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multi-buyer supply chain with credit option
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Int. J. Production Economics 111 (2008) 676–685 www.elsevier.com/locate/ijpe
Coordination of a single-manufacturer/multi-buyer supply chain with credit option S.P. Sarmaha, D. Acharyaa, S.K. Goyalb, a
Department of Industrial Engineering and Management, IIT Kharagpur, Kharagpur 721302, India Faculty of Commerce and Administration, Department of Decision Sciences and MIS, Concordia University, 1455 De Maisonneuve Blvd., West, Montreal, Que., Canada H3G 1M8
b
Received 1 March 2005; accepted 2 April 2007 Available online 18 April 2007
Abstract One of the important issues in supply chain management is the coordination between manufacturers and multi-buyer. A single manufacturer supplying a product to single buyer is hard to find in today’s business environment. This paper investigates a coordination problem in a single-manufacturer and multiple heterogeneous buyers situation. Two typical cases are explored here: (i) ex-site delivery case considering manufacturer dominance where manufacturer with larger production lot size delivers the item to the group of heterogeneous buyers at common replenishment time through common carrier and (ii) ex-factory delivery case with buyer’s dominance and common replenishment time for delivery. The paper develops coordination mechanism that allows improvement of supply chain performance. Acceptance of a coordinated solution depends on the share of surplus generated due to coordination between the parties of the supply chain. The paper also focuses on how negotiation can be carried out to get due share of extra savings after coordination from the business by each party. r 2007 Elsevier B.V. All rights reserved. Keywords: Supply chain; Coordination; Credit; Payment delay; Negotiation; Multiple buyers
1. Introduction In the recent times, people from both academics and industry have shown keen interest on supply chain management (SCM) research realizing its significant potential to improve efficiency of operations and reduced costs. Since the scope for improvement within the organization is gradually decreasing, both academicians and industry leaders Corresponding author. Tel.: +1 514 848 2424x2966; fax: +1 514 848 2824. E-mail address: [email protected] (S.K. Goyal).
have given thought about how to improve the performance of the firm in terms of cost, delivery time, customer satisfaction, etc. They are looking for new alternatives for integrating business operations beyond the organization’s boundary and trying to align and coordinate the business processes and activities of the channel members so that it will improve the performance and effectiveness of the supply chain. The single buyer and the single vendor together constitute a simple two-stage supply chain and the problem is considered as the building block of any supply chain. When a single owner owns both the
0925-5273/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2007.04.003
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business entities then it is a case of centralized supply chain and in the decentralized supply chain case, different individuals own different business entities and most of today’s supply chains belong to the latter category. Thomas and Grifin (1996) in their paper mentioned that effective SCM requires planning and coordination among the various channel members including manufacturers, retailers and intermediaries if any. Further, Mentzer (2001) has mentioned SCM as the systematic, strategic coordination of the traditional business functions within a particular company and across business within the supply chain for the purpose of improving the long-term performance of the individual companies and the supply chain as a whole. In a decentralized supply chain, there may be conflict between the different corporate entities of the supply chain as the members of the supply chain may pursue their own objectives and one party may be dominant over the other to achieve his own goal. Munson et al. (1999) have discussed about misuse of power by the channel leader. Sometimes, it is noticed that a powerful manufacturer in the channel controls dependent suppliers, subcontractors and retailers. However, to improve the performance of the supply chain there is a need to coordinate the decisions between the members of the supply chain and the dominant member if there exists, should take a lead role to make the coordination a win–win proposition for both. In today’s business environment, one often encounters a two-stage supply chain where a manufacturer supplies to multiple buyers and generally these buyers are found to be located in different places. Swenseth and Godfrey (2002) have mentioned that around 50% of total annual logistics cost of a product can be attributed to transportation cost. When a supplier delivers a product to a set of buyers, consolidation of their requirements to form a larger load (a full truck load) makes sense as it leads to cost saving and here one exploits scale economies inherent to transportation of the materials. This paper deals with a manufacturer supplying a product to a group of heterogeneous buyers where the demand of buyers may be different from one another. Each buyer’s demand is constant and satisfied without delay. Two cases are considered here where in the first case transportation cost is borne by the manufacturer and in the second case, it is borne by the buyers. While the former case usually applies to the ex-site delivery condition with transportation cost included
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in the price, the latter is usually the ex-factory delivery condition of supply. In the development of the model here, it is assumed that the buyers are closely located to each other. When acting independently, each buyer orders at economic order quantity to maximize own cost. According to first case, if each order from a buyer is handled independently, maybe in less than full truckload, the transportation cost of the manufacturer increases and reduces his profitability. On the other hand, in the second case, the transportation cost of the buyer increases. A possible method of reducing the adverse effect of transportation cost is freight consolidation. However, this requires coordinated delivery at fixed intervals to the set of buyers who share a common carrier. While in the first case, the manufacturer gains from reduced transportation cost, the customers are benefited from the second case. A coordinated delivery through freight consolidation is possible if the manufacturer induces the buyers to have the delivery of a product at fixed intervals (cycle time) by offering credit (or price discount) to offset their increase in cost and possibly to increase their profit. In the second case, such a coordinated delivery policy can be implemented if the buyers as a group compensate the increase in cost of the manufacturer. Coordinated replenishment has been widely applied in many industries including refrigerated goods replenishment for supermarket chain stores (Hammer, 2001). As far as refrigerated goods are concerned, single-item distribution to a single buyer lacks the economies of scale due to its highly dedicated transportation system and therefore consolidation of demand from number of buyers and supplying it through a full truck load is the best alternative. Hammer (2001) mentioned that General Mills Yougart and Land O_lakes butter delivered by same truck; enroute to the same supermarket is a good example of common replenishment practices. Cetinkaya and Lee (2000) in their work also have justified the benefits generated from such a shipment consolidation arrangement. 2. Review of literature It is rather difficult to separate out the singlemanufacturer/single-buyer literature from the single-manufacturer/multiple-buyers literature as in some cases, authors after developing the model for single-manufacturer and single-buyer case extended it to the multiple-buyers situation and these models
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are mostly confined to a homogeneous group of buyers (Lal and Staelin, 1984; Drezner and Wesolowsky, 1989). A single vendor supplying a product to many different buyers is studied by authors such as Kim and Hwang (1988), Weng and Wong (1993) and recently by Siajidi et al. (2006) and Chan and Kingsman (2007). Zahir and Sarker (1991) have considered a price-dependent demand function for multiple regional wholesalers who are served by a single manufacturer. Woo et al. (2001) have studied an integrated inventory system where a single vendor purchases and processes raw materials in order to deliver it to multiple retailers. Wang (2002) has presented an analysis for a supplier’s quantity discount decision for heterogeneous buyers. With the assumption that the vendor follows a lot-for-lot policy, Viswanathan and Piplani (2001) have shown that a vendor could implement the common replenishment period mechanism by offering price discounts to a set of heterogeneous buyers in a single vendor, multi-buyer supply chain. Mishra (2004) has extended the above model of Viswanathan and Piplani (2001) by considering selective discount policy. Many of these authors assume that the buyers receive items at fixed interval and the manufacturer produces at each interval a batch size that is equal to the sum of the delivery lot sizes of the buyers. This eliminates the problem of finding inventory holding cost of the manufacturer. Banerjee and Burton (1994) in their work on a single-manufacturer and multiple-buyers case have mentioned that when the system consists of a very large number of buyers with relatively small order quantities and cycle time, then the assumption that the manufacturer’s available inventory is depleted at a uniform rate holds good at least approximately. A similar assumption is also made here in the development of the model. In the literature of inventory management, it is seen that the supplier provides incentive to the buyer in the form of discount or credit. The practice of suppliers providing credit to their customers is not new and in competitive business environments, credit is an important form of the financing for business and particularly in developing countries; the role of trade credit is immense where growth of financial institution 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 later date (Issakson, 2002). There are several advantages
of trade credit such as: (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 competitors. The length of the credit period offered by the suppliers 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 of the customers (Wilson et al., 2000). (iii) A supplier may differentiate their offering to the market by extending trade credit to their customers as a commitment to the quality of their products. By offering credit option, customers are allowed time to inspect quality of the product before paying for the goods. (iv) A larger firm has better access to financial institutions. So very often a smaller firm that does not have sufficient funds has to depend on the trade credit offered by the larger business firm. (v) 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. Recognizing the advantages associated with the credit option, many authors have used delay in payment or credit option as an incentive mechanism in their inventory control models. Goyal (1985) suggested a mathematical model for obtaining the economic order quantity for an item for which the supplier permits a fixed delay in settling the account. Kingsman (1983) and Chapman et al. (1985) also have used delay in payment option in the development of their mathematical models. Kim et al. (1995) have determined the optimum length of credit period for the product that the supplier sells to retailers in order to maximize his profit. Shinn and Hwang (2003) have considered order sizedependent delay in payment and have developed a model for determination of retailer’s optimal price and order size simultaneously. Jamal et al. (2000) have studied the optimal cycle and payment time for a retailer in a deteriorating item inventory situation where a wholesaler allows a specified credit period to the retailer for payment without penalty. Moses and Seshadri (2000) also have used credit period in supply chain coordination. Considering the relevance of credit, this paper also uses the credit option as a mechanism to develop coordination between the two parties of the supply chain. Unlike existing inventory models with
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credit option, this paper develops two new models that integrate the transportation cost explicitly in the single vendor multiple-buyers’ situation. The objective in both the models is to view the system as an integrated whole and determine the optimum production and coordinated shipment interval that minimizes the average total cost per unit time. A procedure is devised to coordinate between the different members of the channel and solve the proposed model. Numerical examples are used to show the savings due to coordinated replenishment time. Further, how the extra savings due to coordination can be shared between the two parties is also discussed so that coordination can be a win–win proposition for both the parties. 3. Model formulation for ex-site delivery case
K
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integer lot size multiplier (a positive value, decision variable) production rate of the manufacturer utilization rate where 0ormp1
P (D/ P) ¼ rm Qv economic production quantity of the manufacturer TCBi total relevant cost per unit time of ith buyer TCM total relevant cost per unit time of the manufacturer Ci individual transportation cost per delivery borne by the manufacturer Cc common transportation cost per delivery borne by the manufacturer bm average inventory factor at manufacturer’s side ¼ (K1)(K2)rm
Here, the transportation cost is borne by the manufacturer. The responsibility to deliver the item at the buyer’s site lies with the manufacturer.
For the derivation of the above expression, one can see Joglekar (1988).
3.1. Assumptions and notation
3.2. Individual optimal policy of buyer and the manufacturer
Assumptions: (i) demand of each buyer is known and constant, (ii) no shortages are allowed, (iii) there are large number of buyers with small order size to make uniform depletion of manufacturer’s inventory, (iv) cost of transporting through common carrier to several buyers is assumed to be less than the sum of the costs of supplying to the buyers in separate carriers. Notation: D Sbi Sm hbi hm ti t
tci T
total annual demand ordering cost of ith buyer where i ¼ 1–n setup cost of the manufacturer holding cost of ith buyer where i ¼ 1–n holding cost of the manufacturer individual optimum order interval of ith buyer (decision variable) individual optimum production run length of the manufacturer (decision variable) minimum credit time required by ith buyer (decision variable) common order replenishment time (decision variable)
When there is no coordination, the manufacturer and the set of buyers optimize their cost individually and the manufacturer supplies the item separately to all the buyers. To minimize the total inventoryrelated cost, the buyer orders at his economic order interval and the total relevant cost (ordering and holding cost) of the ith buyer is given as TCBi ¼
S bi hbi ti Di . þ ti 2
(1)
The economic order interval of the ith buyer at which the buyer will place the order to the manufacturer can be determined as sffiffiffiffiffiffiffiffiffiffiffi 2S bi ti ¼ . (2) Di hbi The corresponding optimal cost of the ith buyer is given as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCBi ¼ 2Di Sbi hbi . (3) The total optimal relevant cost of all the buyers can be written as TCBbc ¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2Di S bi hbi .
(4)
i¼1
The economic order interval of each buyer will be different; therefore, the manufacturer will receive
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orders at non-uniform intervals from the buyers. With the assumptions of large number of buyers, the demand rate, D, of the manufacturer can be assumed constant and the same is given as D¼
X
Di ,
where Di is the individual buyer’s demand. The economic production quantity of the manufacturer can be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S D m . Qv ¼ h m 1 rm
individual transportation cost þ Inventory holding cost
Sm þ t
i¼1
Qi
Sm C c 1 þ þ hm DT ðK 1Þ ðK 2Þrm 2 KT T pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n X S bi 1 þ hbi Di T 2S bi Di hbi . þ 2 T i¼1
(5)
TCMbc ¼ Setup cost þ Order processing cost including
¼
TCMðK; T Þ ¼
ð7Þ
Manufacturer’s total relevant cost per unit time before coordination can be written as
n X C i Di
time T considering compensation cost to all the buyers is given as: TCMac ¼ Setup cost + Common transportation cost including order processing cost + Inventory holding cost + Amount of compensation given to the buyers for their increased cost.
1 þ hm DT 1 rm . 2
ð6Þ
3.3. Optimal policy with coordinated common replenishment time implemented by the manufacturer and the common shipment carrier for all the buyers Here the manufacturer is considered to be a stronger party compared to the set of buyers and initiates the coordination process. The manufacturer supplies all the buyers simultaneously at a common replenishment time T and thereby the manufacturer takes advantage of common carrier for transporting the material to all the buyers to save cost. This is valid with the assumption that common transportation cost is less than the sum of the cost incurred for transporting individually to each buyer. Further, when the buyer is replenished at common interval T, manufacturer may produce at an interval of KT times (where K is a positive integer) to reduce the number of setups. Due to the change of the optimal replenishment time from ti of the ith buyer to T, buyer’s inventoryrelated cost will increase. The manufacturer compensates this increase in cost of the buyer due to the change of its ordering policy by giving incentive in the form of credit to the buyer from the extra savings. The total relevant cost of the manufacturer under common order replenishment
Differentiating Eq. (7) with respect to T keeping K fixed, it yields the following expression: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2 ðS m =KÞ þ C c þ ni¼1 Sbi P T ðK Þ ¼ . (8) hm Dbm þ ni¼1 hbi Di The second derivative of Eq. (7) with respect to T keeping K fixed, one gets: n q2 TCMðK; TÞ 2S m 2C c X 2Sbi ¼ þ 3 þ 40, qT 2 KT 3 T T3 i¼1 which shows Eq. (7) is convex with respect to T for all T40. Substituting the value of T (K) in Eq. (7), and after simplification, it yields the following expression: TCMðK Þ ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !ffi u n n u Sm X X t2 hm Dbm þ Sbi hbi Di þ Cc þ K i¼1 i¼1
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2Sbi Di hbi .
ð9Þ
i¼1
Minimizing TCM (K) is equivalent to minimizing (
) ! n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n X X Sm 2Sbi Di hbi ¼2 Sbi þ Cc þ K i¼1 i¼1 ! n X hm Dbm þ hbi Di .
TCMðKÞ þ
i¼1
Considering only the term involving K in the above equation and by simplification, one gets " ( ) n X 1 S m hm D 2rm 1 þ Sm Z ðK Þ ¼ 2 hbi Di K i¼1 ! # n X þKhm D C c þ S bi 1 rm . ð10Þ i¼1
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The second derivative of Eq. (10) with respect to K, one obtains ( ) n X q2 ZðKÞ 4Sm ¼ 3 hm D 2rm 1 þ hbi Di . K qK 2 i¼1 Since lim q2 Z 4S m ¼ 3 rm ! 0 qK 2 K
hm D þ
n X
! hbi Di .
i¼1
2 It is evident from the above Pn equation that ðq ZðKÞ= 2 qK Þ40 holds only if i¼1 ðhbi hm ÞDi 40. However, value is added to the product as it moves from upstream member to the downstream member of the supply chain; therefore, it is reasonable to assume that holding cost of the buyer is higher than that of the manufacturer. Under the above assumption, Eq. (9) is convex with respect to K for all K40. In order to minimize Z(K), we select K ¼ K0 such that Z(K0)pZ(K01) and Z(K0)pZ(K0+1). On substituting the relevant values in Eq. (10) and after simplification, the expression becomes P S m ðð ni¼1 hbi Di Þ=ðhm DÞÞ þ 2rm 1 P K 0 ðK 0 1Þp 1 rm C c þ m i¼1 S bi
pK 0 ðK 0 þ 1Þ.
ð11Þ
Optimum integer value K0 is determined from the above inequality (11) and substituting the value of K0 in Eq. (8), the optimum common replenishment time T ðKÞ is determined. Subsequently, the optimum total relevant cost of manufacturer is determined by substituting the optimal value of T ðKÞ in Eq. (7). 3.3.1. Determination of minimum and maximum credit time The total cost of all the buyers without coordinated common replenishment time is given as TCBbc ¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2S bi Di hbi .
(12)
i¼1
The total cost of all the buyers with coordinated common replenishment time is given as n X Sbi 1 þ Di hbi T . (13) TCBac ¼ 2 T i¼1 A buyer will accept the coordinated common replenishment time T if the manufacturer compensates the extra cost of the buyer by providing credit
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period of tci such that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S bi 1 þ Di hbi T 2Sbi Di hbi . 2 T For the minimum credit period given to the buyer, one can write pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sbi 1 þ Di hbi T 2S bi Di hbi . (14) Di tci hbi ¼ 2 T Substituting the common replenishment time T by optimum T , the minimum credit time required by the ith buyer will be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 S bi 1 D ðtci Þmin ¼ þ h T 2S bi Di hbi . i bi Di hbi T 2 (15) Di tci hbi X
The minimum credit time required for each buyer can be determined by using Eq. (15). The minimum credit time acceptable to all buyers tcmin is given by tcmin ¼ Maximumfðtci Þmin g
for fi ¼ 1; 2; . . . ; ng.
Let tcmax be the maximum credit time the manufacturer can offer to any buyer. The manufacturer’s cost at individually optimized order interval time is given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X C i Di 2DS m hm 1 rm þ . (16) Qi i¼1 Again, manufacturer’s minimum cost with common coordinated order interval time is given as Sm Cc 1 þ þ hm DT ðK 1Þ ðK 2Þrm KT T 2 Sm Cc 1 ¼ þ þ hm DT bm . 2 KT T
ð17Þ
Equating to zero the difference of the manufacturer’s cost without coordinated common replenishment time given by Eq. (16) with coordinated common interval time given by Eq. (17), the expression of maximum credit time that a manufacturer can offer is obtained as follows: " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X 1 C i Di tcmax ¼ Pn 2DS m hm 1 rm þ Qi i¼1 Di hi i¼1
Sm Cc 1 þ þ hm DT bm . ð18Þ KT T 2 The manufacturer can offer the credit time to the set of buyers between the upper and lower bound of credit time tcmax and tcmin. When the manufacturer offers the minimum credit time to the set of buyers as determined by Eq. (15), the benefit of
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coordination will be with the manufacturer whereas when the manufacturer offers the maximum credit time as determined by Eq. (18), all the benefits of coordination will be transferred to the set of buyers. For heterogeneous group of buyers’ case, each individual buyer will require a different credit time from the manufacturer to compensate their increase in costs. Therefore, determination of a uniform minimum credit time for all buyers is essential. Similarly, the manufacturer can give a maximum credit time to the buyers as long as manufacturer’s extra profit due to the changed order policy becomes zero. Stepwise procedure for assigning credit time to the buyers: Step 1: Determine minimum credit time required for each buyer by using Eq. (15). Step 2: Determine the value of maximum credit time from the set of minimum credit times of the buyers i.e. tcmin ¼ max{min(tci)} where i ¼ 1, 2, y, n. Step 3: Determine maximum credit time tcmax a manufacturer can offer by using Eq. (18). Step 4: If tcmax4tcmin, then assign credit time tcmin to all the buyers. This will ensure that no buyer is at least losing. Otherwise, it will not be possible to have coordinated supply policy for all buyers without someone loosing. 4. Numerical example Numerical example considers the data of Banerjee and Burton (1994). Some additional data are also assumed. Buyer Demand Ordering Holding (Units/ cost ($/ cost ($/ day) day) unit/ day)
Transportation cost for individual shipment ($/ shipment)
1 2 3 4 5
40 40 40 40 40
8 15 10 5 20
20 15 6 10 18
.008 .009 .01 .01 .007
Sm ¼ 250 $per setup; hm ¼ .005$ per unit per day; assumed Cc ¼ 100$ per shipment; total demand SDi ¼ 58, rm ¼ .3.
Using Eq. (5), manufacturer’s optimum lot size Qv ¼ 2878 units. The optimum value of K0 is determined using Eq. (11) and K0 ¼ 2 will satisfy the inequality. The optimum value of coordinated order replenishment time is determined using Eq. (8) as T ¼ 27:47 days. When material is transported separately to each buyer, the cost of manufacturer before coordination is determined by using Eq. (6) and is ¼ 157.43$ per week. From Eq. (15), the minimum credit time required for buyer 1, 2, 3, 4 and 5 to compensate their increase in cost is .11, 2.86, 4.96, 1.01 and 2.37 days, respectively. Now, the manufacturer gives the maximum of this minimum credit time to all buyers and therefore, the credit time given to all buyers is equal to 4.96 days. When supply is made by the manufacturer at coordinated common replenishment time of 27.47 days giving minimum credit time equal to 4.96 days to all the buyers, the manufacturer’s total cost is ¼ 98.0$ per week The savings in cost at the manufacturer’s side(157.4398.0) ¼ 59.43$ per week. 5. Model formulation for ex-factory case In this section, we have considered that transportation cost is borne by the buyers. The manufacturer sells the product at ex-factory price and the buyer pays for transportation. 5.1. Assumptions and notation All the earlier assumptions are also valid here. The same notation used in the earlier case requires the new explanations below. Ci Cc
transportation cost for individual delivery of the item, paid by the buyer, common transportation cost per delivery borne by the buyer.
5.2. Optimal policy for the buyer and the manufacturer Here, all the buyers individually optimize their order size. Total relevant cost of ith buyer is given by
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TCBbc ¼ Ordering cost+Holding cost+Transportation cost, TCBbc ¼
Ai 1 Ci þ hbi Di ti þ . ti 2 ti
(19)
Total optimal relevant cost for all the buyers is obtained as TCBbc ¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X
2hbi Di ðAi þ C i Þ.
(21)
i¼1
Considering demand rate of manufacturer to be approximately constant as discussed in the earlier case, the optimum production lot size of the manufacturer is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DS m , Qv ¼ (22) h m 1 rm where D¼
n X
Substituting the value of T in Eq. (24), one gets the optimal total relevant cost of the buyers as TCBac ¼
n X Ai i¼1
Optimal order interval time of the buyer is given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðAi þ C i Þ ti ¼ . (20) hbi Di
Di .
TCMbc ¼
DS m 1 þ h m Q v 1 rm . 2 Qv
TCMac ¼
(23)
Here, all the buyers decide to form an association and go for a common order interval T to share the benefits of common transportation. Due to this common order interval T, total relevant cost of all buyers can be expressed as TCBac ¼
i¼1
T
þ
n 1X
2
i¼1
hbi Di T þ
Cc . T
(24)
Optimum common order replenishment time for all the buyers is obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ffi 2 A þ C i c Pi¼1 . (25) T ¼ n i¼1 hbi Di
n 1X Cc hbi Di T þ . 2 i¼1 T
Sm DS m . ¼ T Q
(26)
(27)
This however increases the manufacturer’s cost. The cost difference DW is obtained from Eqs. (26) and (27) as 1 1 1 DW ¼ DS m hm Qv 1 rm . (28) Q Qv 2 The above expression is the minimum compensation that the manufacturer requires to offset his increase in cost. On the other hand, due to common carrier, the maximum savings to buyers can be written as the difference of Eqs. (21) and (26) and is as follows: DW max ¼
n X i¼1
5.3. Coordinated common order interval time implemented by the group of buyers
n X Ai
T
þ
Since order comes to the manufacturer at fixed interval T and supplies at fixed interval of T , the manufacturer’s inventory holding cost may be assumed to be zero. Thus, total relevant cost of the manufacturer can be found as
i¼1
Total optimal relevant cost of the manufacturer is
683
2Di hi ðAi þ C i Þ
n X Ai i¼1
T
n 1X Cc hbi Di T . 2 i¼1 T
ð29Þ
When DWmax4DWmin, then the benefits accrued in the channel due to common transportation can be distributed between the association of buyers and the manufacturer. 5.4. Negotiation based on order interval time In this section, a negotiation approach for division of surplus generated due to follow up of a coordinated policy between the manufacturer and the set of buyers in the supply chain is discussed. The order replenishment time T between the association of buyers and the manufacturer can be negotiated so that both the parties get certain extra benefits. Initially, association of buyers decides the common order interval T and request the manufacturer to supply the item. As a result, there will be
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increase in cost at the manufacturer’s side. Minimum amount required by the manufacturer to compensate his increase in cost is given as S m DS m 1 (30) DW ¼ hm Qv 1 rm . 2 T Qv Due to order replenishment time T , maximum savings to the association of buyers is given by the expression of Eq. (29). The common order replenishment time is acceptable to both the parties when they get certain extra benefit. The association of buyers as a stronger party in the channel can offer an amount equal to DW and ask the manufacturer to supply the item at time T . The manufacturer may agree or may not agree with the proposal. If the manufacturer accepts the proposal, the manufacturer does not gain any extra profit. To gain a certain amount of extra benefit, the manufacturer may ask the buyers the same amount of compensation DW for a different order interval time T such that T4T . This reduces manufacturer’s cost and can gain certain extra benefit. With the new offer from the manufacturer, the association of buyers calculates their new savings in cost from Eq. (29). If this new savings in cost is more than the amount of compensation required to the manufacturer, i.e. DW, the offer is acceptable to both the parties with a certain amount of benefits to each one. Otherwise, buyers may propose a new order replenishment time to the manufacturer so that savings in cost is more than DW. Manufacturer may accept the offer or otherwise the process may be repeated for an acceptable order replenishment time where both the parties get certain benefit for common transportation of the item. 6. Numerical example The same numerical data are considered here. Sm ¼ 250$ per setup; hm ¼ .005$ per unit per day; Cc ¼ 100$ per shipment; Total demand D ¼ 58; rm ¼ .5. At individually optimized order replenishment time, total relevant cost of buyers is obtained by using Eq. (21) and is ¼ 111.37$ per week Using Eq. (22), the manufacturer’s optimum lot size is Qv ¼ 3405 units. Substituting it into Eq. (23) yields the optimum cost of the manufacturer TCMbc ¼ 59.57$ per week. The common order replenishment time decided by the association of buyers from Eq. (25) is obtained as T ¼ 26:29 days. Substituting it into
Eq. (26) yields the total cost of all the buyers after coordinated order replenishment time TCBac ¼ 89.95$ per week. From Eq. (27), the manufacturer’s total relevant cost at common order interval T is TCMac ¼ 66.5$ per week. The savings in cost to the association of buyers is equal to 21.42$ per week, and the increase in cost at manufacturer’s side is equal to 6.93$ per week. The additional profit available with the association of buyers is equal to 14.49$ per week. 7. Conclusions In this paper, two models of coordination are developed where multiple buyers receive supplies from a manufacturer. In the first model, the transportation cost is borne by the manufacturer whereas in the second model, transportation cost is borne by the buyers. Considering the benefits of common transportation cost in the first case, manufacturer coordinates with all the buyers to have orders at a common replenishment time so that the item can be supplied through common carrier. The increase in cost to the buyers due to coordinated ordering policy is compensated through a uniform credit policy and in the paper, a procedure is suggested to determine the uniform credit time for all the buyers. In the second model, in ex-factory situation, the transportation cost is borne by the buyers. All the buyers located in one region agree to order at a common replenishment time to save in transportation cost. Due to coordinated order policy adopted by the association of buyers, the manufacturer’s cost increases. The buyer and the manufacturer can use the models developed in this paper as a quantitative tool for coordination and negotiation purpose. The stronger party in the channel can take the initiative for coordination and encourage the other party in the channel to adopt a coordinated common order replenishment time to take advantage of common transport. The model can be used to estimate the minimum amount of compensation necessary for coordination and the maximum amount that is feasible. The study revealed that delivering the item at common order interval would result in savings in cost at the stronger party of the channel. The extra savings from the use of common carrier policy to supply the item can be divided between the manufacturer and the association of the buyers providing a potential benefit for both the manufac-
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turer and the set of buyers. The division of savings through negotiation can be a win–win proposition for both the parties and this can lead to a better supply chain coordination. In the development of the model, manufacturer and the buyers must be ready to share information and their sincerity in this direction can be a problem for some organizations. However, the expected benefit received from the coordinated shipment policy may resolve this problem. Acknowledgment The authors are extremely grateful to the two anonymous referees for their very helpful comments and suggestions. References Banerjee, A., Burton, J.S., 1994. Coordinated vs. independent inventory replenishment policies for a vendor and multiple buyers. International Journal of Production Economics 35 (1-3), 215–222. Cetinkaya, S., Lee, C.Y., 2000. Stock replenishment and shipment scheduling for vendor managed inventory systems. Management Science 46 (2), 217–232. Chan, C.K., Kingsman, B.G., 2007. Coordination in a singlevendor multi-buyer supply chain by synchronizing delivery and production cycles. Transportation Research, Part E 43, 90–111. Chapman, C.B., Ward, S.C., Cooper, D.F., Page, M.J., 1985. Credit policy and inventory control. Journal of Operational Research Society 35 (12), 1055–1065. Drezner, Z., Wesolowsky, G.O., 1989. Multi buyers discount pricing. European Journal of Operational Research 40 (1), 38–42. Goyal, S.K., 1985. Economic order quantity under condition of permissible delay in payments. Journal of Operation Research Society 36 (4), 335–338. Hammer, M., 2001. The super efficient company. Harvard Business Review 79 (8), 82–91. Issakson, A., 2002. Trade credit in Kenyan Manufacturing: evidence from plant level data. Working paper, Source: /www.unido.org/userfiles/PuffK/SIN_WPS04.pdfS. Jamal, A.M.M., Sarker, R.B., Wang, S., 2000. Optimal payment time for a retailer under permitted delay of payment by the wholesaler. International Journal of Production Economics 66 (1), 59–66. Joglekar, P.N., 1988. A quantity discount pricing model to increase vendor profits. Management Science 34 (11), 1391–1398. Kim, K.H., Hwang, H., 1988. An incremental discount pricing schedule with multiple customers and single price break. European Journal of Operational Research 35 (1), 71–79.
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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. Kingsman, G.B., 1983. The effect of payment rules on ordering and stockholding in purchasing. Journal of Operational Research Society 34 (11), 1085–1098. Lal, R., Staelin, R., 1984. An approach for developing an optimal discount pricing policy. Management Science 30 (12), 1524–1539. Mentzer, T.J., 2001. Supply Chain Management. Sage Publisher, London. Mishra, A.K., 2004. Selective discount for supplier buyer coordination using common replenishment epochs. European Journal of Operational Research 153, 751–756. Moses, M., Seshadri, S., 2000. Policy mechanisms for supply chain co-ordination. IIE Transactions 32 (3), 245–262. Munson, L.C., Rosenblatt, J.M., Rosenblatt, Z., 1999. The use and abuse of power in supply chains. Business Horizon 42 (1), 55–65. 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. Siajidi, H., Ibrahim, R.F.N., Lochert, P.B., 2006. A single-vendor multiple-buyer inventory model with a multiple-shipment policy. The International Journal of Advanced Manufacturing Technology 27 (9–10), 1030–1037. Swenseth, R.S., Godfrey, R.M., 2002. Incorporating transportation costs into inventory replenishment decisions. International Journal of Production Economics 77 (2), 113–130. Thomas, D.J., Grifin, P.J., 1996. Coordinated supply chain management. European Journal of Operational Research 94 (1), 1–15. Viswanathan, S., Piplani, R., 2001. Coordinating supply chain inventories through common replenishment epoch. European Journal of Operational Research 129 (2), 277–286. Wang, Q., 2002. Determination of Supplier’s optimal quantity discount scehedules with heterogenous buyers. Naval Research Logistics 49 (1), 46–59. Weng, Z.K., Wong, R.T., 1993. General models for the supplier’s all unit quantity discount policy. Naval Research Logistics 40 (7), 971–991. Wilson, N., Wetherhill, P., Summers, B., 2000. Business to Business credit: A strategic tool for the ‘New Economy’, Source: /http://cmrc.vsp.co.ukS, Credit Manangement 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), 203–215. Zahir, S., Sarker, R., 1991. Joint economic ordering policies of multiple wholesalers and a single-manufacturer with price dependent demand functions. Journal of Operational Research Society 42, 157–164.
Int. J. Production Economics 111 (2008) 676–685 www.elsevier.com/locate/ijpe
Coordination of a single-manufacturer/multi-buyer supply chain with credit option S.P. Sarmaha, D. Acharyaa, S.K. Goyalb, a
Department of Industrial Engineering and Management, IIT Kharagpur, Kharagpur 721302, India Faculty of Commerce and Administration, Department of Decision Sciences and MIS, Concordia University, 1455 De Maisonneuve Blvd., West, Montreal, Que., Canada H3G 1M8
b
Received 1 March 2005; accepted 2 April 2007 Available online 18 April 2007
Abstract One of the important issues in supply chain management is the coordination between manufacturers and multi-buyer. A single manufacturer supplying a product to single buyer is hard to find in today’s business environment. This paper investigates a coordination problem in a single-manufacturer and multiple heterogeneous buyers situation. Two typical cases are explored here: (i) ex-site delivery case considering manufacturer dominance where manufacturer with larger production lot size delivers the item to the group of heterogeneous buyers at common replenishment time through common carrier and (ii) ex-factory delivery case with buyer’s dominance and common replenishment time for delivery. The paper develops coordination mechanism that allows improvement of supply chain performance. Acceptance of a coordinated solution depends on the share of surplus generated due to coordination between the parties of the supply chain. The paper also focuses on how negotiation can be carried out to get due share of extra savings after coordination from the business by each party. r 2007 Elsevier B.V. All rights reserved. Keywords: Supply chain; Coordination; Credit; Payment delay; Negotiation; Multiple buyers
1. Introduction In the recent times, people from both academics and industry have shown keen interest on supply chain management (SCM) research realizing its significant potential to improve efficiency of operations and reduced costs. Since the scope for improvement within the organization is gradually decreasing, both academicians and industry leaders Corresponding author. Tel.: +1 514 848 2424x2966; fax: +1 514 848 2824. E-mail address: [email protected] (S.K. Goyal).
have given thought about how to improve the performance of the firm in terms of cost, delivery time, customer satisfaction, etc. They are looking for new alternatives for integrating business operations beyond the organization’s boundary and trying to align and coordinate the business processes and activities of the channel members so that it will improve the performance and effectiveness of the supply chain. The single buyer and the single vendor together constitute a simple two-stage supply chain and the problem is considered as the building block of any supply chain. When a single owner owns both the
0925-5273/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2007.04.003
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business entities then it is a case of centralized supply chain and in the decentralized supply chain case, different individuals own different business entities and most of today’s supply chains belong to the latter category. Thomas and Grifin (1996) in their paper mentioned that effective SCM requires planning and coordination among the various channel members including manufacturers, retailers and intermediaries if any. Further, Mentzer (2001) has mentioned SCM as the systematic, strategic coordination of the traditional business functions within a particular company and across business within the supply chain for the purpose of improving the long-term performance of the individual companies and the supply chain as a whole. In a decentralized supply chain, there may be conflict between the different corporate entities of the supply chain as the members of the supply chain may pursue their own objectives and one party may be dominant over the other to achieve his own goal. Munson et al. (1999) have discussed about misuse of power by the channel leader. Sometimes, it is noticed that a powerful manufacturer in the channel controls dependent suppliers, subcontractors and retailers. However, to improve the performance of the supply chain there is a need to coordinate the decisions between the members of the supply chain and the dominant member if there exists, should take a lead role to make the coordination a win–win proposition for both. In today’s business environment, one often encounters a two-stage supply chain where a manufacturer supplies to multiple buyers and generally these buyers are found to be located in different places. Swenseth and Godfrey (2002) have mentioned that around 50% of total annual logistics cost of a product can be attributed to transportation cost. When a supplier delivers a product to a set of buyers, consolidation of their requirements to form a larger load (a full truck load) makes sense as it leads to cost saving and here one exploits scale economies inherent to transportation of the materials. This paper deals with a manufacturer supplying a product to a group of heterogeneous buyers where the demand of buyers may be different from one another. Each buyer’s demand is constant and satisfied without delay. Two cases are considered here where in the first case transportation cost is borne by the manufacturer and in the second case, it is borne by the buyers. While the former case usually applies to the ex-site delivery condition with transportation cost included
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in the price, the latter is usually the ex-factory delivery condition of supply. In the development of the model here, it is assumed that the buyers are closely located to each other. When acting independently, each buyer orders at economic order quantity to maximize own cost. According to first case, if each order from a buyer is handled independently, maybe in less than full truckload, the transportation cost of the manufacturer increases and reduces his profitability. On the other hand, in the second case, the transportation cost of the buyer increases. A possible method of reducing the adverse effect of transportation cost is freight consolidation. However, this requires coordinated delivery at fixed intervals to the set of buyers who share a common carrier. While in the first case, the manufacturer gains from reduced transportation cost, the customers are benefited from the second case. A coordinated delivery through freight consolidation is possible if the manufacturer induces the buyers to have the delivery of a product at fixed intervals (cycle time) by offering credit (or price discount) to offset their increase in cost and possibly to increase their profit. In the second case, such a coordinated delivery policy can be implemented if the buyers as a group compensate the increase in cost of the manufacturer. Coordinated replenishment has been widely applied in many industries including refrigerated goods replenishment for supermarket chain stores (Hammer, 2001). As far as refrigerated goods are concerned, single-item distribution to a single buyer lacks the economies of scale due to its highly dedicated transportation system and therefore consolidation of demand from number of buyers and supplying it through a full truck load is the best alternative. Hammer (2001) mentioned that General Mills Yougart and Land O_lakes butter delivered by same truck; enroute to the same supermarket is a good example of common replenishment practices. Cetinkaya and Lee (2000) in their work also have justified the benefits generated from such a shipment consolidation arrangement. 2. Review of literature It is rather difficult to separate out the singlemanufacturer/single-buyer literature from the single-manufacturer/multiple-buyers literature as in some cases, authors after developing the model for single-manufacturer and single-buyer case extended it to the multiple-buyers situation and these models
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are mostly confined to a homogeneous group of buyers (Lal and Staelin, 1984; Drezner and Wesolowsky, 1989). A single vendor supplying a product to many different buyers is studied by authors such as Kim and Hwang (1988), Weng and Wong (1993) and recently by Siajidi et al. (2006) and Chan and Kingsman (2007). Zahir and Sarker (1991) have considered a price-dependent demand function for multiple regional wholesalers who are served by a single manufacturer. Woo et al. (2001) have studied an integrated inventory system where a single vendor purchases and processes raw materials in order to deliver it to multiple retailers. Wang (2002) has presented an analysis for a supplier’s quantity discount decision for heterogeneous buyers. With the assumption that the vendor follows a lot-for-lot policy, Viswanathan and Piplani (2001) have shown that a vendor could implement the common replenishment period mechanism by offering price discounts to a set of heterogeneous buyers in a single vendor, multi-buyer supply chain. Mishra (2004) has extended the above model of Viswanathan and Piplani (2001) by considering selective discount policy. Many of these authors assume that the buyers receive items at fixed interval and the manufacturer produces at each interval a batch size that is equal to the sum of the delivery lot sizes of the buyers. This eliminates the problem of finding inventory holding cost of the manufacturer. Banerjee and Burton (1994) in their work on a single-manufacturer and multiple-buyers case have mentioned that when the system consists of a very large number of buyers with relatively small order quantities and cycle time, then the assumption that the manufacturer’s available inventory is depleted at a uniform rate holds good at least approximately. A similar assumption is also made here in the development of the model. In the literature of inventory management, it is seen that the supplier provides incentive to the buyer in the form of discount or credit. The practice of suppliers providing credit to their customers is not new and in competitive business environments, credit is an important form of the financing for business and particularly in developing countries; the role of trade credit is immense where growth of financial institution 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 later date (Issakson, 2002). There are several advantages
of trade credit such as: (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 competitors. The length of the credit period offered by the suppliers 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 of the customers (Wilson et al., 2000). (iii) A supplier may differentiate their offering to the market by extending trade credit to their customers as a commitment to the quality of their products. By offering credit option, customers are allowed time to inspect quality of the product before paying for the goods. (iv) A larger firm has better access to financial institutions. So very often a smaller firm that does not have sufficient funds has to depend on the trade credit offered by the larger business firm. (v) 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. Recognizing the advantages associated with the credit option, many authors have used delay in payment or credit option as an incentive mechanism in their inventory control models. Goyal (1985) suggested a mathematical model for obtaining the economic order quantity for an item for which the supplier permits a fixed delay in settling the account. Kingsman (1983) and Chapman et al. (1985) also have used delay in payment option in the development of their mathematical models. Kim et al. (1995) have determined the optimum length of credit period for the product that the supplier sells to retailers in order to maximize his profit. Shinn and Hwang (2003) have considered order sizedependent delay in payment and have developed a model for determination of retailer’s optimal price and order size simultaneously. Jamal et al. (2000) have studied the optimal cycle and payment time for a retailer in a deteriorating item inventory situation where a wholesaler allows a specified credit period to the retailer for payment without penalty. Moses and Seshadri (2000) also have used credit period in supply chain coordination. Considering the relevance of credit, this paper also uses the credit option as a mechanism to develop coordination between the two parties of the supply chain. Unlike existing inventory models with
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credit option, this paper develops two new models that integrate the transportation cost explicitly in the single vendor multiple-buyers’ situation. The objective in both the models is to view the system as an integrated whole and determine the optimum production and coordinated shipment interval that minimizes the average total cost per unit time. A procedure is devised to coordinate between the different members of the channel and solve the proposed model. Numerical examples are used to show the savings due to coordinated replenishment time. Further, how the extra savings due to coordination can be shared between the two parties is also discussed so that coordination can be a win–win proposition for both the parties. 3. Model formulation for ex-site delivery case
K
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integer lot size multiplier (a positive value, decision variable) production rate of the manufacturer utilization rate where 0ormp1
P (D/ P) ¼ rm Qv economic production quantity of the manufacturer TCBi total relevant cost per unit time of ith buyer TCM total relevant cost per unit time of the manufacturer Ci individual transportation cost per delivery borne by the manufacturer Cc common transportation cost per delivery borne by the manufacturer bm average inventory factor at manufacturer’s side ¼ (K1)(K2)rm
Here, the transportation cost is borne by the manufacturer. The responsibility to deliver the item at the buyer’s site lies with the manufacturer.
For the derivation of the above expression, one can see Joglekar (1988).
3.1. Assumptions and notation
3.2. Individual optimal policy of buyer and the manufacturer
Assumptions: (i) demand of each buyer is known and constant, (ii) no shortages are allowed, (iii) there are large number of buyers with small order size to make uniform depletion of manufacturer’s inventory, (iv) cost of transporting through common carrier to several buyers is assumed to be less than the sum of the costs of supplying to the buyers in separate carriers. Notation: D Sbi Sm hbi hm ti t
tci T
total annual demand ordering cost of ith buyer where i ¼ 1–n setup cost of the manufacturer holding cost of ith buyer where i ¼ 1–n holding cost of the manufacturer individual optimum order interval of ith buyer (decision variable) individual optimum production run length of the manufacturer (decision variable) minimum credit time required by ith buyer (decision variable) common order replenishment time (decision variable)
When there is no coordination, the manufacturer and the set of buyers optimize their cost individually and the manufacturer supplies the item separately to all the buyers. To minimize the total inventoryrelated cost, the buyer orders at his economic order interval and the total relevant cost (ordering and holding cost) of the ith buyer is given as TCBi ¼
S bi hbi ti Di . þ ti 2
(1)
The economic order interval of the ith buyer at which the buyer will place the order to the manufacturer can be determined as sffiffiffiffiffiffiffiffiffiffiffi 2S bi ti ¼ . (2) Di hbi The corresponding optimal cost of the ith buyer is given as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCBi ¼ 2Di Sbi hbi . (3) The total optimal relevant cost of all the buyers can be written as TCBbc ¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2Di S bi hbi .
(4)
i¼1
The economic order interval of each buyer will be different; therefore, the manufacturer will receive
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orders at non-uniform intervals from the buyers. With the assumptions of large number of buyers, the demand rate, D, of the manufacturer can be assumed constant and the same is given as D¼
X
Di ,
where Di is the individual buyer’s demand. The economic production quantity of the manufacturer can be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S D m . Qv ¼ h m 1 rm
individual transportation cost þ Inventory holding cost
Sm þ t
i¼1
Qi
Sm C c 1 þ þ hm DT ðK 1Þ ðK 2Þrm 2 KT T pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n X S bi 1 þ hbi Di T 2S bi Di hbi . þ 2 T i¼1
(5)
TCMbc ¼ Setup cost þ Order processing cost including
¼
TCMðK; T Þ ¼
ð7Þ
Manufacturer’s total relevant cost per unit time before coordination can be written as
n X C i Di
time T considering compensation cost to all the buyers is given as: TCMac ¼ Setup cost + Common transportation cost including order processing cost + Inventory holding cost + Amount of compensation given to the buyers for their increased cost.
1 þ hm DT 1 rm . 2
ð6Þ
3.3. Optimal policy with coordinated common replenishment time implemented by the manufacturer and the common shipment carrier for all the buyers Here the manufacturer is considered to be a stronger party compared to the set of buyers and initiates the coordination process. The manufacturer supplies all the buyers simultaneously at a common replenishment time T and thereby the manufacturer takes advantage of common carrier for transporting the material to all the buyers to save cost. This is valid with the assumption that common transportation cost is less than the sum of the cost incurred for transporting individually to each buyer. Further, when the buyer is replenished at common interval T, manufacturer may produce at an interval of KT times (where K is a positive integer) to reduce the number of setups. Due to the change of the optimal replenishment time from ti of the ith buyer to T, buyer’s inventoryrelated cost will increase. The manufacturer compensates this increase in cost of the buyer due to the change of its ordering policy by giving incentive in the form of credit to the buyer from the extra savings. The total relevant cost of the manufacturer under common order replenishment
Differentiating Eq. (7) with respect to T keeping K fixed, it yields the following expression: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2 ðS m =KÞ þ C c þ ni¼1 Sbi P T ðK Þ ¼ . (8) hm Dbm þ ni¼1 hbi Di The second derivative of Eq. (7) with respect to T keeping K fixed, one gets: n q2 TCMðK; TÞ 2S m 2C c X 2Sbi ¼ þ 3 þ 40, qT 2 KT 3 T T3 i¼1 which shows Eq. (7) is convex with respect to T for all T40. Substituting the value of T (K) in Eq. (7), and after simplification, it yields the following expression: TCMðK Þ ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! !ffi u n n u Sm X X t2 hm Dbm þ Sbi hbi Di þ Cc þ K i¼1 i¼1
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2Sbi Di hbi .
ð9Þ
i¼1
Minimizing TCM (K) is equivalent to minimizing (
) ! n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n X X Sm 2Sbi Di hbi ¼2 Sbi þ Cc þ K i¼1 i¼1 ! n X hm Dbm þ hbi Di .
TCMðKÞ þ
i¼1
Considering only the term involving K in the above equation and by simplification, one gets " ( ) n X 1 S m hm D 2rm 1 þ Sm Z ðK Þ ¼ 2 hbi Di K i¼1 ! # n X þKhm D C c þ S bi 1 rm . ð10Þ i¼1
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The second derivative of Eq. (10) with respect to K, one obtains ( ) n X q2 ZðKÞ 4Sm ¼ 3 hm D 2rm 1 þ hbi Di . K qK 2 i¼1 Since lim q2 Z 4S m ¼ 3 rm ! 0 qK 2 K
hm D þ
n X
! hbi Di .
i¼1
2 It is evident from the above Pn equation that ðq ZðKÞ= 2 qK Þ40 holds only if i¼1 ðhbi hm ÞDi 40. However, value is added to the product as it moves from upstream member to the downstream member of the supply chain; therefore, it is reasonable to assume that holding cost of the buyer is higher than that of the manufacturer. Under the above assumption, Eq. (9) is convex with respect to K for all K40. In order to minimize Z(K), we select K ¼ K0 such that Z(K0)pZ(K01) and Z(K0)pZ(K0+1). On substituting the relevant values in Eq. (10) and after simplification, the expression becomes P S m ðð ni¼1 hbi Di Þ=ðhm DÞÞ þ 2rm 1 P K 0 ðK 0 1Þp 1 rm C c þ m i¼1 S bi
pK 0 ðK 0 þ 1Þ.
ð11Þ
Optimum integer value K0 is determined from the above inequality (11) and substituting the value of K0 in Eq. (8), the optimum common replenishment time T ðKÞ is determined. Subsequently, the optimum total relevant cost of manufacturer is determined by substituting the optimal value of T ðKÞ in Eq. (7). 3.3.1. Determination of minimum and maximum credit time The total cost of all the buyers without coordinated common replenishment time is given as TCBbc ¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 2S bi Di hbi .
(12)
i¼1
The total cost of all the buyers with coordinated common replenishment time is given as n X Sbi 1 þ Di hbi T . (13) TCBac ¼ 2 T i¼1 A buyer will accept the coordinated common replenishment time T if the manufacturer compensates the extra cost of the buyer by providing credit
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period of tci such that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S bi 1 þ Di hbi T 2Sbi Di hbi . 2 T For the minimum credit period given to the buyer, one can write pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sbi 1 þ Di hbi T 2S bi Di hbi . (14) Di tci hbi ¼ 2 T Substituting the common replenishment time T by optimum T , the minimum credit time required by the ith buyer will be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 S bi 1 D ðtci Þmin ¼ þ h T 2S bi Di hbi . i bi Di hbi T 2 (15) Di tci hbi X
The minimum credit time required for each buyer can be determined by using Eq. (15). The minimum credit time acceptable to all buyers tcmin is given by tcmin ¼ Maximumfðtci Þmin g
for fi ¼ 1; 2; . . . ; ng.
Let tcmax be the maximum credit time the manufacturer can offer to any buyer. The manufacturer’s cost at individually optimized order interval time is given as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X C i Di 2DS m hm 1 rm þ . (16) Qi i¼1 Again, manufacturer’s minimum cost with common coordinated order interval time is given as Sm Cc 1 þ þ hm DT ðK 1Þ ðK 2Þrm KT T 2 Sm Cc 1 ¼ þ þ hm DT bm . 2 KT T
ð17Þ
Equating to zero the difference of the manufacturer’s cost without coordinated common replenishment time given by Eq. (16) with coordinated common interval time given by Eq. (17), the expression of maximum credit time that a manufacturer can offer is obtained as follows: " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n X 1 C i Di tcmax ¼ Pn 2DS m hm 1 rm þ Qi i¼1 Di hi i¼1
Sm Cc 1 þ þ hm DT bm . ð18Þ KT T 2 The manufacturer can offer the credit time to the set of buyers between the upper and lower bound of credit time tcmax and tcmin. When the manufacturer offers the minimum credit time to the set of buyers as determined by Eq. (15), the benefit of
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coordination will be with the manufacturer whereas when the manufacturer offers the maximum credit time as determined by Eq. (18), all the benefits of coordination will be transferred to the set of buyers. For heterogeneous group of buyers’ case, each individual buyer will require a different credit time from the manufacturer to compensate their increase in costs. Therefore, determination of a uniform minimum credit time for all buyers is essential. Similarly, the manufacturer can give a maximum credit time to the buyers as long as manufacturer’s extra profit due to the changed order policy becomes zero. Stepwise procedure for assigning credit time to the buyers: Step 1: Determine minimum credit time required for each buyer by using Eq. (15). Step 2: Determine the value of maximum credit time from the set of minimum credit times of the buyers i.e. tcmin ¼ max{min(tci)} where i ¼ 1, 2, y, n. Step 3: Determine maximum credit time tcmax a manufacturer can offer by using Eq. (18). Step 4: If tcmax4tcmin, then assign credit time tcmin to all the buyers. This will ensure that no buyer is at least losing. Otherwise, it will not be possible to have coordinated supply policy for all buyers without someone loosing. 4. Numerical example Numerical example considers the data of Banerjee and Burton (1994). Some additional data are also assumed. Buyer Demand Ordering Holding (Units/ cost ($/ cost ($/ day) day) unit/ day)
Transportation cost for individual shipment ($/ shipment)
1 2 3 4 5
40 40 40 40 40
8 15 10 5 20
20 15 6 10 18
.008 .009 .01 .01 .007
Sm ¼ 250 $per setup; hm ¼ .005$ per unit per day; assumed Cc ¼ 100$ per shipment; total demand SDi ¼ 58, rm ¼ .3.
Using Eq. (5), manufacturer’s optimum lot size Qv ¼ 2878 units. The optimum value of K0 is determined using Eq. (11) and K0 ¼ 2 will satisfy the inequality. The optimum value of coordinated order replenishment time is determined using Eq. (8) as T ¼ 27:47 days. When material is transported separately to each buyer, the cost of manufacturer before coordination is determined by using Eq. (6) and is ¼ 157.43$ per week. From Eq. (15), the minimum credit time required for buyer 1, 2, 3, 4 and 5 to compensate their increase in cost is .11, 2.86, 4.96, 1.01 and 2.37 days, respectively. Now, the manufacturer gives the maximum of this minimum credit time to all buyers and therefore, the credit time given to all buyers is equal to 4.96 days. When supply is made by the manufacturer at coordinated common replenishment time of 27.47 days giving minimum credit time equal to 4.96 days to all the buyers, the manufacturer’s total cost is ¼ 98.0$ per week The savings in cost at the manufacturer’s side(157.4398.0) ¼ 59.43$ per week. 5. Model formulation for ex-factory case In this section, we have considered that transportation cost is borne by the buyers. The manufacturer sells the product at ex-factory price and the buyer pays for transportation. 5.1. Assumptions and notation All the earlier assumptions are also valid here. The same notation used in the earlier case requires the new explanations below. Ci Cc
transportation cost for individual delivery of the item, paid by the buyer, common transportation cost per delivery borne by the buyer.
5.2. Optimal policy for the buyer and the manufacturer Here, all the buyers individually optimize their order size. Total relevant cost of ith buyer is given by
ARTICLE IN PRESS S.P. Sarmah et al. / Int. J. Production Economics 111 (2008) 676–685
TCBbc ¼ Ordering cost+Holding cost+Transportation cost, TCBbc ¼
Ai 1 Ci þ hbi Di ti þ . ti 2 ti
(19)
Total optimal relevant cost for all the buyers is obtained as TCBbc ¼
n pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X
2hbi Di ðAi þ C i Þ.
(21)
i¼1
Considering demand rate of manufacturer to be approximately constant as discussed in the earlier case, the optimum production lot size of the manufacturer is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DS m , Qv ¼ (22) h m 1 rm where D¼
n X
Substituting the value of T in Eq. (24), one gets the optimal total relevant cost of the buyers as TCBac ¼
n X Ai i¼1
Optimal order interval time of the buyer is given as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðAi þ C i Þ ti ¼ . (20) hbi Di
Di .
TCMbc ¼
DS m 1 þ h m Q v 1 rm . 2 Qv
TCMac ¼
(23)
Here, all the buyers decide to form an association and go for a common order interval T to share the benefits of common transportation. Due to this common order interval T, total relevant cost of all buyers can be expressed as TCBac ¼
i¼1
T
þ
n 1X
2
i¼1
hbi Di T þ
Cc . T
(24)
Optimum common order replenishment time for all the buyers is obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ffi 2 A þ C i c Pi¼1 . (25) T ¼ n i¼1 hbi Di
n 1X Cc hbi Di T þ . 2 i¼1 T
Sm DS m . ¼ T Q
(26)
(27)
This however increases the manufacturer’s cost. The cost difference DW is obtained from Eqs. (26) and (27) as 1 1 1 DW ¼ DS m hm Qv 1 rm . (28) Q Qv 2 The above expression is the minimum compensation that the manufacturer requires to offset his increase in cost. On the other hand, due to common carrier, the maximum savings to buyers can be written as the difference of Eqs. (21) and (26) and is as follows: DW max ¼
n X i¼1
5.3. Coordinated common order interval time implemented by the group of buyers
n X Ai
T
þ
Since order comes to the manufacturer at fixed interval T and supplies at fixed interval of T , the manufacturer’s inventory holding cost may be assumed to be zero. Thus, total relevant cost of the manufacturer can be found as
i¼1
Total optimal relevant cost of the manufacturer is
683
2Di hi ðAi þ C i Þ
n X Ai i¼1
T
n 1X Cc hbi Di T . 2 i¼1 T
ð29Þ
When DWmax4DWmin, then the benefits accrued in the channel due to common transportation can be distributed between the association of buyers and the manufacturer. 5.4. Negotiation based on order interval time In this section, a negotiation approach for division of surplus generated due to follow up of a coordinated policy between the manufacturer and the set of buyers in the supply chain is discussed. The order replenishment time T between the association of buyers and the manufacturer can be negotiated so that both the parties get certain extra benefits. Initially, association of buyers decides the common order interval T and request the manufacturer to supply the item. As a result, there will be
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increase in cost at the manufacturer’s side. Minimum amount required by the manufacturer to compensate his increase in cost is given as S m DS m 1 (30) DW ¼ hm Qv 1 rm . 2 T Qv Due to order replenishment time T , maximum savings to the association of buyers is given by the expression of Eq. (29). The common order replenishment time is acceptable to both the parties when they get certain extra benefit. The association of buyers as a stronger party in the channel can offer an amount equal to DW and ask the manufacturer to supply the item at time T . The manufacturer may agree or may not agree with the proposal. If the manufacturer accepts the proposal, the manufacturer does not gain any extra profit. To gain a certain amount of extra benefit, the manufacturer may ask the buyers the same amount of compensation DW for a different order interval time T such that T4T . This reduces manufacturer’s cost and can gain certain extra benefit. With the new offer from the manufacturer, the association of buyers calculates their new savings in cost from Eq. (29). If this new savings in cost is more than the amount of compensation required to the manufacturer, i.e. DW, the offer is acceptable to both the parties with a certain amount of benefits to each one. Otherwise, buyers may propose a new order replenishment time to the manufacturer so that savings in cost is more than DW. Manufacturer may accept the offer or otherwise the process may be repeated for an acceptable order replenishment time where both the parties get certain benefit for common transportation of the item. 6. Numerical example The same numerical data are considered here. Sm ¼ 250$ per setup; hm ¼ .005$ per unit per day; Cc ¼ 100$ per shipment; Total demand D ¼ 58; rm ¼ .5. At individually optimized order replenishment time, total relevant cost of buyers is obtained by using Eq. (21) and is ¼ 111.37$ per week Using Eq. (22), the manufacturer’s optimum lot size is Qv ¼ 3405 units. Substituting it into Eq. (23) yields the optimum cost of the manufacturer TCMbc ¼ 59.57$ per week. The common order replenishment time decided by the association of buyers from Eq. (25) is obtained as T ¼ 26:29 days. Substituting it into
Eq. (26) yields the total cost of all the buyers after coordinated order replenishment time TCBac ¼ 89.95$ per week. From Eq. (27), the manufacturer’s total relevant cost at common order interval T is TCMac ¼ 66.5$ per week. The savings in cost to the association of buyers is equal to 21.42$ per week, and the increase in cost at manufacturer’s side is equal to 6.93$ per week. The additional profit available with the association of buyers is equal to 14.49$ per week. 7. Conclusions In this paper, two models of coordination are developed where multiple buyers receive supplies from a manufacturer. In the first model, the transportation cost is borne by the manufacturer whereas in the second model, transportation cost is borne by the buyers. Considering the benefits of common transportation cost in the first case, manufacturer coordinates with all the buyers to have orders at a common replenishment time so that the item can be supplied through common carrier. The increase in cost to the buyers due to coordinated ordering policy is compensated through a uniform credit policy and in the paper, a procedure is suggested to determine the uniform credit time for all the buyers. In the second model, in ex-factory situation, the transportation cost is borne by the buyers. All the buyers located in one region agree to order at a common replenishment time to save in transportation cost. Due to coordinated order policy adopted by the association of buyers, the manufacturer’s cost increases. The buyer and the manufacturer can use the models developed in this paper as a quantitative tool for coordination and negotiation purpose. The stronger party in the channel can take the initiative for coordination and encourage the other party in the channel to adopt a coordinated common order replenishment time to take advantage of common transport. The model can be used to estimate the minimum amount of compensation necessary for coordination and the maximum amount that is feasible. The study revealed that delivering the item at common order interval would result in savings in cost at the stronger party of the channel. The extra savings from the use of common carrier policy to supply the item can be divided between the manufacturer and the association of the buyers providing a potential benefit for both the manufac-
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turer and the set of buyers. The division of savings through negotiation can be a win–win proposition for both the parties and this can lead to a better supply chain coordination. In the development of the model, manufacturer and the buyers must be ready to share information and their sincerity in this direction can be a problem for some organizations. However, the expected benefit received from the coordinated shipment policy may resolve this problem. Acknowledgment The authors are extremely grateful to the two anonymous referees for their very helpful comments and suggestions. References Banerjee, A., Burton, J.S., 1994. Coordinated vs. independent inventory replenishment policies for a vendor and multiple buyers. International Journal of Production Economics 35 (1-3), 215–222. Cetinkaya, S., Lee, C.Y., 2000. Stock replenishment and shipment scheduling for vendor managed inventory systems. Management Science 46 (2), 217–232. Chan, C.K., Kingsman, B.G., 2007. Coordination in a singlevendor multi-buyer supply chain by synchronizing delivery and production cycles. Transportation Research, Part E 43, 90–111. Chapman, C.B., Ward, S.C., Cooper, D.F., Page, M.J., 1985. Credit policy and inventory control. Journal of Operational Research Society 35 (12), 1055–1065. Drezner, Z., Wesolowsky, G.O., 1989. Multi buyers discount pricing. European Journal of Operational Research 40 (1), 38–42. Goyal, S.K., 1985. Economic order quantity under condition of permissible delay in payments. Journal of Operation Research Society 36 (4), 335–338. Hammer, M., 2001. The super efficient company. Harvard Business Review 79 (8), 82–91. Issakson, A., 2002. Trade credit in Kenyan Manufacturing: evidence from plant level data. Working paper, Source: /www.unido.org/userfiles/PuffK/SIN_WPS04.pdfS. Jamal, A.M.M., Sarker, R.B., Wang, S., 2000. Optimal payment time for a retailer under permitted delay of payment by the wholesaler. International Journal of Production Economics 66 (1), 59–66. Joglekar, P.N., 1988. A quantity discount pricing model to increase vendor profits. Management Science 34 (11), 1391–1398. Kim, K.H., Hwang, H., 1988. An incremental discount pricing schedule with multiple customers and single price break. European Journal of Operational Research 35 (1), 71–79.
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