the following notations, subscript m and b represent manufacturer and buyer respectively, Si hi1 hi2 hi YNPi TCb
setup/ordering cost of the firm i, i ¼ fm; bg, ($ per setup) capital cost of the firm i, i ¼ fm; bg, ($ per unit per time period) other inventory cost the firm i, i ¼ fm; bg, ($ per unit per time period) holding cost of the firm i, =hi1 + hi2 where i ¼ fm; bg, ($ per unit per time period) yearly net profit of the firm i, i ¼ fm; bg total cost of the buyer
3.2. Case I: No pre-decided individual target profit for the two members In this case, both the parties of the supply chain do not have any pre-decided target profit. The model assumes that the two parties are willing to adopt a common order quantity Q that is larger than the economic order quantity of the buyer. Due to the adoption of larger order quantity Q, the manufacturer will generate additional profit. This additional profit generated by the manufacturer should be shared equitably through credit option mechanism. Initially, the buyer orders the EOQ and the total annual inventory related cost to the buyer is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCb ¼ 2DS b hb ; ð1Þ when buyer orders at EOQ, the manufacturer’s profit consists of gross revenue minus setup cost is YNPm ¼ ðP CÞD
DS m : Qb
ð2Þ
Since the manufacturer’s setup cost is higher than that of buyer’s ordering cost, the manufacturer will induce the buyer to order in quantities larger than buyer’s EOQ, and any increase in cost at buyer’s side is compensated through credit option. The buyer’s total inventory related cost is the sum of ordering cost and holding cost. If Q > Qb, the ordering quantity of the buyer, then the buyer’s total annual inventory related cost is given by TCb1 ¼
DS b Q þ hb : 2 Q
ð3Þ
Increase in buyer’s cost is the difference of the cost at buyer’s new ordering quantity Q and the cost at the EOQ and is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DS b Q þ hb 2DS b hb : ð4Þ Z¼ 2 Q This increase in cost of the buyer is to be compensated by the manufacturer. Manufacturer’s yearly net profit (YNPm) due to changed ordering policy is YNPm ¼ Gross revenue Setup cost Amount of compensation to the buyer equal to the increase in cost of the buyer pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DS m DS b Q YNPm ¼ ðP CÞD hb þ 2DS b hb : 2 Q Q ð5Þ Differentiating Eq. (5) with respect to Q and equating to zero provides the optimum value of order quantity Q and is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðS b þ S m Þ Q ¼ : ð6Þ hb The manufacturer compensates any increase in buyer’s cost due to changed ordering policy through credit. The buyer will be interested to order a lot size Q larger than the buyer’s EOQ, provided buyer’s cost does not increase from the initial cost, which is the optimum cost for the buyer. Minimum credit time required by the buyer that compensates the increase in cost due to coordinated order quantity from Eq. (4) is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DS b Q hb 2DS b hb ¼ Dtmin hb : þ Q 2
ð7Þ
Eq. (7) results the minimum credit time required by the buyer to compensate buyer’s increase in cost and is given by sffiffiffiffiffiffiffiffi Sb Q 2S b tmin ¼ þ : ð8Þ Q hb 2D Dhb When manufacturer offers minimum credit time to the buyer, manufacturer’s maximum yearly net profit is YNPm max ¼ ðP CÞD
DS m Dtmin hm : Q
ð9Þ
The manufacturer can give credit to the buyer as long as manufacturer’s cost with credit due to coor-
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dinated order quantity Q* does not exceed the cost without credit, i.e. cost incurred when buyer is ordering at EOQ. From the difference of Eqs. (9) and (2), we get the following condition DS m DS m þ Dtmax hm ¼ : Q Qb The maximum credit time that manufacturer can offer to the buyer is given by Sm 1 1 tmax ¼ : ð10Þ hm Q b Q The two values tmin and tmax are the two bounds of the credit period over which negotiation between the two parties can take place to transfer the benefit of profit due to coordination. The additional profit available at manufacturer’s side is the difference of profit at the new order quantity of the buyer to the profit when the buyer orders at EOQ. The difference of Eqs. (9) and (2) gives the maximum extra profit manufacturer obtains due to the change of the ordering policy of the buyer. With the assumption that the surplus is to be divided equitably between the two parties by giving extra credit period to the buyer and let this extra credit period be equal to Dt. For equal distribution, the expression can be written as 1 1 2DtDhb ¼ S m D Dtmin hm Qb Q Sm 1 1 tmin hm ) Dt ¼ : ð11Þ 2hb Qb Q 2hb Through the credit giving mechanism, manufacturer is not only compensating the buyer’s extra cost but also at the same time shares equitably the surplus generated due to coordination by giving credit for an extra period of time Dt and now total credit time t becomes t ¼ tmin þ Dt:
ð12Þ
After sharing the surplus equitably, buyer’s profit is YNPb ¼ ðR P ÞD
DS b Q hb þ Dðtmin þ DtÞhb : Q 2 ð13Þ
Similarly, manufacturer’s profit after sharing the surplus equitably is given by DS m YNPm ¼ ðP CÞD Dðhm tmin þ hb DtÞ: Q ð14Þ
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3.3. Case II: With pre-decided target profit for the two members The buyer and the manufacturer of a supply chain depending on their power structure in the channel may have some target profit from the business and their target profits may be different. In the development of the model, it is presumed that both the parties are rational in deciding their target profits. Also it is assumed that both the parties know each other’s target profit and the parties are willing to coordinate as long as they achieve their target profits. The buyer’s target profit, Tb can be expressed as T b ¼ ð1 þ xÞT b min ; T b min
where DS b Qb hb and x P 0: ¼ ðR P ÞD Qb 2
ð15Þ
Similarly, target profit of the manufacturer is expressed as T m ¼ ð1 þ yÞT m min ; T m min
where DS m ¼ ðP CÞD and y P 0: Qb
ð16Þ
The profit obtained in the channel is maximum for centralized coordinated solution and it can be expressed as CP max ¼ ðR CÞD þ Dtðhb hm Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðS b þ S m Þhb : ð17Þ
It is observed from Eq. (17) that centralized profit is an increasing function of credit time and it will be maximum when t ¼ tmax . When the sum of target profit of buyer and manufacturer is higher than that of maximum profit achievable through centralized solution i.e. T b þ T m > CP max , then no solution is possible that satisfies the target profit of both the members. In such cases, the target profits of both the members should be lowered proportionately. The revised target profit of the buyer and the manufacturer under such situation can be written as Tb CP max ; Tb þ Tm Tm ¼ CP max : Tb þ Tm
T 0b ¼ T 0m
ð18Þ ð19Þ
However, when T b þ T m 6 CP max , then there exists a solution. When sum of the target profit of two
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members is less than profit obtained due to centralized solution, extra profit left over after achieving each one’s individual target profit is divided equally between the two parties. Thus, new profit of the manufacturer with initial target profit Tm can be written as 1 T m1 ¼ T m þ ½CP max ðT b þ T m Þ: 2
ð20Þ
Similarly, new profit of the buyer with initial target profit Tb of buyer becomes 1 T b1 ¼ T b þ ½CP max ðT b þ T m Þ: 2
ð21Þ
Therefore, it can be said that when T b1 ¼ kCP max , where k is a fractional value between 0 and 1, credit time required for the buyer to achieve the profit is t¼
kCP max þ
DS b Q
þ
Q 2
hb ðR P ÞD
Dhb
age cost, cost of handling, insurance, deterioration, etc. Further, cost of capital considered here refers to cost of borrowed capital to finance the inventory. 4.1. All unit quantity discount policy Many authors have used quantity discount as coordination mechanisms for supply chain coordination where the manufacturer provides discount to the buyer for ordering quantity larger than buyer’s EOQ. When the buyer’s orders quantity Q is larger than EOQ, manufacturer provides quantity discount d per unit to compensate the increase in cost of the buyer. Buyer’s profit with order quantity of Q and discount d is YNPb1 ¼ ðR P ÞD
:
ð22Þ
Correspondingly, the manufacturer’s profit is given by ð1 kÞCP max . 4. A comparison of credit period and quantity discount policies for two-stage supply chain coordination The purpose of this section is to make a comparison of the two incentive mechanisms namely credit giving policy and quantity discount policy as coordination mechanism for two-stage supply chain coordination. Arcelus and Srinivasan (1990) considered delay of payment and price discount for the case of extra ordinary purchases by the buyer and derived the relationship between credit period and discount from buyer’s perspective and mentioned that sometimes it is not possible to go for discount policy for fear of future retaliation from other manufactures. On the other hand, the buyer with less cash on hand may not be able to accept a discount offer from the manufacturer, even if it is beneficial. In such a situation, credit may be an acceptable alternative option. The equivalence of the two policies is studied here for supply chain coordination when both the parties belong to two different organizations. The two components of the holding costs are considered here separately and one component is the cost of capital and the other one is the cost of holding the item physically in the store. This includes stor-
DS b Q hb þ dPD: 2 Q
ð23Þ
Manufacturer’s total profit corresponding to the above discount is given by YNPm1 ¼ ðP CÞD
DS m dPD: Q
ð24Þ
4.2. Credit option policy For the benefit of the reader and continuity, the profit equations of buyer and manufacturer are rewritten here again. Manufacturer provides credit period t to the buyer for order quantity of Q larger than buyer’s EOQ and buyer’s profit is given by YNPb2 ¼ ðR P ÞD
DS b Q hb þ Dthb : 2 Q
ð25Þ
Manufacturer’s profit corresponding to the given credit period t is YNPm2 ¼ ðP CÞD
DS m Dthm : Q
ð26Þ
4.3. Comparison of the two coordination policies Manufacturer will be indifferent to offer credit for t time period or discount d per unit provided YNPm1 ¼ YNPm2 thm : d¼ P
which implies that ð27Þ
Similarly, the buyer will be indifferent to get credit for t time period or discount d per unit provided
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YNPb1 ¼ YNPb2 thb : d¼ P
which implies that ð28Þ
From (27) and (28), it is found that for both the policies to be equivalent it must be hm ¼ hb :
ð29Þ
Eq. (29) can be written as hm1 þ hm2 ¼ hb1 þ hb2 :
ð30Þ
Normally, there is no appreciable difference in the physical holding cost component of the two parties and the difference is mainly due to the capital cost component. Therefore, one gets hm1 ¼ hb1 :
ð31Þ
It is also well known that the cost of borrowed capital of a financially weak organization is higher than that of a financially strong organization. Therefore, the parties will be indifferent to two policies provided both the parties are either financially strong or financially weak, i.e. when hm1 ¼ hb1 . Two cases are considered here where hm1 6¼ hb1 , i.e hm 6¼ hb Case 1. When cost of borrowed capital of manufacturer is higher than that of buyer i.e. hm > hb . Proposition 1. For higher cost of borrowed capital for the manufacturer compared to the buyer, the manufacturer prefers discount policy to credit option. Proof. For a common coordinated order quantity Q(>EQOb) of the two parties, for an acceptable amount of benefit of coordination that the manufacturer is willing to give to the buyer through discount mechanism is dPD. For the same amount of benefit, the credit period t1 that manufacturer offers satisfies the equation dPD ¼ Dt1 hm :
ð32Þ
Similarly, to avail the same benefit of dPD from the manufacturer, the buyer expects a credit period t2 such that dPD ¼ Dt2 hb :
ð33Þ
From Eqs. (32) and (33); it is seen that t 1 hm ¼ t 2 h b :
ð34Þ
Since, hm > hb , hence, t1 < t2 . Therefore, manufacturer will prefer to give quantity discount to the buyer as this will increase
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manufacturer’s profit by an amount equal to (t2 t1)Dhm. h Case 2. When cost of borrowed capital of manufacturer is lower than that of buyer, i.e. hm1 < hb1 . Proposition 2. For the lower cost of borrowed capital of the manufacturer compared to the buyer, the manufacturer prefers credit policy to discount option. Proof. Following the similar line of argument as in Case 1, it can be easily established that in such situation, t1 > t2 . Thus, when the manufacturer is financially strong compared to the buyer (i.e hm < hb ) manufacturer may prefer to offer credit option to the buyer, as this will increase manufacturer’s profit by an amount = (t1 t2)Dhm. h In this situation, the buyer should also prefer this option, as it will bring down his capital needs.
5. Numerical study First, we have studied a base case considering the following data to examine the proposed procedure for division of surplus generated due to coordination and then an extensive numerical experiment is carried out. Annual demand D ¼ 1000 units, Setup cost of manufacturer S m ¼ $400 per setup, Ordering cost of the buyer S b ¼ $25 per order, Holding cost of buyer per unit per year hb ¼ $5, Holding cost of manufacturer per unit per year hm ¼ $4. Contract price between the buyer and the manufacturer P ¼ $120 per unit. Cost of acquiring the item by the manufacturer C ¼ $100 per unit, sell price of the item by the buyer R ¼ $140 per unit. Economic order quantity of the buyer Qb ¼ ffiffiffiffiffiffiffiffi q 2DS b ¼ 100 units. hb Channel optimum order quantity from Eq. (6) is Q ¼ 412:3. Minimum credit time offered by manufacturer tmin ¼ 0:0327 year. Maximum credit time manufacturer can offer tmax ¼ :757 year. Total channel profit is obtained from Eq. (17). When manufacturer offers minimum credit period to the buyer CP min ¼ $17; 971:25.
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When manufacturer offers maximum credit period to the buyer CP max ¼ $18; 695:45. Before coordination between the parties b Profit of the buyer = ðR P ÞD DS Q2b hb ¼ Qb $19; 500. m Profit of the manufacturer = ðP CÞD DS ¼ Qb $16; 000. We have considered here the case of equitable distribution of profit between the members of supply chain. When the manufacturer offers minimum credit time and there is no pre decided target profit between the partners, using Eqs. (13) and (14) respectively, one gets Profit of the buyer YNPb ¼ $20; 521:87, i.e. an increase of 5% from without coordinated situation. Profit of the manufacturer YNPm ¼ $17; 449:12, i.e. an increase of 8.3% from without coordinated situation. Further, to test the practical performance of the proposed procedure for division of surplus after coordination and the impact of the various parameters value on the results, we conducted a computational study including 243 problem instances. These problems were generated using three different settings for each parameter and the numerical experiments were carried out using Matlab 7. Experimental factors considered in the analysis are given in Table 1. Since most of buyer vendor coordination models assume that hb > hm (because of the added value to items as they are carried through the supply chain) we have also followed here the same logic in considering the value of hm and hb respectively. Further, various prices considered in the model are constant and since demand is deterministic, we have set prices at the following level for the numerical experiment C ¼ 100; P ¼ 120; R ¼ 140. From Eq. (17) and the results of the numerical experiments performed; some interesting findings can be summarized as follows: Table 1 Sm Sb hm hb D
175 20 0.5 4 200
350 100 1 8 650
Total number of problem instances
700 150 2 16 1200
3 3 3 3 3
values values values values values
243
1. There is an increase in total channel profit from minimum credit time offered by the manufacturer to the maximum credit time offered for channel coordination. 2. When inventory carrying cost (hm) of the manufacturer increases with the values of D, Sm, Sb and hb fixed at a particular level, it is found that in all the cases, total channel profit decreases. It means that when cost of borrowed capital of manufacturer increases to maintain inventory then with credit option channel profit starts decreasing. 3. With the increase of inventory carrying cost of buyer (hb), the size of the buyer’s EOQ and coordinated order quantity of the channel decreases. As the value of hb increases for the fixed value of D, Sm, Sb and hm at a particular level, it is found that in 48 cases out of the total 243 cases buyer’s individual profit increases and in rest of the cases, buyer’s individual profit decreases. In all the 48 cases where individual profit of buyer increases, it is seen that setup cost of manufacturer is substantially higher than the ordering cost of the buyer. Other way round, one can say that when the ratio of ordering cost of buyer and setup cost of manufacturer is significantly low then as holding cost of buyer (hb) increases, value of individual profit of buyer (YNPb) also increases. However the total channel profit and individual profit of the manufacturer decreases in all the cases. Due to decrease in the lot size, there is a negative impact on the profit of the manufacturer as well as on total channel profit, which is largely dependent on the lot size. 6. Summary and conclusions In this paper, we present credit scheme for coordination between two parties of supply chain and develop a simple procedure for equitable distribution of profit generated in the channel through credit mechanism when both the parties have certain amount of target profit from the business and when there is no target profit. The range of credit time over which coordination between the two members can take place and improves the channel profit has been derived here. We have also compared two incentive mechanisms, i.e. credit and discount policy and derive the conditions under which both discount and credit policies, as incentive mechanisms for coordination are equivalent. The result shows that when cost of borrowed capital
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for inventory for both the buyer and the manufacturer are same then both the policies are equivalent. When manufacturer is financially strong, manufacturer prefers credit policy to discount option and when financially weak, the reverse is true. The numerical study conducted on 243 problem shows that the coordination mechanism enhances the individual profit of both the members. We can summarize the following conclusions: First, there is an increase in channel profit when the credit time offered by the manufacturer increases from the lower bound to the upper bound. Second, there is a decrease in the total channel profit with increase in the holding cost of the manufacturer. As the cost of borrowed capital increases, the benefits accrued due to coordination through credit option decreases. Thirdly, as the inventory holding cost of buyer increases, it is observed that when the ratio of ordering cost of buyer and setup cost of manufacturer is low, individual profit of buyer increases. On the other hand, as the ratio increases, individual profit of the buyer decreases. However, in all the cases individual profit of manufacturer and total channel profit decreases. There is a negative impact on profit of the manufacturer as well in the channel due to decrease in the lot sizes. In this paper, it is assumed that both the parties share all the information. However, in practice, the members of supply chain may not be interested to disclose all information and in future, models should be developed to consider the case of division of profit under imperfect information sharing between the parties of a supply chain. Acknowledgement The authors are grateful to the anonymous reviewers for their helpful comments and suggestions. References Abad, P.L., Jaggi, C.K., 2003. A joint approach for setting unit price and the length of the credit period for a seller when end demand is price sensitive. International Journal of Production Economics 83, 115–122. Aggarwall, S.P., Jaggi, C.K., 1995. Ordering policies of deteriorating items under permissible delay in payment. Journal of Operational Research Society 46 (6), 658–662. Arcelus, F.J., Srinivasan, G., 1990. Delay of payment vs price discount for extra ordinary purchases: The buyers perspective. Engineering cost and Production Economics 19 (1–3), 273– 279.
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