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Selective discount for supplier–buyer coordination using common replenishment epochs
European Journal of Operational Research 153 (2004) 751–756 www.elsevier.com/locate/dsw
Production, Manufacturing and Logistics
Selective discount for supplier–buyer coordination using common replenishment epochs Ajay K. Mishra
*
School of Management, State University of New York, Binghamton, NY 13902, USA Received 27 July 2001; accepted 16 September 2002
Abstract A supplier may reduce its order costs by providing an incentive in the form of price discounts to buyers to restrict their replenishment intervals to multiples of a common replenishment epoch (CRE). This coordination mechanism was studied by Viswanathan and Piplani who suggested that the supplier offer a discount that is the maximum of the discount required by all buyers. We generalize their model to allow for a selective discount policy that, if beneficial, excludes some buyers to minimize the supplierÕs total cost. Using a computational study, we observe that offering discounts to buyers selectively, if necessary, by segmenting them and offering multiple CRE, reduces the supplierÕs cost in many scenarios. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Inventory; Replenishment; Coordination
1. Selective discount model One of the important streams of research in supply chain management is the coordination between suppliers and buyers by varying the lot-size (or, delivery schedule) and the purchase price (e.g., see Goyal and Gupta, 1989; Munson and Rosenblatt, 1998; Tsay et al., 1998). We consider the situation in which a supplier sells an item to m buyers, where buyer i ¼ 1; . . . ; m, has demand Di , order cost Ki , and holding cost rate hi , all known to the supplier. The purchase price P of the item is the same for all buyers. The buyers manage their
*
Tel.: +1-607-777-4214; fax: +1-607-777-4422. E-mail address: [email protected] (A.K. Mishra).
inventory using the EOQ model. We ignore the supplierÕs inventory holding cost (a common assumption when dealing with multiple buyers, e.g., see Weng and Wong, 1993). The supplier purchases the item and, in the absence of coordination with the buyers, procures each buyerÕs order individually, in a lot-for-lot policy. The supplier incurs an order processing cost of Asi and a delivery cost of Ai for buyer iÕs order, independent of the order quantity. In order to save on its order costs, the supplier wishes to pool the orders of the buyers. To this end, it proposes a common replenishment epoch (CRE) and gives the same incentive, a price discount d (0 6 d < 1) on the purchase price, to all the buyers to order in integer multiples of the CRE. For example, if the CRE were two weeks, a participating buyer could choose to order once
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00811-1
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A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
every two weeks, or every four weeks, and so on. We assume that the supplier incurs a joint order processing cost of AS for the buyers who agree to order using the CRE, irrespective of the number of buyers. This simple and elegant coordination mechanism was recently proposed by Viswanathan and Piplani (2001), henceforth referred to as VP. An implicit assumption in their model is that all the buyers participate in this coordination scheme and hence the supplier gives a discount that is the maximum of the discount required by each buyer for participation; we term this inclusive discount (ID). In this paper, we investigate coordination using CRE by allowing the possibility of some buyers participating in the coordination scheme while other buyers continuing to order as earlier; we call this selective discount (SD). We denote the no coordination (implying no discount) situation as ND. SD subsumes the ND and ID policies. We observe that SD decreases the supplierÕs total cost in many situations as compared to ND and ID. In addition, we allow a general P and AS 6¼ Asi (VP assume P ¼ 1 and AS ¼ Asi ), which are minor differences that primarily impact the computational insights. For the convenience of the readers we use the same notation as VP except for a few changes. The supplier selects a CRE, T0 2 X . When buyer i deviates from the EOQ model to order in integer multiples ni P 1 of the CRE, its cost, the sum of order and holding costs, increases. The buyer accepts the discount d and participates in the coordination scheme only if the reduction in purchase cost results in at least 100 S% savings; S can be customized to each buyer as Si without any additional modeling effort. Please note that we model d and S as the same for all participants since this is in keeping with regulations in many countries, e.g., the Robinson–Patman act in the US. Below, we generalize the model in VP. 1.1. Before coordination For buyer i, following the well known EOQ model, the order interval tiU , the sum of order and holding cost giU p , and the total TiU are given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi cost pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U ti ¼ ð1=Di Þ ð 2Ki Di =hi P Þ ¼ 2Ki =hi PDi , giU ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKi =tiU Þ þ ð1=2Þhi PDi tiU ¼ 2Ki hi PDi , and TiU ¼ PDi þ giU : The supplier incurs an order and delivery cost of Ai þ Asi for P buyer i leading to the total m cost gsU given by gsU ¼ i¼1 ðAi þ Asi Þ=tiU . 1.2. After coordination Let C be the set of buyers that accept the discount and use the CRE to place orders; C could be a null set. The order interval for buyer i 2 C is tiC ¼ ni T0 and the sum of order and holding cost, giC , and total cost TiC are given by giC ¼ ðKi =ni T0 Þ þ ð1=2Þhi PDi ni T0 , and TiC ¼ ð1 dÞPDi þ giC . Since giC is convex with respect to ni , to obtain the minimum giC ðni Þ, we select the optimal ni , denoted as ni , such that giC ðni Þ is smaller than both giC ðni 1Þ and giC ðni þ 1Þ, to satisfy ni ðni 1Þ 6 2Ki =hi PDi T02 6 ni ðni þ 1Þ: Hence, ni ¼ bð1=2Þð1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð8Ki =hi PDi T02 ÞÞc, where bxc is the largest integer 6 x. For a given T0 , the discount acceptable to buyer i, qi , can be given by Ki 1 ð1 SÞ 1 qi ¼ þ hi ni T0 ð1 SÞtiU U ti 2 PDi ni T0 ð1Þ where (1) splits the discount sought by the buyer into two components to compensate for order cost and holding cost. Buyer i will not accept the coordination scheme if the discount d < qi . When C is empty, we set q0 ¼ 0. Let C represent the set of buyers who do not accept the discount and continue to place orders as earlier. We notice that for identical T0 , S, h, and P , the discount sought by a buyer is a nonlinear function of Ki =Di . The supplierÕs minimization problem to determine T0 , d, and C is given by AS X Ai C Min gs ¼ þ dPDi þ T0 ni T 0 i2C X ðAj þ Asj Þ þ tjU j2C
s:t: T0 2 X ; d P qi ; i 2 C; and ni P 1; and integer; i 2 C:
ð2Þ
For a given T0 , we will determine d to minimize gsC by using the property that the discount offered by
A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
the supplier will either be zero or be binding for one of the m buyers. As observed by VP, the objective function above need not be convex in the values of T0 . For a given T0 , we find that the objective function is not convex for different values of d ¼ qi , i ¼ 0; 1; . . . ; m. Hence, we must conduct an exhaustive search over T0 and qi , i ¼ 0; 1; . . . ; m.
753
varying Asi , AS , and Ai ; we set AS ¼ aAsi , where a is an integer constant. Table 2 shows the results for Asi ¼ 10; 100; 200; 500; 1000, Ai ¼ 10; 100; 200; 500; 1000, i ¼ 1; . . . ; 15, and a ¼ 1, 3. We see that when Asi is very low, SD is the same as ND, from the supplierÕs perspective. Of the 50 scenarios in Table 2, ND is the same as SD in three while ND is preferred to ID in 20 scenarios. Offering selective discounts is a better policy than ND or ID in 44 of the 50 scenarios. We found that if Asi or Ai were very high (2000, 5000, etc.), SD is the same as ID, and hence we did not include it in Table 2. When a ¼ 1, on the average SD reduces supplierÕs cost by about 38.9% and 17.7% over ND and ID, respectively, and at the same time average buyerÕs savings is 0.1% for the SD policy while it is 0.2% for the ID policy; when a ¼ 3, these numbers are 32.1% and 12%, 0.12%, and 0.2%, respectively. We also implemented a ¼ 5. We noticed that as a increases ND is more attractive for low values of Asi and SD is the same as ID for lower values of Asi or Ai when a ¼ 5 rather than when a ¼ 1. qi is also a nonlinear function of P and hi . For the above data, with hi ¼ 0:15, S ¼ 10%, and a ¼ 1, for P ¼ 1; 2; 10; 100; 1000, we observed that the average savings with SD over ID were 17.7%, 18.3%, 18.5%, 13.8%, and 0.9%, respectively; the savings for SD over ND were 38.9%, 38.4%, 37%, 34.8%, and 25.9%, respectively. For very high P , everything else being the same, the SD policy is effectively the ID policy. We also set P ¼ 1, S ¼ 10%, and a ¼ 1 for hi ¼ 0:1; 0:15; 0:2; 0:3, and observed that the average savings with SD over ID were 16.9%, 17.7%, 16%, and 18.3%, respectively, while the average savings for SD over ND were about 37–38%. It appears that the policies are not sensitive to the holding cost rate, for the instances tested.
Algorithm For x 2 X , set T0 ¼ x Determine discount qi , i ¼ 1; . . . ; m, from (1). For d ¼ qi ; i ¼ 0; 1; . . . ; m, determine the objective function gsC from (2). Select the T0 and d pair that minimizes the objective function gsC . Since there are jX j (i.e., cardinality of X ) major loops and m minor loops, for practical values of jX j and m it is not computationally intensive.
2. Computational study We conducted computations to obtain insights on coordination using CREs and determine situations when it is preferable for a supplier to give discount to a few (SD) rather than to none (ND) or to everyone (ID). We implemented ID and SD such that the supplierÕs savings were calculated only when they were positive. We used the data set for buyers shown in Table 1. We used X ðin weeksÞ ¼ f1=7; 3=7; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13g to obtain more precise optimal T0 for situations in which the supplier has a choice. Some of the key insights follow. With P ¼ 1, S ¼ 10%, and hi ¼ 0:15, i ¼ 1; . . . ; 15, we determined the optimal solution by
Table 1 Data for buyers, sorted by replenishment interval prior to any coordination Buyer
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Order cost (in $) Annual demand (in millions) Replenishment interval (in weeks; price ¼ 1, holding cost rate ¼ 0:15)
50 2
50 1
150 2
50 0.5
150 1
100 0.5
500 2
500 1
1500 2
500 0.5
1500 1
1000 0.5
5000 2
3000 1
3000 0.5
0.9
1.3
1.6
1.9
2.3
2.7
3.0
4.2
5.2
6.0
7.4
8.5
9.5
10.4
14.7
754
Computational study with P ¼ 1, hi ¼ 0:15, and S ¼ 0:1 Serial #
As
Asi
Inclusive discount (ID)
Ai
CRE
Discount (%)
SupplierÕs savings over ND (%) – – –
Selective discount (SD)
SD with Two CRE BuyerÕs savings over ND (%)
System savings over ND (%)
SupplierÕs savings–Improvement over SD (%)
CRE– Discount (%)–# Included
0.00 0.83 6.20
– 0.01 0.03
– 0.01 0.05
– – 0.03
15.36 25.22
15.36 5.70
0.11 0.11
0.23 0.49
– –
– – 2–0.11–5; 3–0.08–1 – –
0.11 0.11 0.11
31.73 23.83 20.90
31.73 23.83 20.90
0.03 0.03 0.03
0.08 0.10 0.12
– – 1.07
10 10
0.23 0.23
25.58 29.99
21.19 4.26
0.11 0.11
0.34 0.60
– –
2 2 2
8 8 8
0.12 0.12 0.12
48.58 38.66 32.87
48.58 38.66 32.87
0.05 0.05 0.05
0.21 0.23 0.25
– – 0.85
SupplierÕs savings over ND (%)
BuyerÕs savings over ND (%)
System savings over ND (%)
CRE
# Included
Discount (%)
– – –
– – –
– 2 2
– 3 5
– 0.07 0.11
– 0.83 6.20
– 0.34
– 0.64
3 3
10 10
0.23 0.23
– – –
– – –
2 2 2
6 6 5
0.34 0.34
0.38 0.77
3 3
– – –
– – –
SupplierÕs savings–Improvement over ID (%)
1 2 3
10 10 10
10 10 10
10 100 200
– – –
– – –
4 5
10 10
10 10
500 1000
– 5
– 0.50
6 7 8
100 100 100
100 100 100
10 100 200
– – –
– – –
9 10
100 100
100 100
500 1000
5 5
0.50 0.50
11 12 13
200 200 200
200 200 200
10 100 200
– – –
– – –
14 15
200 200
200 200
500 1000
5 5
0.50 0.50
18.51 32.65
0.34 0.34
0.53 0.92
3 3
10 10
0.23 0.23
33.85 34.44
18.83 2.67
0.11 0.11
0.47 0.73
– –
16
500
500
10
3
0.43
38.36
0.37
0.66
2
10
0.17
63.27
40.41
0.10
0.59
5.06
17
500
500
100
4
0.44
39.16
0.33
0.68
3
10
0.23
57.72
30.50
0.11
0.63
0.23
18
500
500
200
4
0.44
39.88
0.33
0.74
3
10
0.23
54.51
24.34
0.11
0.69
0.24
– 20.70 – – – 5.57 26.87 – – –
– – 2–0.11–5; 4–0.14–3 – – – – 2–0.117– 5; 4– 0.14–3 – – 2–0.12–8; 8–0.25–3 2–0.12–8; 8–0.25–3 3–0.23– 10; 9– 0.26–2
A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
Table 2
500
500
500
5
0.50
41.80
0.34
0.96
3
12
0.26
48.86
12.14
0.15
0.88
0.26
20
500
500
1000
5
0.50
45.34
0.34
1.34
4
14
0.36
45.41
0.11
0.24
1.25
–
3–0.23– 10; 9– 0.26–2 –
21 22 23 24 25
1000 1000 1000 1000 1000
1000 1000 1000 1000 1000
10 100 200 500 1000
4 4 4 5 5
0.44 0.44 0.44 0.50 0.50
66.45 64.62 62.92 59.91 58.04
0.33 0.33 0.33 0.34 0.34
1.33 1.38 1.45 1.66 2.04
3 3 3 3 5
12 12 12 12 15
0.26 0.26 0.26 0.26 0.50
74.72 71.54 68.57 62.03 58.04
24.66 19.56 15.24 5.29 0.00
0.15 0.15 0.15 0.15 0.34
1.28 1.32 1.38 1.53 2.04
– – – – –
– – – – –
26 27 28 29 30
30 30 30 30 30
10 10 10 10 10
10 100 200 500 1000
– – – – 5
– – – – 0.50
– – – – 20.62
– – – – 0.34
– – – – 0.64
– – 2 3 3
– – 5 10 10
– – 0.11 0.23 0.23
– – 5.28 15.11 25.09
0.00 0.00 5.28 15.11 5.63
– – 0.03 0.11 0.11
– – 0.04 0.23 0.49
– – – – –
– – – – –
31 32 33 34 35
300 300 300 300 300
100 100 100 100 100
10 100 200 500 1000
– – – 5 5
– – – 0.50 0.50
– – – 4.28 26.17
– – – 0.34 0.34
– – – 0.37 0.76
2 2 2 3 3
6 6 5 10 10
0.11 0.11 0.11 0.23 0.23
14.20 14.19 14.47 23.43 28.82
14.20 14.19 14.47 20.01 3.5 8
0.03 0.03 0.03 0.11 0.11
0.05 0.07 0.09 0.32 0.58
– – – – –
– – – – –
36 37 38 39 40
600 600 600 600 600
200 200 200 200 200
10 100 200 500 1000
– – – 5 5
– – – 0.50 0.50
– – – 16.30 31.36
– – – 0.34 0.34
– – – 0.50 0.89
2 2 3 3 3
8 8 10 10 10
0.12 0.12 0.23 0.23 0.23
30.22 25.80 26.35 30.18 32.30
30.22 25.80 26.35 16.57 1.37
0.05 0.05 0.11 0.11 0.11
0.15 0.17 0.27 0.43 0.69
– – – – –
– – – – –
41 42 43 44 45
1500 1500 1500 1500 1500
500 500 500 500 500
10 100 200 500 1000
4 4 4 5 5
0.44 0.44 0.44 0.50 0.50
28.83 31.13 32.99 37.94 42.77
0.33 0.33 0.33 0.34 0.34
0.55 0.61 0.67 0.90 1.28
3 3 3 3 5
10 10 10 12 15
0.23 0.23 0.23 0.26 0.50
49.07 47.00 45.33 42.44 42.77
28.44 23.05 18.41 7.24 0.00
0.11 0.11 0.11 0.15 0.34
0.49 0.54 0.59 0.78 1.28
– – – – –
– – – – –
46 47 48 49 50
3000 3000 3000 3000 3000
1000 1000 1000 1000 1000
10 100 200 500 1000
4 4 5 5 5
0.44 0.44 0.50 0.50 0.50
56.90 55.86 55.36 54.77 54.19
0.33 0.33 0.34 0.34 0.34
1.18 1.24 1.32 1.55 1.93
3 3 4 4 5
12 12 14 14 15
0.26 0.26 0.36 0.36 0.50
61.99 59.86 58.46 55.58 54.19
11.81 9.06 6.96 1.79 0.00
0.15 0.15 0.24 0.24 0.34
1.09 1.13 1.29 1.48 1.93
– – – – –
– – – – –
A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
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A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
We also observed the three policies for the data in VP. Of the 48 scenarios reported by them (VP, Table 2, pp. 282–283), ND dominated ID in twenty-three scenarios and SD in three (when Asi ¼ 0, Ai ¼ 100, hi ¼ 0:1; 0:2; 0:3). SD was the same as ID in two others (when hi ¼ 0:3, Ai ¼ 1000, Asi ¼ 200; 500). Thus, SDs were preferred in 43 of 48 instances reported in their paper. 2.1. Selective discount with two CRE This concept and model can be further extended by segmenting the buyers and offering different discounts for multiple CREs. It allows giving less discount to some who were getting more when only one CRE was offered or attracting more buyers to coordinate using CRE, but this benefit must be balanced against a higher order processing cost. We implemented SD with two CRE, each with different discounts, such that buyers had the option to choose either CRE or none. The optimal solution for the supplier was determined by extending the algorithm in the previous section and using an exhaustive search. We found that when a ¼ 1, SD with two CRE improved over SD with one CRE in 7 of 25 scenarios. The last two columns in Table 2 show the increase in supplierÕs savings over SD (with one CRE), the two CRE proposed, and the corresponding discounts offered. When a ¼ 3 or 5, SD with two CRE was never the cheapest option for the supplier.
3. Summary and conclusions When a supplier purchases and delivers an item to several buyers it could use a CRE strategy and provide a discount to the participants. Viswanathan and Piplani (2001) first studied this model and suggested that the supplier offer a discount that is the maximum of the discount required by all buyers to participate. We propose a selective discount policy in which the goal is to minimize supplierÕs cost; the discount and CRE are chosen so that, if necessary, some buyers are excluded. From the model and computational experience
reported above, we find that in most practical situations it makes sense to coordinate with at least a few buyers and, in some situations, it might be beneficial to segment the buyers and offer multiple CRE; this paper provides a methodology to decide the discount. Together with inclusive discounts, the CRE strategy dominates no coordination situations. Hence, the strategy merits further investigation. We conjecture that it will be more effective when the replenishment interval of buyers is spread out and there are a large number of buyers. When there are a small number of buyers this approach might still be effective but the objective function (2) does not calculate the cost precisely because the first factor (AS =T0 ) assumes that an order is placed every T0 periods. When the replenishment interval of the buyers are close by, CRE with ID should be the optimal choice. Acknowledgements I thank the reviewers and the editor for their careful reading and insightful suggestions that improved the paper. I also gratefully acknowledge research support for this work through a summer research grant from the School of Management, State University of New York, Binghamton.
References Goyal, S.K., Gupta, Y.P., 1989. Integrated inventory models: The buyer–vendor coordination. European Journal of Operational Research 41, 261–269. Munson, C.L., Rosenblatt, M.J., 1998. Theories and realities of quantity discounts: An exploratory study. Production and Operations Management 7 (4), 352–369. Tsay, A.A., Nahmias, S., Agrawal, N., 1998. Modeling supply chain contracts: A review. In: Tayur, S., Ganeshan, R., Magazine, M. (Eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Boston, MA, pp. 299–336. Viswanathan, S., Piplani, R., 2001. Coordinating supply chain inventories through common replenishment epochs. European Journal of Operational Research 129, 277–286. Weng, Z.K., Wong, R.T., 1993. General models for the supplierÕs all-unit quantity discount policy. Naval Research Logistics 40, 971–991.
Production, Manufacturing and Logistics
Selective discount for supplier–buyer coordination using common replenishment epochs Ajay K. Mishra
*
School of Management, State University of New York, Binghamton, NY 13902, USA Received 27 July 2001; accepted 16 September 2002
Abstract A supplier may reduce its order costs by providing an incentive in the form of price discounts to buyers to restrict their replenishment intervals to multiples of a common replenishment epoch (CRE). This coordination mechanism was studied by Viswanathan and Piplani who suggested that the supplier offer a discount that is the maximum of the discount required by all buyers. We generalize their model to allow for a selective discount policy that, if beneficial, excludes some buyers to minimize the supplierÕs total cost. Using a computational study, we observe that offering discounts to buyers selectively, if necessary, by segmenting them and offering multiple CRE, reduces the supplierÕs cost in many scenarios. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Supply chain management; Inventory; Replenishment; Coordination
1. Selective discount model One of the important streams of research in supply chain management is the coordination between suppliers and buyers by varying the lot-size (or, delivery schedule) and the purchase price (e.g., see Goyal and Gupta, 1989; Munson and Rosenblatt, 1998; Tsay et al., 1998). We consider the situation in which a supplier sells an item to m buyers, where buyer i ¼ 1; . . . ; m, has demand Di , order cost Ki , and holding cost rate hi , all known to the supplier. The purchase price P of the item is the same for all buyers. The buyers manage their
*
Tel.: +1-607-777-4214; fax: +1-607-777-4422. E-mail address: [email protected] (A.K. Mishra).
inventory using the EOQ model. We ignore the supplierÕs inventory holding cost (a common assumption when dealing with multiple buyers, e.g., see Weng and Wong, 1993). The supplier purchases the item and, in the absence of coordination with the buyers, procures each buyerÕs order individually, in a lot-for-lot policy. The supplier incurs an order processing cost of Asi and a delivery cost of Ai for buyer iÕs order, independent of the order quantity. In order to save on its order costs, the supplier wishes to pool the orders of the buyers. To this end, it proposes a common replenishment epoch (CRE) and gives the same incentive, a price discount d (0 6 d < 1) on the purchase price, to all the buyers to order in integer multiples of the CRE. For example, if the CRE were two weeks, a participating buyer could choose to order once
0377-2217/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-2217(02)00811-1
752
A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
every two weeks, or every four weeks, and so on. We assume that the supplier incurs a joint order processing cost of AS for the buyers who agree to order using the CRE, irrespective of the number of buyers. This simple and elegant coordination mechanism was recently proposed by Viswanathan and Piplani (2001), henceforth referred to as VP. An implicit assumption in their model is that all the buyers participate in this coordination scheme and hence the supplier gives a discount that is the maximum of the discount required by each buyer for participation; we term this inclusive discount (ID). In this paper, we investigate coordination using CRE by allowing the possibility of some buyers participating in the coordination scheme while other buyers continuing to order as earlier; we call this selective discount (SD). We denote the no coordination (implying no discount) situation as ND. SD subsumes the ND and ID policies. We observe that SD decreases the supplierÕs total cost in many situations as compared to ND and ID. In addition, we allow a general P and AS 6¼ Asi (VP assume P ¼ 1 and AS ¼ Asi ), which are minor differences that primarily impact the computational insights. For the convenience of the readers we use the same notation as VP except for a few changes. The supplier selects a CRE, T0 2 X . When buyer i deviates from the EOQ model to order in integer multiples ni P 1 of the CRE, its cost, the sum of order and holding costs, increases. The buyer accepts the discount d and participates in the coordination scheme only if the reduction in purchase cost results in at least 100 S% savings; S can be customized to each buyer as Si without any additional modeling effort. Please note that we model d and S as the same for all participants since this is in keeping with regulations in many countries, e.g., the Robinson–Patman act in the US. Below, we generalize the model in VP. 1.1. Before coordination For buyer i, following the well known EOQ model, the order interval tiU , the sum of order and holding cost giU p , and the total TiU are given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi cost pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U ti ¼ ð1=Di Þ ð 2Ki Di =hi P Þ ¼ 2Ki =hi PDi , giU ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðKi =tiU Þ þ ð1=2Þhi PDi tiU ¼ 2Ki hi PDi , and TiU ¼ PDi þ giU : The supplier incurs an order and delivery cost of Ai þ Asi for P buyer i leading to the total m cost gsU given by gsU ¼ i¼1 ðAi þ Asi Þ=tiU . 1.2. After coordination Let C be the set of buyers that accept the discount and use the CRE to place orders; C could be a null set. The order interval for buyer i 2 C is tiC ¼ ni T0 and the sum of order and holding cost, giC , and total cost TiC are given by giC ¼ ðKi =ni T0 Þ þ ð1=2Þhi PDi ni T0 , and TiC ¼ ð1 dÞPDi þ giC . Since giC is convex with respect to ni , to obtain the minimum giC ðni Þ, we select the optimal ni , denoted as ni , such that giC ðni Þ is smaller than both giC ðni 1Þ and giC ðni þ 1Þ, to satisfy ni ðni 1Þ 6 2Ki =hi PDi T02 6 ni ðni þ 1Þ: Hence, ni ¼ bð1=2Þð1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ð8Ki =hi PDi T02 ÞÞc, where bxc is the largest integer 6 x. For a given T0 , the discount acceptable to buyer i, qi , can be given by Ki 1 ð1 SÞ 1 qi ¼ þ hi ni T0 ð1 SÞtiU U ti 2 PDi ni T0 ð1Þ where (1) splits the discount sought by the buyer into two components to compensate for order cost and holding cost. Buyer i will not accept the coordination scheme if the discount d < qi . When C is empty, we set q0 ¼ 0. Let C represent the set of buyers who do not accept the discount and continue to place orders as earlier. We notice that for identical T0 , S, h, and P , the discount sought by a buyer is a nonlinear function of Ki =Di . The supplierÕs minimization problem to determine T0 , d, and C is given by AS X Ai C Min gs ¼ þ dPDi þ T0 ni T 0 i2C X ðAj þ Asj Þ þ tjU j2C
s:t: T0 2 X ; d P qi ; i 2 C; and ni P 1; and integer; i 2 C:
ð2Þ
For a given T0 , we will determine d to minimize gsC by using the property that the discount offered by
A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
the supplier will either be zero or be binding for one of the m buyers. As observed by VP, the objective function above need not be convex in the values of T0 . For a given T0 , we find that the objective function is not convex for different values of d ¼ qi , i ¼ 0; 1; . . . ; m. Hence, we must conduct an exhaustive search over T0 and qi , i ¼ 0; 1; . . . ; m.
753
varying Asi , AS , and Ai ; we set AS ¼ aAsi , where a is an integer constant. Table 2 shows the results for Asi ¼ 10; 100; 200; 500; 1000, Ai ¼ 10; 100; 200; 500; 1000, i ¼ 1; . . . ; 15, and a ¼ 1, 3. We see that when Asi is very low, SD is the same as ND, from the supplierÕs perspective. Of the 50 scenarios in Table 2, ND is the same as SD in three while ND is preferred to ID in 20 scenarios. Offering selective discounts is a better policy than ND or ID in 44 of the 50 scenarios. We found that if Asi or Ai were very high (2000, 5000, etc.), SD is the same as ID, and hence we did not include it in Table 2. When a ¼ 1, on the average SD reduces supplierÕs cost by about 38.9% and 17.7% over ND and ID, respectively, and at the same time average buyerÕs savings is 0.1% for the SD policy while it is 0.2% for the ID policy; when a ¼ 3, these numbers are 32.1% and 12%, 0.12%, and 0.2%, respectively. We also implemented a ¼ 5. We noticed that as a increases ND is more attractive for low values of Asi and SD is the same as ID for lower values of Asi or Ai when a ¼ 5 rather than when a ¼ 1. qi is also a nonlinear function of P and hi . For the above data, with hi ¼ 0:15, S ¼ 10%, and a ¼ 1, for P ¼ 1; 2; 10; 100; 1000, we observed that the average savings with SD over ID were 17.7%, 18.3%, 18.5%, 13.8%, and 0.9%, respectively; the savings for SD over ND were 38.9%, 38.4%, 37%, 34.8%, and 25.9%, respectively. For very high P , everything else being the same, the SD policy is effectively the ID policy. We also set P ¼ 1, S ¼ 10%, and a ¼ 1 for hi ¼ 0:1; 0:15; 0:2; 0:3, and observed that the average savings with SD over ID were 16.9%, 17.7%, 16%, and 18.3%, respectively, while the average savings for SD over ND were about 37–38%. It appears that the policies are not sensitive to the holding cost rate, for the instances tested.
Algorithm For x 2 X , set T0 ¼ x Determine discount qi , i ¼ 1; . . . ; m, from (1). For d ¼ qi ; i ¼ 0; 1; . . . ; m, determine the objective function gsC from (2). Select the T0 and d pair that minimizes the objective function gsC . Since there are jX j (i.e., cardinality of X ) major loops and m minor loops, for practical values of jX j and m it is not computationally intensive.
2. Computational study We conducted computations to obtain insights on coordination using CREs and determine situations when it is preferable for a supplier to give discount to a few (SD) rather than to none (ND) or to everyone (ID). We implemented ID and SD such that the supplierÕs savings were calculated only when they were positive. We used the data set for buyers shown in Table 1. We used X ðin weeksÞ ¼ f1=7; 3=7; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13g to obtain more precise optimal T0 for situations in which the supplier has a choice. Some of the key insights follow. With P ¼ 1, S ¼ 10%, and hi ¼ 0:15, i ¼ 1; . . . ; 15, we determined the optimal solution by
Table 1 Data for buyers, sorted by replenishment interval prior to any coordination Buyer
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Order cost (in $) Annual demand (in millions) Replenishment interval (in weeks; price ¼ 1, holding cost rate ¼ 0:15)
50 2
50 1
150 2
50 0.5
150 1
100 0.5
500 2
500 1
1500 2
500 0.5
1500 1
1000 0.5
5000 2
3000 1
3000 0.5
0.9
1.3
1.6
1.9
2.3
2.7
3.0
4.2
5.2
6.0
7.4
8.5
9.5
10.4
14.7
754
Computational study with P ¼ 1, hi ¼ 0:15, and S ¼ 0:1 Serial #
As
Asi
Inclusive discount (ID)
Ai
CRE
Discount (%)
SupplierÕs savings over ND (%) – – –
Selective discount (SD)
SD with Two CRE BuyerÕs savings over ND (%)
System savings over ND (%)
SupplierÕs savings–Improvement over SD (%)
CRE– Discount (%)–# Included
0.00 0.83 6.20
– 0.01 0.03
– 0.01 0.05
– – 0.03
15.36 25.22
15.36 5.70
0.11 0.11
0.23 0.49
– –
– – 2–0.11–5; 3–0.08–1 – –
0.11 0.11 0.11
31.73 23.83 20.90
31.73 23.83 20.90
0.03 0.03 0.03
0.08 0.10 0.12
– – 1.07
10 10
0.23 0.23
25.58 29.99
21.19 4.26
0.11 0.11
0.34 0.60
– –
2 2 2
8 8 8
0.12 0.12 0.12
48.58 38.66 32.87
48.58 38.66 32.87
0.05 0.05 0.05
0.21 0.23 0.25
– – 0.85
SupplierÕs savings over ND (%)
BuyerÕs savings over ND (%)
System savings over ND (%)
CRE
# Included
Discount (%)
– – –
– – –
– 2 2
– 3 5
– 0.07 0.11
– 0.83 6.20
– 0.34
– 0.64
3 3
10 10
0.23 0.23
– – –
– – –
2 2 2
6 6 5
0.34 0.34
0.38 0.77
3 3
– – –
– – –
SupplierÕs savings–Improvement over ID (%)
1 2 3
10 10 10
10 10 10
10 100 200
– – –
– – –
4 5
10 10
10 10
500 1000
– 5
– 0.50
6 7 8
100 100 100
100 100 100
10 100 200
– – –
– – –
9 10
100 100
100 100
500 1000
5 5
0.50 0.50
11 12 13
200 200 200
200 200 200
10 100 200
– – –
– – –
14 15
200 200
200 200
500 1000
5 5
0.50 0.50
18.51 32.65
0.34 0.34
0.53 0.92
3 3
10 10
0.23 0.23
33.85 34.44
18.83 2.67
0.11 0.11
0.47 0.73
– –
16
500
500
10
3
0.43
38.36
0.37
0.66
2
10
0.17
63.27
40.41
0.10
0.59
5.06
17
500
500
100
4
0.44
39.16
0.33
0.68
3
10
0.23
57.72
30.50
0.11
0.63
0.23
18
500
500
200
4
0.44
39.88
0.33
0.74
3
10
0.23
54.51
24.34
0.11
0.69
0.24
– 20.70 – – – 5.57 26.87 – – –
– – 2–0.11–5; 4–0.14–3 – – – – 2–0.117– 5; 4– 0.14–3 – – 2–0.12–8; 8–0.25–3 2–0.12–8; 8–0.25–3 3–0.23– 10; 9– 0.26–2
A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
Table 2
500
500
500
5
0.50
41.80
0.34
0.96
3
12
0.26
48.86
12.14
0.15
0.88
0.26
20
500
500
1000
5
0.50
45.34
0.34
1.34
4
14
0.36
45.41
0.11
0.24
1.25
–
3–0.23– 10; 9– 0.26–2 –
21 22 23 24 25
1000 1000 1000 1000 1000
1000 1000 1000 1000 1000
10 100 200 500 1000
4 4 4 5 5
0.44 0.44 0.44 0.50 0.50
66.45 64.62 62.92 59.91 58.04
0.33 0.33 0.33 0.34 0.34
1.33 1.38 1.45 1.66 2.04
3 3 3 3 5
12 12 12 12 15
0.26 0.26 0.26 0.26 0.50
74.72 71.54 68.57 62.03 58.04
24.66 19.56 15.24 5.29 0.00
0.15 0.15 0.15 0.15 0.34
1.28 1.32 1.38 1.53 2.04
– – – – –
– – – – –
26 27 28 29 30
30 30 30 30 30
10 10 10 10 10
10 100 200 500 1000
– – – – 5
– – – – 0.50
– – – – 20.62
– – – – 0.34
– – – – 0.64
– – 2 3 3
– – 5 10 10
– – 0.11 0.23 0.23
– – 5.28 15.11 25.09
0.00 0.00 5.28 15.11 5.63
– – 0.03 0.11 0.11
– – 0.04 0.23 0.49
– – – – –
– – – – –
31 32 33 34 35
300 300 300 300 300
100 100 100 100 100
10 100 200 500 1000
– – – 5 5
– – – 0.50 0.50
– – – 4.28 26.17
– – – 0.34 0.34
– – – 0.37 0.76
2 2 2 3 3
6 6 5 10 10
0.11 0.11 0.11 0.23 0.23
14.20 14.19 14.47 23.43 28.82
14.20 14.19 14.47 20.01 3.5 8
0.03 0.03 0.03 0.11 0.11
0.05 0.07 0.09 0.32 0.58
– – – – –
– – – – –
36 37 38 39 40
600 600 600 600 600
200 200 200 200 200
10 100 200 500 1000
– – – 5 5
– – – 0.50 0.50
– – – 16.30 31.36
– – – 0.34 0.34
– – – 0.50 0.89
2 2 3 3 3
8 8 10 10 10
0.12 0.12 0.23 0.23 0.23
30.22 25.80 26.35 30.18 32.30
30.22 25.80 26.35 16.57 1.37
0.05 0.05 0.11 0.11 0.11
0.15 0.17 0.27 0.43 0.69
– – – – –
– – – – –
41 42 43 44 45
1500 1500 1500 1500 1500
500 500 500 500 500
10 100 200 500 1000
4 4 4 5 5
0.44 0.44 0.44 0.50 0.50
28.83 31.13 32.99 37.94 42.77
0.33 0.33 0.33 0.34 0.34
0.55 0.61 0.67 0.90 1.28
3 3 3 3 5
10 10 10 12 15
0.23 0.23 0.23 0.26 0.50
49.07 47.00 45.33 42.44 42.77
28.44 23.05 18.41 7.24 0.00
0.11 0.11 0.11 0.15 0.34
0.49 0.54 0.59 0.78 1.28
– – – – –
– – – – –
46 47 48 49 50
3000 3000 3000 3000 3000
1000 1000 1000 1000 1000
10 100 200 500 1000
4 4 5 5 5
0.44 0.44 0.50 0.50 0.50
56.90 55.86 55.36 54.77 54.19
0.33 0.33 0.34 0.34 0.34
1.18 1.24 1.32 1.55 1.93
3 3 4 4 5
12 12 14 14 15
0.26 0.26 0.36 0.36 0.50
61.99 59.86 58.46 55.58 54.19
11.81 9.06 6.96 1.79 0.00
0.15 0.15 0.24 0.24 0.34
1.09 1.13 1.29 1.48 1.93
– – – – –
– – – – –
A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
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A.K. Mishra / European Journal of Operational Research 153 (2004) 751–756
We also observed the three policies for the data in VP. Of the 48 scenarios reported by them (VP, Table 2, pp. 282–283), ND dominated ID in twenty-three scenarios and SD in three (when Asi ¼ 0, Ai ¼ 100, hi ¼ 0:1; 0:2; 0:3). SD was the same as ID in two others (when hi ¼ 0:3, Ai ¼ 1000, Asi ¼ 200; 500). Thus, SDs were preferred in 43 of 48 instances reported in their paper. 2.1. Selective discount with two CRE This concept and model can be further extended by segmenting the buyers and offering different discounts for multiple CREs. It allows giving less discount to some who were getting more when only one CRE was offered or attracting more buyers to coordinate using CRE, but this benefit must be balanced against a higher order processing cost. We implemented SD with two CRE, each with different discounts, such that buyers had the option to choose either CRE or none. The optimal solution for the supplier was determined by extending the algorithm in the previous section and using an exhaustive search. We found that when a ¼ 1, SD with two CRE improved over SD with one CRE in 7 of 25 scenarios. The last two columns in Table 2 show the increase in supplierÕs savings over SD (with one CRE), the two CRE proposed, and the corresponding discounts offered. When a ¼ 3 or 5, SD with two CRE was never the cheapest option for the supplier.
3. Summary and conclusions When a supplier purchases and delivers an item to several buyers it could use a CRE strategy and provide a discount to the participants. Viswanathan and Piplani (2001) first studied this model and suggested that the supplier offer a discount that is the maximum of the discount required by all buyers to participate. We propose a selective discount policy in which the goal is to minimize supplierÕs cost; the discount and CRE are chosen so that, if necessary, some buyers are excluded. From the model and computational experience
reported above, we find that in most practical situations it makes sense to coordinate with at least a few buyers and, in some situations, it might be beneficial to segment the buyers and offer multiple CRE; this paper provides a methodology to decide the discount. Together with inclusive discounts, the CRE strategy dominates no coordination situations. Hence, the strategy merits further investigation. We conjecture that it will be more effective when the replenishment interval of buyers is spread out and there are a large number of buyers. When there are a small number of buyers this approach might still be effective but the objective function (2) does not calculate the cost precisely because the first factor (AS =T0 ) assumes that an order is placed every T0 periods. When the replenishment interval of the buyers are close by, CRE with ID should be the optimal choice. Acknowledgements I thank the reviewers and the editor for their careful reading and insightful suggestions that improved the paper. I also gratefully acknowledge research support for this work through a summer research grant from the School of Management, State University of New York, Binghamton.
References Goyal, S.K., Gupta, Y.P., 1989. Integrated inventory models: The buyer–vendor coordination. European Journal of Operational Research 41, 261–269. Munson, C.L., Rosenblatt, M.J., 1998. Theories and realities of quantity discounts: An exploratory study. Production and Operations Management 7 (4), 352–369. Tsay, A.A., Nahmias, S., Agrawal, N., 1998. Modeling supply chain contracts: A review. In: Tayur, S., Ganeshan, R., Magazine, M. (Eds.), Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Boston, MA, pp. 299–336. Viswanathan, S., Piplani, R., 2001. Coordinating supply chain inventories through common replenishment epochs. European Journal of Operational Research 129, 277–286. Weng, Z.K., Wong, R.T., 1993. General models for the supplierÕs all-unit quantity discount policy. Naval Research Logistics 40, 971–991.