European Journal of Operational Research 180 (2007) 601–616
www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
A quantity discount approach to supply chain coordination
Hojung Shin
a,1
, W.C. Benton
b,*
a
b
Department of LSOM, Business School, Korea University, Anam-dong, Seongbuk-gu, Seoul 136-701, Korea
Department of Management Sciences, Fisher College of Business, The Ohio State University, Columbus, OH 43210, United States
Received 5 March 2004; accepted 2 April 2006
Available online 23 June 2006
Abstract
Quantity discounts provide a practical foundation for inventory coordination in supply chains. However, typical supply
chain participants may encounter difficulties in implementing the coordination policy simply because (1) specified lot size
adjustments may deviate from the economic lot sizes and (2) the buying firm may face amplified overstocking risks related
to increased order quantities. The main objective of this study is to develop a quantity discount model that resolves the
practical challenges associated with implementing quantity discount policies for supply chain coordination between a supplier and a buyer. The proposed Buyer’s Risk Adjustment (B-RA) model allows the supplier to offer discounts that capitalize on the original economic lot sizes and share the buyer’s risk of temporary overstocking under uncertain demand.
The analytical results suggest that the proposed B-RA discount approach is a feasible alternative for supply chain coordination under uncertain demand conditions.
2006 Elsevier B.V. All rights reserved.
Keywords: Quantity discounts; Inventory coordination; Supply management
1. Introduction
The popularity of quantity discounts in practice stems from the fact that suppliers’ discount offers can influence buying firms’ purchasing behavior by providing economic incentives to the buying firms. The terms,
‘‘buying firm’’ and ‘‘buyer’’ will be used interchangeably. Since Monahan (1984), the literature has shown that
the application of all-units discounts contributes to reducing the buyer’s inventory cost and improving the supplier’s profit simultaneously (Rubin and Benton, 2003; Yano and Gilbert, 2004). The literature includes
Rosenblatt and Lee (1985), Lee and Rosenblatt (1986), Kim and Hwang (1989), Weng and Wong (1993),
Weng (1995), Klastorin et al. (2002), Rubin and Benton (2003) and Shin and Benton (2004). This research
herein is an extension of Lee and Rosenblatt (1986) and Weng and Wong (1993) which model the use of
all-units discounts to coordinate inventory decisions between a single buyer and a single supplier. For a more
*
1
Corresponding author. Tel.: +1 614 292 8868; fax: +1 614 292 1272.
E-mail addresses: [email protected] (H. Shin), [email protected] (W.C. Benton).
Tel.: +82 2 3290 2813.
0377-2217/$ - see front matter 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2006.04.033
602
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
comprehensive review of the literature, readers may refer to the review articles in this subject area, including
Rubin and Benton (2003) and Yano and Gilbert (2004).
The motivation for this research is driven by the shortcomings of the traditional quantity discount literature. Specifically, there are many hidden costs associated with the operational characteristics of the traditional
discount approach when supply chain relationships are considered. First, a common assumption made in the
literature is that the supplier and the buyer are willing to deviate from their current order quantities as a quantity discount policy is adopted. In practice, however, the current order quantities may represent the firms’ economic lot sizes which minimize the combination of inventory carrying cost, ordering cost, and transportation
cost. Chen (2000) addressed that in practice materials flows from one stage to another usually occur in the
form of fixed lot sizes, such as full truckloads and full containers, to achieve transportation economies of scale.
Under these circumstances, any lot size adjustment may force the buyer to carry less-than-truck loads, resulting in hidden freight costs that were not considered in the conventional quantity discount models. Economies
of freight lots are one of the primary concerns of logistics managers (Rabinovich et al., 1999), and it is possible
that the freight rate condition may mitigate any discount savings.
Second, most quantity discount models consider deterministic demand, assuming that the buyer faces no
risk when accepting quantity discounts from the supplier. Conventionally, deterministic quantity discount
models are designed to maximize the supplier’s profit. In other words, these deterministic models schedule
a minimum level of discounts which barely compensates the buying firm for its increased cycle-inventory holding cost, only to leave marginal or zero cost savings to the buying firm (Rubin and Benton, 2003, p. 182).
When demand is uncertain, however, this minimum discount approach may not be acceptable to the buyer
because an increase in order quantity in exchange for quantity discounts escalates the buyer’s temporary overstocking risk and cost. If the realized demand becomes lower than the expected demand for a period, the buyer
eventually must carry a larger volume of cycle inventory for a longer period under the quantity discount policy
than under the no-discount policy. Thus, it is likely that the buyer will reject the supplier’s discount offer, and
the supplier’s attempt for supply chain inventory coordination will fail.
Given the limitations in the literature, the primary objective of this paper is to suggest a more equitable
quantity discount model under stationary stochastic demand (uncertain demand hereafter). In our Buyer’ Risk
Adjustment (B-RA) modeling approach, we intend to improve both supplier’s and buyer’s profitability without changing the basis of both parties’ economic lot sizes. In other words, the economic lot sizes for both the
buying and selling firms are kept intact under the proposed B-RA modeling approach.
In addition, the B-RA discount approach allows for a supplier’s discount schedule that considers the
buyer’s amplified overstocking risk under uncertain demand. The objective is to ensure that the supplier offers
a discount schedule that is acceptable to the buyer cost and risk structure. The B-RA model is useful for supply chains where the supplier and the buyer are independent entities and have little motivation to sacrifice their
own profitability, yet they share enough information to achieve a degree of coordination for better supply
chain performance. The results from the numerical experiments verify that the proposed B-RA approach
can be a more practical supply chain coordination mechanism under uncertain demand conditions.
2. The model
In most instances, all-units discount models for supply chain inventory coordination have evolved from the
supplier’s perspective (Benton and Park, 1996; Rubin and Benton, 2003). The supplier is usually the active
party who offers the discount schedule to the buyer in order to entice the buyer to place a larger order quantity. The modeling tradition has been to determine an appropriate single price break in order to maximize the
profitability of the supplying firm. As an example, if the discount is set too high, the supplier experiences on
the average decreased profits. If the discount is set too low, the buyer will ignore the supplier’s discount policy.
To reconcile the conflicting interests between the buyer and the supplier, it has been suggested that the supplier
find a minimum level of discount, which would compensate for at least the buyer’s increased inventory holding
costs under the discount policy. With the objective stated above, typical all-units discount models evaluate the
optimal values that include (1) supplier’s economic lot size, (2) discount per unit, and (3) buyer’s corresponding order quantity given the discount. The structure of the proposed B-RA model is consistent with the conventional all-units discount models. However there are two points of departure. They are:
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
603
• The relaxation of the deterministic constant demand assumption.
• The relaxation of the constant use of single price break assumption.
In the proposed B-RA model, the supplier can use discounts intermittently with which the buyer’s and the
supplier’s economic lot sizes are reserved. To illustrate, Fig. 1 is drawn based on Condition [03] in Table 1 with
an assumption that demand rate is constant. If the buyer’s and the supplier’s inventory systems are sequentially optimized without using any discount policy, the buyer’s optimal order quantity is Q (usually EOQ)
and the supplier’s economic lot size is 5Q as shown in Fig. 1(a). Weng and Wong (1993) show that the system
cost can be minimized by the constant use of quantity discounts if the buyer’s and the supplier’s lot sizes
increase to 1.8Q and 5.4Q respectively, as shown in Fig. 1(b). However, Weng and Wong’s solution may
not be practical and cost effective because it may force the buyer to carry 1.8 truckloads rather than a truckload. As a result, the supplier must produce 5.4 truckloads rather than 5 truckloads in order to avoid inventory remnants at the supplier’s side.
In the B-RA model, the supplier’s economic lot size of 5Q is kept intact as shown in Fig. 1(c). Given the
supplier’s economic lot size, the buyer’s optimal order sequence within the supplier’s replenishment cycle is
determined as 2Q–2Q–1Q, and the supplier’s quantity discount is applied intermittently only when the buyer’s
order quantity is doubled as 2Q. In this context, the B-RA discount approach is a natural outcome driven by
the condition that both supplier’s and buyer’s economic lot sizes are reserved. The buyer’s optimal order
sequence will be rolled over and maintained in each of the supplier’s inventory cycles. Therefore, our use
of temporary discounts must be understood within the context of supply chain coordination as opposed to
their use in one-time sales models (Arcelus and Srinivasan, 1995; Aull-Hyde, 1992; Ramasesh and Rachamadugu, 2001; Tersine and Barman, 1995).
2.1. Assumptions and notation
The conventional assumptions for quantity discount based inventory coordination models are given in the
following literature (e.g., Monahan, 1984; Lee and Rosenblatt, 1986; Goyal and Gupta, 1989; Weng and Wong,
1993; Weng, 1995; Shin and Benton, 2004; Rubin and Benton, 2003, p.174). The conventional assumptions are:
(1) The buyer orders a fixed order quantity, typically EOQ.
(2) Demand is deterministic and constant.
(3) The buyer’s inventory cost parameters including holding cost and ordering cost are known to the
supplier.
(4) Replenish lead times are deterministic. Thus, the supplier’s inventory is replenished instantly upon the
buyer’s order placement (in the case of zero lead time), thus no stockout will occur at the supplier’s side.
(5) Once scheduled, a single price break of all-units discounts is constantly used on an infinite time horizon.
As specified in Section 2, we relax assumptions 2 and 5 in this paper. It is assumed that demand is uncertain,
following a normal distribution. Demand is not price-sensitive, so the supplier’s discount offer does not
increase overall demand from the end-customers. In this paper the notation used by Monahan (1984) and
Lee and Rosenblatt (1986) is retained for consistency.
mKQ
rKQ
C
Cv
D
a
dK
H1
H 02
H2
actual demand per buyer’s expected order cycle, a random variable
standard deviation of demand per buyer’s expected order cycle
supplier’s unit cost of producing or acquiring the product
standard coefficient for demand variation, standard deviation divided by the mean
expected annual demand
expected demand per period used in the simulation experiments
discount amount per unit (all-units discount)
buyer’s inventory holding cost rate (%)
supplier’s nominal inventory holding cost rate (%) expressed as a proportion of C
supplier’s inventory holding cost rate (%) expressed as a proportion of P, H 2 P ¼ H 02 C
604
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
(a) Two-Stage EOQ Policy
•
•
5Q
Buyer’s order quantity (Q) = EOQ
Supplier’s lot size (kQ) = 5Q
Q
Supplier’s
Lot Size
4Q
Q
3Q
Q
2Q
Q
1Q
Q
Time
(b) Weng & Wong (1993)
•
•
Buyer’s order quantity = [1.8 x EOQ] = 1.8Q
Supplier’s lot size = [5.4 x EOQ] = 5.4Q
5.4Q
Supplier’s
Lot Size
1.8Q
3.6Q
1.8Q
1.8Q
1.8Q
Time
(c) The Proposed B-RAModel
•
•
Buyer’s order quantity (KQ) = EOQ or [2 x EOQ]
Supplier’s economic lot size (kQ) = 5Q
5Q
Supplier’s
Lot Size
2Q
3Q
2Q
1Q
1Q
Time
Fig. 1. Comparison of supplier’s inventory behavior under quantity discount policies.
K
k
factor to determine the buyer’s increased order quantity
integer factor to determine the supplier’s economic lot size; k P K
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
605
Table 1
Experimental parameters
Controlled parameters
Unit price (P)
Unit cost (C)
Number of periods per year
Supplier’s nominal holding cost rate (H 02 )
Supplier’s holding cost rate (H2)
Lead times (week)
Coefficient for demand variation (Cv)
Annual fill rate
Shortage cost per unit (pS)
100
70
50
0.25
0.175
1
0.1, 0.2, and 0.3 (of a)
0.9999
pc
Condition
[01]
[02]
[03]
Expected demand rate per period (a)
Expected annual demand (D)
Buyer’s holding cost rate (H1)
Buyer’s order cost (S1)
Supplier’s setup cost (S2)
Expected buyer’s EOQ
50
2500
0.15
100
1000
183
600
30,000
0.20
500
7500
1225
100
5000
0.25
1000
20,000
632
O
P
Q
r
S1
S2
si
ti
u
Z
buyer’s overstock per expected order cycle, a random variable
unit price paid by the buyer before discounts
buyer’s current order quantity
number of lots (out of k) shipped to the buyer until the buyer’s new order moment
buyer’s fixed ordering cost per order
supplier’s fixed setup cost
the expected moment of buyer’s order arrival for ith lot (out of k) within the supplier’s inventory cycle
deterministic representation of si in the solution algorithm presented in Section 2.7
a standard normal variable to represent realized demand, (1, 0]
a decision variable to represent a certain proportion of buyer’s expected overstock to be compensated
for by the supplier
In Sections 2.2–2.6, the details of the modeling framework are presented to derive the three decision variables – the supplier’s lot size factor (k), the all-units discount (dK), and the buyer’s corresponding lot size factor
(K).
2.2. Supplier’s predetermined economic lot size
The objective in this section is to find the supplier’s economic lot size (kQ) as an integer multiple (k) of the
buyer’s current order quantity (Q). In a two-stage supply chain structure composed of a single buyer and a
single supplier, the supplier’s inventory level is a function of buyer’s order quantity and follows a stepwise
function (Peterson and Silver, 1979, pp. 475–478). Rosenblatt and Lee (1985, p. 390) demonstrate that the supplier’s inventory cost is minimized when the supplier’s lot size is strictly an integer multiple (k) of the buyer’s
order quantity (Q) and that the supplier’s average inventory level is determined by [(k 1)Q/2]. Joglekar
(1988) shows that the supplier’s inventory level can be modified to [(k 1) (k 2)D/R](Q/2) if the supplier
manufactures the item with an annual capacity of R. Since modeling properties remain identical in both cases,
one can use [(k 1)Q/2] in order to maintain simplicity in symbolic iteration (Weng, 1995).
Then, the question becomes as to whether [(k 1)Q/2] can be used for an uncertain demand environment.
Indeed, the supplier’s expected cycle inventory level is [(k 1)Q/2] even under uncertain demand, if the
expected demand rate (D) is static on a continuous time horizon. The detailed proof is given in the Appendix.
Given the expected inventory level of [(k 1)Q/2], the supplier’s inventory cost can be represented by the
cost function given in Eq. (1)
606
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
D
ðk 1ÞQ
þ H 2P
;
TC S ðkQÞ ¼ S 2
kQ
2
ð1Þ
k P 1:
In order to determine the supplier’s optimal lot size factor (k*), we relax the assumption that k is an integer.
If the buyer’s order quantity (Q) is fixed to EOQ and k is a real number on a continuous function, Eq. (1)
transforms into a typical inventory cost function which is continuous rather than discrete – see Fig. 2. The
necessary condition for the minimum value of Eq. (1) can be derived by taking the first derivative of Eq.
(1) with respect to k and setting the first derivative to zero as follows:
dTC S ðkQÞ
S2D
QH 2 P 1
¼ 2 þ
¼ 0;
dk
2
ðk QÞ
ð2Þ
k>0
or
ð1=2Þ
k 0 ¼ ½1=Q½ð2S 2 DÞ=ðH 2 P Þ
¼ EOQS =EOQB ;
ð3Þ
k > 0:
The sufficient condition for the minimum value of Eq. (1) can be derived from Eq. (4) which shows that the
second derivative of Eq. (1) should be greater than zero
d2 TC S ðkQÞ
¼ ð2S 2 DÞ=ðk 3 QÞ > 0;
dk 2
ð4Þ
k > 0:
As shown in (4), the value of the second derivative of Eq. (1), d2TCS(kQ)/dk2, is always non-negative since
k is greater than zero. Thus, as illustrated in Fig. 2, the supplier’s total cost function is convex with respect to
k, and the optimal integer k* is one of the two integers around the unique k0, which minimizes the value of (1).
The unique k0 given in (3) is intuitive since it suggests that the supplier’s optimal lot size factor (k*) be estimated directly by dividing supplier’s EOQ by buyer’s EOQ.
2.3. Supplier’s near optimal discount schedule with the buyer’s continuous review policy
In this section, we show how the conventional all-units discount models (e.g., Monahan, 1984; Rosenblatt
and Lee, 1985; Lee and Rosenblatt, 1986; Weng and Wong, 1993) can be modified when the buyer uses a continuous review (Q, R) system. This is a preliminary step to develop the B-RA model in Section 2.5. For a continuous review system, the buyer’s total cost function is commonly formulated as (5). By using an iterative
approach, the optimal solution for (5) can be found as the partial derivatives of (5) with respect to order quantity (Q) and reorder point (R) (Hax and Candea, 1984)
DS 1
Q
D
TC E ðQ; RÞ ¼ PD þ
þ ss þ E½C S ;
þ PH 1
ð5Þ
2
Q
Q
TC($)
TCS (kQ)
(k)
k =0
k =1
k°
k*=2
k =3
k =4
Fig. 2. Illustration of supplier’s optimal lot size factor (k*) identification.
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
607
where
ss = safety stock; E[CS] = the expected shortage cost per cycle, which is
R 1 R = reorder point;
pS R ðn RÞf ðnÞdðnÞ ; n = demand during the lead time; and pS = the buyer’s shortage penalty cost per unit.
If the same cycle service level is maintained by the buyer under the discount policy, the buyer’s expected
total cost function with the discount is given as (6)
DS 1
KQ
D
TC E ðKQ; RÞ ¼ ðP d K ÞD þ
þ ss þ
E½C S :
þ ðP d K ÞH 1
ð6Þ
2
KQ
KQ
Given (5) and (6), the supplier’s objective is to identify the minimum level of discount that would maximize the
supplier’s profit and at the same time secure the buyer’s inventory cost savings. Therefore, in scheduling the
minimum level of discount, the condition given in (7) must be satisfied
TC E ðQ; RÞ P TC E ðKQ; RÞ () TC E ðQ; RÞ TC E ðKQ; RÞ P 0:
ð7Þ
The inequality condition in (7) indicates that the buyer’s total cost under the quantity discount policy,
TCE(KQ, R), should be less than or equal to the buyer’s total cost under no-discount policy, TCE(Q, R). Otherwise, the buyer would not accept the supplier’s discount offer because the buyer’s cost under the discount policy is greater than that of no-discount policy.
Let us assume that the safety stock should be purchased at the regular price, i.e., Eq. (8) holds true.
PH 1 ðssÞ ðP d K ÞH 1 ðssÞ:
ð8Þ
This assumption is reasonable because the buyer’s safety stock can be purchased in advance before implementing the coordinated quantity discount policy. This assumption reduces computational complexity considerably by eliminating the need for the supplier to adjust discount schedule based on the buyer’s safety
stock amount, but the resulting difference in the actual discount amounts is marginal. Then, substituting
(5) and (6) into (7) and solving (7) with respect to dK, the near optimal discount schedule can be identified
as (9)
2
d K ðKQ; RÞ P
2Dð1 KÞ½S 1 þ EðC S Þ þ P ðQÞ H 1 ðK 2 KÞ
:
KQð2D þ KQH 1 Þ
ð9Þ
The discount schedule in Eq. (9) will be smaller than the deterministic counterpart proposed by Lee
and Rosenblatt (1986) or Weng and Wong (1993) because the buyer’s shortage cost savings under the discount policy are also extracted in favor of the supplier. Therefore, the supplier only compensates for the
net amount of the buyer’s inventory cost increase (=holding cost increase setup cost savings shortage
cost savings). The main problem with the discount schedule given in (9) is that the buyer’s risk of temporary overstocking is not considered. If the buyer’s actual demand is lower than the expected demand
rate, the buyer will encounter a situation in which the inventory cost increase under the discount policy
exceeds the savings from the discounted purchase cost. Because of this limitation, the supplier must take
the buyer’s temporary overstocking risk into account in scheduling quantity discounts for supply chain
coordination.
2.4. The B-RA model: Buyer’s cost function
In the proposed B-RA modeling, the buyer’s total cost function given in (6) should be modified into a cost
function for the buyer’s increased order cycle of E[KQ/D]. This step is necessary because the supplying firm is
allowed to use quantity discounts intermittently within its replenishment cycle. Therefore, the buyer’s inventory cost must be evaluated within the supplier’s replenishment cycle as well. The buyer’s cost per order cycle
can be found by multiplying (6) and (KQ/D), and the result is given in (10)
DS 1 ðP d K ÞH 1 KQ
D
0
CIC E ðKQÞ ¼ ðP d K ÞD þ
þ PH 1 ðssÞ þ
E½C S ðKQ=DÞ
þ
2
KQ
KQ
¼ ðP d K ÞKQ þ S 1 þ
ðP d K ÞH 1 ðKQÞ2 PH 1 ðssÞKQ
þ E½C S :
þ
D
2D
ð10Þ
608
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
Given the buyer’s temporary overstocking risk described in the introduction section, the supplier must
share the buyer’s expected overstocking cost in order to make the discount attractive to the buyer. The following procedure illustrates how to incorporate the buyer’s overstocking cost to improve upon the deterministic
discount schedule in the literature.
Let demand (mKQ) per buyer’s expected order cycle (E[KQ/D]) be a normally distributed independent random variable. With the buyer’s increased order quantity of KQ, the buyer’s possible overstock per replenishment cycle is determined by (11)
KQ mKQ if mKQ < KQ;
O¼
ð11Þ
0
if mKQ P KQ:
Given the conditions in (11), the buyer’s expected overstocking cost can be evenly deducted from the unit
price as a part of all-units discounts. However, the supplier need not compensate the buyer for all the possible
overstocking conditions, especially when the buyer’s overstock possibility is marginal (may be due to less variation in demand). To illustrate this issue, Fig. 3 is provided. As shown in Fig. 3, the buyer’s temporary overstock occurs when the actual demand is less than the order quantity (KQ), 1264 units in this example. With this
probabilistic demand pattern, the supplier can decide a certain proportion of the buyer’s overstock which will
be compensated for by the scheduled all-units discounts. In other words, the supplier may decide to share the
buyer’s overstocking cost only for the probability that the overstock exceeds a certain number of units. This
cutoff point can be represented as a standardized Z score as shown in Fig. 3. If we introduce the standard
decision variable Z, (11) can be modified into (12)
ðKQ ZrKQ Þ mKQ if mKQ < ðKQ ZrKQ Þ;
OZ ¼
ð12Þ
0
if mKQ P ðKQ ZrKQ Þ:
Technically, OZ is analogous to estimation errors, the gap between the expectation and the realization of
demand. The errors are approximately normal for a wide range of demand distributions for the fast moving
items (Hax and Candea, 1984, p. 196). Therefore, the normality assumption can be extended to other types of
demand distributions. Given (12), the estimate of expected overstock (E[OZ]) is straightforward as shown in
(13)
Z KQZr
E½OZ ¼
½ðKQ ZrKQ Þ mKQ f ðmKQ ÞdðmKQ Þ:
ð13Þ
1
With the normality assumption, (13) can be standardized by a unit normal variable (u), yielding (14)
Z Z
mKQ KQ
Z
ðZ uÞf ðuÞdðuÞ; where u ¼
N ð0; 1Þ:
ð14Þ
E½O ¼ rKQ
rKQ
1
Fig. 3. Buyer’s temporary overstock risk and supplier’s risk sharing.
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
609
To simplify the expected value in (14), we introduce an integral function given in (15)
Z Z
GðuÞ ¼
ðZ uÞf ðuÞdðuÞ:
ð15Þ
1
In its structure, the function G(u) is similar to the standard loss function that measures the expected shortage
of units in determining safety stock and reorder point. The value of G(u) depends on Z, which reflects the supplier’s willingness to cover a certain magnitude of overstock. By substituting (15) into (14), (14) can be simplified into (16)
ð16Þ
E½OZ ¼ rKQ GðuÞ:
Then, the buyer’s expected overstocking cost (CE[OZ]) per cycle (E[KQ/D]) is determined as (17)
ðP d K ÞH 1 KQ
Z
C E ½O ¼ rKQ GðuÞ
:
D
ð17Þ
By adding (17) to (10), the buyer’s cost function is finalized as (18)
ðP d K ÞH 1 KQ
CIC E ðKQ; RÞ ¼ ðP d K ÞKQ þ S 1 þ ðKQ þ 2rKQ GðuÞÞ
2D
PH 1 ðssÞKQ
þ E½C S for K > 1:
ð18Þ
þ
D
Unlike Eq. (6), Eq. (18) is not the buyer’s real cost function under uncertain demand. Instead, Eq. (18) should
be understood as the buyer’s cost components that must be compensated for by the supplier’s discount schedule to secure the buyer’s cost savings under uncertain demand.
2.5. The B-RA Model: Intermittent all-units discount with reserved economic lot-sizes
In this section, we develop an all-units discount schedule that covers not only the buyer’s increased holding
cost but also the buyer’s expected overstocking cost. The scheduled discount is an important component in
evaluating the supplier profit per replenishment cycle. When the buyer adheres to EOQ without discounts,
the sum of the buyer’s cost during the period of (KQ/D) can be determined as (19)
DS 1 PH 1 Q
D
CIC E ðQ; RÞ ¼ PD þ
þ PH 1 ðssÞ þ E½C S ðKQ=DÞ
þ
2
Q
Q
¼ PKQ þ KS 1 þ
PH 1 KQ2 PH 1 ðssÞKQ
þ KðE½C S Þ:
þ
D
2D
ð19Þ
If total cost from the EOQ policy, CICE(Q, R) given in (19), is less than the cost under the quantity discount
policy, CICE(KQ, R) given in (18), the buyer will not accept the supplier’s discount offer. Thus, the supplier’s
discount must satisfy the inequality given in (20), which imposes that the buyer’s cost under the discount
policy should be less than the buyer’s cost under the no-discount policy
CIC E ðQ; RÞ P CIC E ðKQ; RÞ () CIC E ðQ; RÞ CIC E ðKQ; RÞ P 0:
ð20Þ
By substituting (18) and (19) into (20), we obtain (21). Rearranging (21) with respect to (dK), the discount
amount per unit can be computed as (22)
H 1 KQ½ðKQ þ 2rKQ GðuÞÞðP d K Þ PQ
P 0;
2D
PH 1 KQ½KQ Q þ 2rKQ GðuÞ 2DðK 1ÞðS 1 þ E½C S Þ
dK P
for K > 1:
KQ½2D þ H 1 KQ þ 2H 1 rKQ GðuÞ
d K ðKQÞ þ ðK 1ÞðS 1 þ E½C S Þ ð21Þ
ð22Þ
It is noteworthy that the discount amount given in (22) is feasible only when K > 1. If the buyer’s lot size
factor is equal to or less than one (i.e., K 6 1), it means that the buyer adheres to its original economic order
quantity (Q) or reduces its order quantity. In this case, the value of (22) must be set to zero since the supplier is
not obliged to offer a discount.
610
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
2.6. The B-RA model: Supplier’s expected profit function
Without discounts offered, the supplier’s expected profit function per inventory cycle (kQ/D) can be modified from (1), and the result is given by (23)
kðk 1ÞQ2
p0SIC ¼ ðP CÞkQ S 2 ð23Þ
ðPH 2 Þ:
2D
As the supplier offers quantity discounts intermittently within the supplier’s inventory cycle, it is necessary
to separate (23) into a group of sub-profit functions because the supplier’s inventory reduction rate changes,
depending on whether or not the quantity discounts are applied to the buyer’s certain orders. Technically,
these sub-functions are for the period with the discount applied (pdK) and for the periods before and after
the discount period (pBdK, pAdK). Thus, Eq. (23) must be reformulated as (24)
pSIC ðsi Þ ¼ pBdKðiÞ þ pdKðiÞ þ pAdKðiÞ ;
ð24Þ
where si represents the buyer’s order moment within the supplier’s replenishment cycle, and i = 0, . . . , k 1.
In Eq. (24), the first sub-profit function (pBdK) reflects a proportion of the supplier’s expected profit to be
made before the supplier offers discounts. Since the supplier’s expected inventory level estimated in Section 2.2
(i.e., (k 1)Q/2) is no longer plausible, we must configure the supplier’s expected average inventory for all the
sub-profit functions included in (24). The supplier’s expected average inventory (IBdK) between the beginning
of the inventory cycle (s0) and the moment of discount (si) can be represented as (25)
I BdK ¼
ðk 1ÞQ þ ðk 2ÞQ þ þ ðk rÞQ Qð2k r 1Þ
¼
;
r
2
ð25Þ
where r is the number of lots (out of k) shipped to the buyer until si.
The supplier’s expected average inventory (IBdK) given in (25) is the same as the supplier’s inventory level
under deterministic conditions. Although the buyer’s time-between-orders varies due to the fixed order quantity under uncertain demand, the supplier’s expected average inventory remains the same mainly because the
buyer’s expected order moment (E[si]) under stochastic demand is the same as the deterministic order moment.
Therefore, the proof provided in the Appendix can be applied to this case as well. The proof in the Appendix
implies that the supplier’s deterministic inventory level can be used to approximate the supplier’s average
inventory level for a stochastic environment. Given (25), the supplier’s profit before the discount period
can be formulated as (26)
ð2k r 1Þ rQ2
pBdKðiÞ ¼ ðP CÞrQ S 2 ð26Þ
ðPH 2 Þ:
2
D
Note that the supplier’s setup cost (S2) is only included in Eq. (26) because pBdK occurs prior to pdK and
pAdK. If the supplier offers discounts at the beginning of its inventory cycle (s0) (i.e., r becomes zero), the supplier’s profit during the period (pBdK) automatically becomes negative (i.e., S2). This implies that the supplier
only pays off its setup cost and no positive profit was made during this period.
The supplier’s expected average inventory during the discount period (IdK) can be represented as (27).
Therefore, the second sub-profit function (pdK) given in (28) represents the supplier’s expected profit given
the discount offer. The supplier’s setup cost is not included in (28) because it is already included in (26)
I dK ¼ ðk r KÞQ;
pdKðiÞ ¼ ðP C d k ÞKQ ðk r KÞ
2
KQ
ðPH 2 Þ;
D
ð27Þ
ð28Þ
where (dK) is defined in (22).
The third sub-profit function (pAdK) must include the expected inventory cost for the post-discount period
in the supplier’s inventory cycle. At the beginning of this period, the buyer would place a new order after the
end-customers consumed all the discounted products. The supplier’s expected inventory (IAdK) during the
remaining period can be given by (29)
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
611
ðk r K 1ÞQ þ ðk r K 2ÞQ þ þ ðk r K ðk r KÞÞQ
ðk r KÞ
Qðk r K 1Þ
:
ð29Þ
¼
2
Since the remaining inventory cycle is [(k r K)Q/D], the supplier’s expected profit for the post-discount
period is determined as (30)
ðk r K 1Þ ðk r KÞ
pAdKðiÞ ¼ ðP CÞðk r KÞQ ð30Þ
ðQ2 PH 2 Þ:
2
D
I AdK ¼
In conclusion, Eq. (24) is an additive function which combines (26), (28) and (30). The major difference
between (24) and the supplier’s profit functions in other research (e.g., Monahan, 1984; Lee and Rosenblatt,
1986; Weng and Wong, 1993) is that (30) estimates the supplier’s profit per replenishment cycle under uncertain demand whereas others estimate the annual profit under deterministic conditions.
2.7. Solution procedure
The following algorithm has been operationalized in this research.
[1] Define Sj = {K(ti1)} as a set of feasible integer K’s that would maximize the value of (24).
i = 1, . . . , k* and j = 1, . . . , n where k* is the supplier’s economic lot size factor and n is the total number
of possible solution sets.
[2] Start with j = 1 and i = 1, so that K(ti1) = K(t0) = 1.
[3] Add K(tP
i) to {K(ti1)}, then Sj = {K(ti)}. Sj 5 Sj if j 5 j.
[4] If k fKðti1 Þg > 1, go to step [5], or otherwise go to step [6].
[5] Given
the current Sj, search the next feasible K(ti1) over the range of (ti1) in which
P
½ fKðti1 Þg þ 1 6 i 6 k . Go to step [4].
[6] If i = k*, go to step [7]. Otherwise, i = i + 1 and go to step [3].
[7] If j = n, go to step [8]. Otherwise, j = j + 1 and go to step [3].
[8] From the group of Sj, sort out the optimal S j containing a set of K*(ti1) and the corresponding dKj that
maximizes (18).
[9] Stop.
The solution algorithm in general identifies the supplier economic lot size factor (k*) defined in (3) and a
set of buyer’s lot size factors (K) and discounts per unit (dK) that could maximize the value of the supplier’s
profit function give in (24). Since both Kand k are integers and K 6 k, a grid search procedure is more effective in determining the optimal solutions than the bounding procedures suggested in Lee and Rosenblatt
(1986, L&R hereafter) or Weng and Wong (1993, W&W hereafter). Specifically, the algorithm first searches
for a feasible combination of K(t0) and dK(t0), which imply that the supplier discount offer is made at the
beginning of the supplier’s replenishment cycle. Then, pSIC(t0) is continually updated with the next feasible
combination K(ti) and dK(ti) at each end of the buyer’s order cycle. The solution procedure starts over with
a new starting point of pSIC(t1) and continues to pSIC(tk1) until it exhausts all the possible combinations of
solutions. Out of all possible solutions, the optimal combination of K(ti)’s and dK(ti)’s is sorted out as the
final solutions.
Consider three conditions in Table 1. The problems were solved by EOQ (two-stage EOQ model with no
discount offered), L&R, W&W, and the proposed B-RA model (B-RA hereafter). In obtaining the solutions
for B-RA, the algorithm described above was applied with an assumption that the supplier would compensate
the buyer’s expected overstocking cost at the maximum level, i.e., Z = 0. Demand parameters were approxi1=2
mated from the experimental parameters. For example, rKQ was estimated by aðCvÞðKQ=aÞ . The results are
summarized in Table 2. Note that L&R and W&W provide the same discounts regardless of demand variation
level while the discount from B-RA increases as the buyer’s risk with uncertain demand (approximated by Cv)
increases.
612
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
Table 2
Summary of the solutions
Ex. no.
Solution categories
EOQ
L&R (Lee and
Rosenblatt, 1986)
W&W (Weng and
Wong, 1993)
B-RA
Coefficient of variation
0.1
0.2
0.3
[01]
Buyer’s optimal K
Buyer’s order Qty.
Supplier’s k
Supplier’s lot size
Discount per unit
1
183
3
549
0.00
3.32
608
1
608
0.90
3.35a
613
1
613
0.89
3
549
3b
549
0.75
3
549
3b
549
0.79
3
549
3b
549
0.83
[02]
Buyer’s optimal K
Buyer’s order Qty.
Supplier’s k
Supplier’s lot size
Discount per unit
1
1225
4
4900
0.00
2.13
2609
2
5218
0.24
4.41a
4964
1
4964
0.92
4
4900
4b
4900
0.94
4
4900
4b
4900
0.98
3-1
3675 or 1225
4b
4900
0.65
[03]
Buyer’s optimal K
Buyer’s order Qty.
Supplier’s k
Supplier’s lot size
Discount per unit
1
632
5
3160
0.00
1.79
1129
3
3387
0.55
1.80a
1136
3
3408
0.54
2-2-1
1264 or 632
5b
3160
0.83
2-2-1
1264 or 632
5b
3160
0.89
2-2-1
1264 or 632
5b
3160
0.96
a
Weng and Wong (1993) directly estimates the buyer’s order quantity without adopting the buyer’s lot size factor (K). Thus, the buyer’s
lot size factor was reverse-calculated from the buyer’s order quantity solutions.
b
The supplier’s k in the proposed model (B-RA) is a factor for the buyer’s EOQ whereas k is a factor for K in the other models.
For Condition [01], as shown in Table 2, the solution by B-RA suggests a lot-for-lot policy in which the
buyer’s order quantity is the same as that of the supplier’s. In this case, B-RA adopts a constant discount
approach as used in L&R and W&W. For Condition [02], B-RA provides two different lot-size solutions.
When the value of Cv is 0.1 or 0.2, B-RA suggests a lot-for-lot policy with the buyer’s order quantity of
4Q. With a Cv of 0.3, the buyer’s maximum order quantity drops to 3Q, leading to a smaller amount of quantity discounts (0.65) as indicated in Table 2. Note that in this case the optimal order sequence with respect to K
is determined as 3Q–1Q (given the supplier’s lot size of 4Q) from many possible sequences such as 1Q–2Q–1Q
and 2Q–2Q. It was found that the buyer’s order sequence 3Q–1Q and 1Q–3Q equally maximize the supplier’s
profit. Of the two, however, the 3Q–1Q sequence is chosen as the final solution over 1Q–3Q with the reasoning
that the supplier can reduce the risk of holding larger inventories by offering the discount at the beginning
stage of its inventory cycle. In other words, if the supplier is risk-averse, the supplier would attempt to reduce
its inventory level as early as possible. This rule is incorporated explicitly in the solution algorithm. Accordingly,
for Condition [03], the buyer’s order sequence of 2Q–2Q–1Q was chosen over 1Q–2Q–2Q or 2Q–1Q–2Q,
as shown in Table 2.
3. Numerical experiments, results, and implications
Four alternative models listed in Table 2 were tested by discrete-event terminating simulation. The purpose
of the simulation experiment is to evaluate the consequences of implementing quantity discount policies in a
non-biased environment and to determine whether or not the proposed B-RA quantity discount policy (compared with other discount policies) would be successful in improving the buyer’s cost, supplier’s profit, and
supply chain system’s cost simultaneously.
The simulation experiment represents a two-stage continuous review system in which the buyer’s inventory
behavior is determined by end-customer’s demand and the supplier’s inventory behavior is determined by the
buyer’s ordering policy. Since the aim of quantity discount research for supply chain coordination is to schedule discounts and corresponding order quantities, the models do not provide complete ordering policies necessary for uncertain demand environments. Therefore, the buyer’s decision criteria including reorder points
and safety stocks were reverse calculated using the parameters given in Table 1 and the quantity discount solutions in Table 2.
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
613
The annual service level of 99.99% is adopted throughout all the simulation runs; see Chopra and Meindl
(2001, p. 187) for a detailed definition. Such a high service level may not always be realistic in practice, but it
helps build a non-biased test environment, limiting the subjective use of shortage penalty costs and reducing
the possible impact of stock-outs on the simulation outcome. For example, if an extremely high shortage cost
is used with a fixed cycle service level, a quantity discount policy with the largest order quantity would be
favored due to the inverse relationship between the amount of cycle stock and the number of possible
stock-outs. In case of a stock-out with any policy, the shortage was treated as lost profits, and the related cost
was measured as the gap between unit price (p) and unit cost (c). It is also assumed that the supplier’s inventory would be replenished on time during the constant lead time (the time between the buyer’s order placement
and the buyer’s receipt of shipment), thus no safety lead time buffer is imposed at the supplier’s side.
Three levels of Cv’s (0.1, 0.2, and 0.3) were empirically tested to represent alternative levels of demand
uncertainty, while the demand per period (a) was generated randomly from the normal distribution. When
the expected demand rate per period (
a) is 100, a Cv of 0.1 produces demand ranging from 70 to 130 with
the probability of 0.9973 whereas a Cv of 0.3 generates demand ranging from 10 to 190 at the same probability. Due to the trade-offs between realistic simulation length and reliability of outcome, a fifty-year period
conditioning and another fifty-year simulation were chosen after a number of preliminary trials.
As summarized in Table 3, the experimental results from the simulation are scaled by the relative efficacy
measures – the supply chain system’s inventory cost reduction ratio (SCRR), supplier’s profit improvement
ratio (PIR), and buyer’s cost reduction ratio (CRR), as illustrated in Eqs. (31)–(33)
SCRR ¼ ½STCðQÞ STCðKQÞ=STCðQÞ;
ð31Þ
PIR ¼ ½pðKQÞ pðQÞ=pðQÞ;
CRR ¼ ½TCðQÞ TCðKQÞ=TCðQÞ:
ð32Þ
ð33Þ
As shown in the above equations, SCRR given in (31) represents the supply chain’s relative inventory
cost reduction under the discount policy, [STC(Q) STC(KQ)], scaled by the supply chain’s total inventory
cost under the no-discount policy, STC(Q). Similarly, CRR given in (33) represents the buyer’s relative cost
Table 3
Relative performance of quantity discount policies
Policies
L&R
Coefficient of
variation
0.1
0.2
0.3
Avg
0.1
0.2
0.3
Avg
0.1
0.2
0.3
Avg
13.55
2.28
2.82
0.00
13.37
2.29
3.00
0.00
13.37
2.32
3.57
0.00
13.43
2.30
3.13
0.00
13.53
2.34
1.56
2.00
13.37
2.36
1.88
0.00
13.36
2.37
2.48
0.00
13.42
2.36
1.97
0.67
13.04
2.14
3.66
0.00
12.93
1.97
6.75
0.00
12.80
1.86
9.34
0.00
12.92
1.99
6.58
0.00
[Unit = 1%]
[01]
SCRR
PIR
CRR
FRCRR
W&W
B-RA
[02]
SCRR
PIR
CRR
FRCRR
4.61
0.65
0.71
11.50
4.62
0.64
1.16
6.50
4.67
0.66
1.56
2.50
4.63
0.65
1.14
6.83
5.03
0.75
1.65
2.00
5.26
0.76
2.58
0.00
5.47
0.76
3.37
0.50
5.25
0.76
2.53
0.83
5.05
0.61
5.11
0.00
5.34
0.45
9.23
0.00
3.82
0.24
7.59
0.00
4.74
0.43
7.31
0.00
[03]
SCRR
PIR
CRR
FRCRR
2.58
1.98
0.88
3.00
2.56
1.95
1.09
1.50
2.57
1.90
1.35
2.00
2.57
1.94
1.11
2.17
2.56
2.01
0.27
41.00
2.60
2.00
0.50
30.00
2.66
2.03
0.70
13.00
2.61
2.01
0.49
28.00
2.00
1.34
1.90
0.00
2.03
1.05
3.31
0.00
2.12
0.75
5.05
0.00
2.05
1.05
3.42
0.00
Avg
SCRR
PIR
CRR
FRCRR
6.91
1.64
1.47
4.83
6.85
1.63
1.75
2.67
6.87
1.63
2.16
1.50
6.88
1.63
1.79
3.00
7.04
1.70
1.16
15.00
7.08
1.71
1.65
10.00
7.16
1.72
2.18
4.50
7.09
1.71
1.67
9.83
6.70
1.36
3.56
0.00
6.77
1.16
6.43
0.00
6.25
0.95
7.33
0.00
6.57
1.16
5.77
0.00
The figures in bold represent the total averages.
SCRR: supply chain cost reduction ratio, PIR: supplier’s profit improvement ratio, CRR: buyer’s cost reduction ratio, FRCRR: failure
rates in CRR.
614
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
reduction under the quantity discount policy, [TC(Q) TC(KQ)], scaled by the buyer’s total inventory cost
under the no-discount policy, TC(Q). In contrast, PIR represents the supplier’s relative profit improvement
under the discount policy [p(KQ) p(Q)] in comparison with the supplier’s profit under the no-discount
policy, p(Q).
SCRRs, PIRs, and CRRs reported in Table 3 are the mean values estimated from 200 replications and represent the relative percent improvements under quantity discount policies from the base case two-stage EOQ
policy. For instance, for Condition [01] with the Cv of 0.1, L&R has shown an SCRR of 13.43%. This implies
that the implementation of L&R reduces the system’s inventory cost by 13.43% from the base case, the twostage EOQ policy. Failure rate (FR) is used as a supplementary metric. FR measures the percentage of times
that a certain quantity discount policy fails to provide either cost savings to the buyer, profit improvement to
the supplier, or cost savings to the supply chain as a whole. The evaluation of FR conforms to the notion that
the goal of supply chain coordination is to benefit all the supply chain participants.
The experimental results can be summarized as follows:
• As can be seen in Table 3, all the quantity discount policies for supply chain inventory coordination provide
positive SCRRs, CRRs, and PIRs and confirm that quantity discounts in general can be an effective inventory coordination mechanism even under stochastic demand.
• B-RA has a mean SCRR of 6.57% and seems to work as effectively as L&R (6.88%) and W&W (7.09%) in
improving the supply chain’s inventory cost. The average gap was 0.52% from the best performing model,
W&W, and 0.31% from L&R. It was expected that the SCRR performance of B-RA could be inferior to
that of L&R or W&W since B-RA adheres to the buyer’s and supplier’s original economic lot sizes. However, when the lot size adjustment is too complicated operationally, B-RA could be the most appropriate
because of its practical properties relying on economic lot-sizing.
• Given the limited room to improve the supply chain system’s inventory performance, the choice of quantity
discount policies is an issue of how to allocate the benefits between the buyer and the supplier. L&R and
W&W extract the most surplus out of the buyer to maximize the supplier’s profit, resulting in higher PIRs
than that of B-RA. In contrast, B-RA secures the buyer’s inventory cost savings (5.77% on average compared with 1.79% of L&R and 1.67% in W&W) by offering a larger discount that covers the buyer’s
expected overstocking cost.
• The question as to whether or not the discounts are large enough to attract the buyer can be answered by
analyzing FRs for CRR in Table 3 (see FRCRR). No discount policy fails to improve the supplier’s profit or
to reduce the system’s inventory cost. Thus, no further discussion will be made on PIR and SCRR. Whereas
B-RA never fails to benefit the buyer with FRs of zero, the FRs of W&W are as high as 41.0%, i.e., the
buyer’s probability to face a cost increase reaches 41.0% under W&W. L&R is an improvement over
W&W yet shows the maximum FR of 11.50% in Condition [02]. It appears that the discounts scheduled
by the conventional quantity discount policies, W&W in particular, are too small to always reduce the
buyer’s total costs under stochastic demand. Consequently, the buyer may not accept the discounts proposed by W&W or L&R. In turn, the supplier’s attempt to reduce its own inventory using quantity discounts would fail. The experimental result provides a justification why the deterministic discount
schedule should be improved considering the buyer’s amplified risk of overstocking.
• In conclusion, although B-RA, which relies on the buyer’s and the supplier’s original economic lot sizes,
may not be able to optimize the system, it appears to be as effective as L&R and W&W in improving
the supply chain’s inventory cost. The result confirms the robustness of economic lot sizing. Also, B-RA
may be a more practical and attractive coordination mechanism since it does not fail to secure cost savings
for both supplier and buyer.
4. Concluding remarks
In this research, we developed a quantity discount model that capitalizes on the buyer’s and the supplier’s
economic lot sizes and that allows the supplier to share the buyer’s potential overstocking risk, relaxing the
two common assumptions in the literature. These assumptions are: (1) the supplier and the buyer are willing
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
615
to deviate from their current order quantities as a quantity discount policy is adopted, and (2) demand rate is
deterministic and constant so that the buyer faces no risk on accepting discounts, thus will always accept the
supplier’s discount offer without hesitation. The experimental results provide evidence that even under stochastic demand conditions, our proposed B-RA modeling approach can schedule equitable quantity discounts
that improve the supplier’s profit and reduce the buyers’ inventory cost simultaneously. In this context, it can
be concluded that the B-RA approach is an effective alternative for supply chain coordination.
Since the focus of this research is on scheduling discounts, the proposed B-RA model does not compute all
the inventory decision variables including safety stock, service level, and reorder point. Therefore, future
research may improve upon this work by directly including these inventory decision variables in modeling.
Also, a meaningful extension would be to test the validity of the proposed B-RA model by using actual data
from the industry or by developing a variety of experimental settings in order to extend the generality of the
experimental results.
Appendix
In this appendix, we show that even under stochastic demand the supplier’s average inventory level converges to [(k 1)Q]/2 if the buyer orders a fixed quantity in the two stage continuous review system in which
and supplier’s inventory level is determined by a stepwise function (see Peterson and Silver, 1979, pp. 475–
478).
Suppose that the buyer’s orders of a fixed order quantity (Q) arrive at s0, s1, s2, . . ., and si. In fact, the
buyer’s order arrival moment (si) is a random variable, and the distribution of si can be determined as a function of realized demand (mQ). Without loss of generality, however, it is possible to obtain the first order
moment of si without defining its probability distribution. If we can show that the expected moment for each
of the buyer’s order arrivals under stochastic demand (E[si]) is the same as the buyer’s order arrival under
deterministic demand (Ti), then the supplier’s expected average inventory level should be determined as
[(k 1)Q]/2 as proven in Rosenblatt and Lee (1985).
Let [(Q mQ)/D] represent the absolute deviation of Ti from the location of si. In other words, if the realized demand (mQ) during the expected inventory cycle is smaller (or larger) than the order quantity (Q), the
location of Ti (deterministic order arrival moment) can be reformulated as a relative deviation from the actual
si, i.e., E½si þ ðQ mQ Þ=
a
a ¼ E½si þ E½ðQ mQ Þ=a;
T i ¼ E½si þ ðQ mQ Þ=
where E½ðQ mQ Þ=
a ¼ ð1=
aÞE½Q mQ Z 1
E½Q mQ ¼
ðQ mQ Þf ðmQ ÞdðmQ Þ ¼ 0 () E½ðQ mQ Þ=a ¼ 0;
1
* E½si ¼ T i :
References
Arcelus, F.J., Srinivasan, G., 1995. Discount strategies for one-time-only sales. IIE Transactions 27, 618–624.
Aull-Hyde, R.L., 1992. Evaluation of supplier-restricted purchasing options under temporary price discounts. IIE Transactions 24, 184–
186.
Benton, W.C., Park, S., 1996. A classification of literature on determining the lot size under quantity discounts. European Journal of
Operational Research 92, 219–238.
Chen, F., 2000. Optimal policies for multi-echelon inventory problems with batch ordering. Operations Research 48 (3), 376–389.
Chopra, S., Meindl, P., 2001. Supply Chain Management: Strategy, Planning, and Operation. Prentice Hall, New Jersey.
Goyal, S.K., Gupta, Y.P., 1989. Integrated inventory models: The buyer-vendor coordination. European Journal of Operational Research
41 (3), 261–269.
Hax, A.C., Candea, D., 1984. Production and Inventory Management. Prentice-Hall, Englewood Cliffs, New Jersey.
Joglekar, P., 1988. Comments on a quantity discount pricing model to increase vendor profits. Management Science 34 (11), 1391–
1400.
Kim, K.H., Hwang, H., 1989. Simultaneous improvement of supplier’s profit and buyer’s cost by utilizing quantity discount. Journal of
the Operational Research Society 40 (3), 55–265.
616
H. Shin, W.C. Benton / European Journal of Operational Research 180 (2007) 601–616
Klastorin, T.D., Moinjadea, K., Son, J., 2002. Coordinating orders in supply chains through price discounts. IIE Transactions 34, 679–
689.
Lee, H.L., Rosenblatt, M.J., 1986. A generalized quantity discount pricing model to increase supplier’s profits. Management Science 32
(9), 1177–1185.
Monahan, J.P., 1984. A quantity discount pricing model to increase vendor profits. Management Science 30 (6), 720–726.
Peterson, P., Silver, E., 1979. Decision System for Inventory Management and Production Planning. John Wiley and Sons, New York.
Rabinovich, E., Windle, R., Dresner, M., Corsi, T., 1999. Outsourcing of integrated logistics functions: An examination of industry
practices. International Journal of Physical Distribution and Logistics Management 29 (6), 35–37.
Ramasesh, R.V., Rachamadugu, R., 2001. Lot-sizing decisions under limited-time price reduction. Decision Sciences 32 (1), 125–143.
Rosenblatt, M.J., Lee, H.L., 1985. Improving profitability with quantity discounts under fixed demand. IIE Transactions 17, 388–395.
Rubin, P.A., Benton, W.C., 2003. A generalized framework for quantity discount pricing schedules. Decision Sciences 34 (1), 173–188.
Shin, H., Benton, W.C., 2004. Quantity discount-based inventory coordination: Effectiveness and critical environmental factors.
Production and Operations Management 13 (1), 63–76.
Tersine, R., Barman, S., 1995. Economic purchasing strategies for temporary price discounts. European Journal of Operational Research
80, 328–343.
Weng, Z.K., 1995. Channel coordination and quantity discounts. Management Science 41 (9), 1509–1522.
Weng, Z.K., Wong, R.T., 1993. General models of the supplier’s all-units quantity discount policy. Naval Research Logistics 40, 971–991.
Yano, C.A., Gilbert, S.M., 2004. Coordinated pricing and production/procurement decisions: A Review. In: Chakravarty, A., Eliashberg,
J. (Eds.), Managing Business Interfaces: Marketing, Engineering and Manufacturing Perspectives. Kluwer Academic Publishers.