Trade promotion policies in manufacturer-retailer supply chains

Transportation Research Part E 96 (2016) 20–39

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Trade promotion policies in manufacturer-retailer supply chains Yu-Chung Tsao a,⇑, Jye-Chyi Lu b a b

Department of Industrial Management, National Taiwan University of Science and Technology, Taipei, Taiwan School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, USA

a r t i c l e

i n f o

Article history: Received 20 May 2016 Received in revised form 28 September 2016 Accepted 30 September 2016

Keywords: Supply chain Trade promotion Unsold-discount Target rebate Newsvendor

a b s t r a c t This study considers a manufacturer-retailer supply chain in which the manufacturer provides trade promotions the retailer. We compare with four trade promotions (off-invoice, scan-back, unsold-discount and target rebate). Consider the linear price sensitive and uncertain demand, the results indicate that both manufacturer and retailer benefit from the unsold-discount policy; only manufacturer benefits from the target rebate policy. However, target rebate can benefit both manufacturer and retailer when the wholesale price is determined within the appropriate range of an agreement. Consider the priceelasticity and uncertain demand, both manufacturer and retailer can only benefit from the target rebate policy. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Trade promotion refers to marketing activities that are executed between the manufacturer and the retailer to enhance sales of specific products. Manufacturers of consumer packaged goods are increasing their trade promotions worldwide (Yuan et al., 2013). Between 1978 and 1996 the dollars spent on trade promotions grew on average from 5% of sales to 13% of sales (Trade Promotions Management: An In-depth Review). Trade promotions continue to play an important role in the U.S. supermarket industry as well as other industries (Su and Geuness, 2012). Due to trade promotions are widely used by manufacturers, this may led to inefficient spending practices in trade promotions. Therefore, improving trade promotion by developing decision models to realize what kind of promotions can increase profits is important. Off-invoice and scan-back policies are two commonly used trade promotions. According to the off-invoice policy, manufacturers offer discounts on the order quantity sold to retailers. In off-invoice policy the manufacturer offers a discount to the retailer so that the retailer will offer a price discount to the consumer. According to the scan-back policy, manufacturers offer discounts on the actual quantity that retailers sell to end customers. It is measured by using the retailer’s scanner (Point of Sale, POS) system. In scan-back policy the manufacturer pays only for those units sold on promotion and not additional units purchased by the channel at the discounted price. However, for markets that do not have POS systems, scan back promotion cannot be used. Recently, regarding price-dependent and uncertain demand, Yuan et al. (2013) presented a theoretical analysis and implemented market experiments for two types of market in which trade promotion discounts are offered as either offinvoice or scan-back policies. However, in their theoretical model, both off-invoice and scan-back trade promotions did ⇑ Corresponding author. E-mail addresses: [email protected] (Y.-C. Tsao), [email protected] (J.-C. Lu). http://dx.doi.org/10.1016/j.tre.2016.09.014 1366-5545/Ó 2016 Elsevier Ltd. All rights reserved.

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

not benefit manufacturers and retailers. The influences of trade promotions on decisions should also be discussed furtherly. This raises several questions: Do any trade promotion policies benefit channel members when demand is uncertain? How do trade promotions affect the decisions and profits of manufacturers and retailers? In this study, two new trade promotion policies are considered: unsold-discount and target rebate. We answer the aforementioned questions by comparing two trade promotions (unsold-discount and target rebate) with two common trade promotions (off-invoice and scan-back) and discussing manufacturer-retailer decisions on retail, wholesale price, and order quantity by considering these trade promotions. Blattberg and Neslin (1990) stated that there are thirteen major types of trade promotions from the execution perspective: off-invoice, bill-back, volume rebate, free goods, floor planning, cooperative advertising, display allowances, display allowances, SPIFFS, contests, quota incentives, sales drives, slotting allowances and street money. In our paper we consider trade promotions from the modeling perspective, which manufacturers offer discounts on different quantities of goods in different trade promotions. In the off-invoice policy manufacturers offer discounts on the order quantity sold to retailers; in the scan-back policy manufacturers offer discounts on the quantity that retailers sold to end customers; in the unsold-discount policy manufacturers offer discounts on the unsold quantity that retailers did not sell to end customers; in the target rebate policy manufacturers offer discounts on the items sold above a target quantity. We consider and compare the four different trade promotions which the discount is offered on different quantities. The extant literature has focused on off-invoice and scan-back trade promotions and discussed manufacturer and retailer preferences for these trade promotions. Because off-invoice and scan-back promotions do not increase channel members’ profits under demand uncertainty, our models contributed to the trade promotion literature by providing effective types of trade promotions and demonstrating how such trade promotions affect manufacturer-retailer decisions. In the basic models, we determine the optimally derived prices and quantities under a given trade promotion policy to reflect our theoretic studies for comparing impact from implementing four different trade promotion policies. Consider the linear price sensitive and uncertain demand, we reveal that both the manufacturer’s and retailer’s profits increase through the use of the unsolddiscount policy; and only manufacturer’s profits increase by using the target rebate policy. However, manufacturer’s and retailer’s profits can increase when the wholesale price is determined within the appropriate range of an agreement. This implies that the manufacturer and retailer prefer the unsold-discount and target rebate trade promotions over off-invoice and scan-back trade promotions under demand uncertainty. However, when consider the linear price sensitive and uncertain demand, both manufacturer and retailer can only benefit from the target rebate policy. Furthermore, we consider three extensions of the basic models. First, we consider the market expansion model in which trade promotion discounts influence market expansion, and by which the optimal discount levels can be obtained. Recent empirical research has documented the effective use of trade promotions by manufacturers in relation to the market expansion efforts of retailers (Yuan et al., 2013). Therefore, we extend the basic model by considering trade promotion discount decisions endogenously. In addition, retailers commonly sell products from competing manufacturers. Companies wish to determine which trade promotion is optimal according to price competition. Thereby, we also consider the price competition for various brands. In this study, the demand of a brand are dependent on its retail price and the price of the competing brand. Mishra and Raghunathan (2004), Babich et al. (2007), Karray and Martín-Herrán (2009), Jiang and Wang (2010) and Choi and Fredj (2013) considered manufacturer or brand competition. The model used in this study is appropriate when demand was uncertain, the product life cycle is short, the trade promotion discount influence market expansion, and the price is competitive for various brands. Finally, we also consider other non-uniform distribution demand functions to test the results, e.g., Normal, and non-linear price functions. This paper considers a manufacturer-retailer channel in which the manufacturer determines the wholesale price and trade promotion policy, and the retailer determines the retail price and order quantity to maximize his/her profits when end demand is uncertain. Fig. 1 shows the manufacturer-retailer channel considered in this paper. The remainder of this paper is organized as follows: in Section 2, we review the related studies; Section 3 introduces the basic models for offinvoice and scan-back; in Sections 4 and 5, the development of unsold-discount and target rebate models is described and the four trade promotions are compared; in Section 6, we consider three extensions: market expansion, price competition and different demand functions. Finally, Section 7 presents conclusions on managerial implications and future research. The results can assist business managers in improving their decision making. 2. Literature review Trade promotions are special incentives offered by manufacturers to retailers. Trade promotions are temporary price cuts that manufacturers offer retailers (Kumar et al., 2001). The goal of trade promotion is for the retailer to offer the consumer a price discount (Blattberg and Levin, 1987). Trade promotions account for over 60% of manufacturer marketing budgets for

Trade Promotion

Retail price

Wholesale price Manufacturer

Order quantity

Retailer

Demand

Fig. 1. The manufacturer-retailer channel under trade promotion.

Customers

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

packaged goods (Nijs et al., 2010). Although trade promotions are widely used by manufacturers, other manufacturers may regard them as wasteful efforts (Cui et al., 2008). For example, Drèze and Bell (2003) indicated that only 16% of trade promotions are profitable. Trade promotion has a significant impact on the retail business, particularly in the fashion sectors (Kurata and Yue, 2008). Therefore, determining the profitability of various types of promotion is important. In considering price-dependent demand, Drèze and Bell (2003) compared off-invoice and scan-back trade promotions and determined that retailers prefer off-invoice trade promotions, whereas manufacturers prefer scan-back trade promotions. They also stated that manufacturers and retailers could benefit from using their redesigned scan-back policy. Those results are derived based on the assumption of deterministic demand function. In practice, because customer demand cannot be forecasted accurately, considering consumer demand as an uncertainty and applying probability theory is crucial for companies in making informed business decisions, particularly for companies that sell innovative products, such as fashion goods and high-tech products. Regarding price-dependent and uncertain demand, Yuan et al. (2013) considered the two commonly used trade promotions (off-invoices and scan-backs) and discussed the effects of trade promotions on manufacturer and retailer decisions and profits. They showed that both off-invoice and scan-back trade promotions cannot benefit manufacturers and retailers. Target rebate is a type of channel rebate which is a payment from a manufacturer to a retailer based on the sales of the retailer to the end customers. The rebate is paid for each unit sold beyond the target level. The computer hardware and software industries, in which rebates are frequently used, are characterized by short life cycles and high demand variability (Taylor, 2002). An example of a target rebate is Hewlett-Packard Company paying its resellers a 10% rebate for the sale of servers in volumes that exceed a target level (Zarley, 1998). Numerous companies, such as Compaq, the International Business Machines Corporation, Intel, Microsoft, Novell, and Nissan, provide rebates to their retailers based on the volume of sales to end consumers. Target rebate contracts have been shown to be useful in coordinating supply chains with risk-neutral and risk sensitive agents (Chiu et al., 2011; Wang et al., 2009). Xing and Liu (2012) compared the target rebate contract and the wholesale price discount contract for sales effort coordination. Our paper considers target rebate and unsold-discount as two trade promotions because they are suitably used in a business-to-business environment in the context of uncertain demand and short life cycles. The methodology employed in this article for comparing these four trade promotions is to develop their models and examine their optimal decisions. We compare these decisions in theoretical and numerical studies. For example, Propositions 1–5 provides a theoretical comparison of the first three policies when their optimal decisions are analytically trackable. They are further supported by numerical validations presented in Tables 1 and 2. For the more complicated target-rebate policy, its optimal decision has to be solved numerically, and thus, its properties are compared with others using computational methods. The contributions of this paper to the literature are as follows. This paper develops a theoretical model to examine pricing and ordering behaviors considering trade promotion policies when demand is uncertain. We provide two new trade promotions (unsold-discount and target rebate) and compare with the two common trade promotions (off-invoice and scan-back). This study considers several demand functions and shows that the performances of promotional strategies are strongly influenced by demand function. In addition, this study explores two extensions reflecting real-life situations: when the amount of trade promotion discount influences market expansion and when price is competitive for various brands. Our studies contribute to the trade promotion literature by providing effective types of trade promotions and demonstrating how such trade promotions affect manufacturer-retailer decisions.

Table 1 Predictions of the theoretical model (a = 100; b = 5; c = 1). Trade promotion discount

Wholesale price

Retail price

Order quantity

Manufacturer profits

Retailer profits

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

0.00

6.28 (0%)a 6.78 (7.38%) 7.28 (13.74%) 7.78 (19.29%) 8.28 (24.17%) 8.78 (28.49%) 9.28 (32.34%) 9.78 (35.80%)

6.28 (0%) 6.66 (7.51%) 7.04 (14.20%) 7.43 (20.19%) 7.82 (25.58%) 8.22 (30.41%) 8.63 (34.76%) 9.03 (38.76%)

14.37

14.37

31.72

31.72

167.35

167.35

128.34

128.34

14.37

14.35

31.72

31. 20

167.35

165.13

128.34

127.74

14.37

14.32

31.72

30.68

167.35

162.94

128.34

127.08

14.37

14.30

31.72

30.18

167.35

160.79

128.34

126.37

14.37

14.28

31.72

29.63

167.35

158.67

128.34

125.62

14.37

14.27

31.72

29.22

167.35

156.58

128.34

124.82

14.37

14.25

31.72

28.75

167.35

154.53

128.34

123.98

14.37

14.23

31.72

28.30

167.35

152.52

128.34

123.12

0.50 1.00 1.50 2.00 2.50 3.00 3.50 a

Trade promotion as a percentage of wholesale price.

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39 Table 2 Predictions of the theoretical model under unsold-discount (a = 100; b = 5; c = 1). Trade promotion discount

Wholesale price Unsold-discount

Retail price Unsold-discount

Order quantity Unsold-discount

Manufacturer profits Unsold-discount

Retailer profits Unsold-discount

0.00

6.28 (0%)a 6.40 (7.81%) 6.53 (15.30%) 6.67 (22.48%) 6.82 (29.34%) 6.97 (35.88%) 7.13 (42.09%) 7.30 (47.98%)

14.37

31.72

167.35

128.34

14.39

32.25

169.60

128.56

14.42

32.80

171.87

129.30

14.45

33.35

174.16

129.64

14.47

33.92

176.47

129.88

14.51

34.49

178.78

129.99

14.54

35.07

181.10

129.97

14.58

35.65

183.42

129.79

0.50 1.00 1.50 2.00 2.50 3.00 3.50 a

Trade promotion as a percentage of wholesale price.

3. Basic models for off-invoice and scan-back Let us first introduce the basic models for the two common trade promotions (off-invoice and scan-back). The following notations are used in the models: PO : retail price in the off-invoice case W O : wholesale price in the off-invoice case Q O : retailer order quantity in the off-invoice case a: per-unit discount under the off-invoice policy PS : retail price in the scan-back case W S : wholesale price in the scan-back case Q S : retailer order quantity in the scan-back case b: per-unit discount under the scan-back policy c: constant marginal cost for producing a product (production cost) D: demand function D ¼ f ðpÞ þ U, where f ðpÞ ¼ a  bp is a function of unit retail price p with a > 0; b > 0, and U is a continuous random variable following a uniform distribution on ½ða  bpÞ; a  bp. PMO : manufacturer’s profit in the off-invoice case PRO : retailer’s profit in the off-invoice case PMS : manufacturer’s profit in the scan-back case PRS : retailer’s profit in the scan-back case When the off-invoice policy is used, the trade promotion discount a is offered by the manufacturer as a price reduction on the normal price of goods. The retailer profit in the off-invoice case can be written as

PRO ¼ PO

Z 0

QO

t dt þ P O 2ða  bPO Þ

Z

2ðabPO Þ

QO

QO dt  ðW O  aÞQ O : 2ða  bP O Þ

ð1Þ

The manufacturer’s profit in the off-invoice case is

PMO ¼ ðW O  a  cÞQ O :

ð2Þ

In Eq. (1), the first term is the expected revenue when the demand is less than the order quantity and the second term is the expected revenue when the demand is higher than the order quantity. As shown in the second term, the retailer will not be able to satisfy customer demand if the demand higher than the quantity ordered. In this model, the retailer determines the optimal retail price and order quantity and the manufacturer determines the optimal wholesale price to maximize their own profits. We first find the optimal order quantity Q O and retail price PO by solving the first-order conditions, i.e. @ PRO =@Q O ¼ 0 and @ PRO =@PO ¼ 0, then substitute Q O and P O into PMO and determine the optimal wholesale price W O to maximize PMO . The manufacturer is the leader and the retailer is the follower in this model and solve the problem, the optimal retail price, order quantity, and wholesale price are

PO ¼

aþ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi a2 þ 8abðW O  aÞ ; 4b

ð3Þ

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

Q O

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 a  a þ 8bðW O  aÞ 5a þ 4bðW O  aÞ  3 a2 þ 8abðW O  aÞ ¼ ; ¼ 4 2

ð4Þ

W O

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5a þ 32bc þ 64ba þ 3 17a2 þ 64abc : ¼ 64b

ð5Þ

When the scan-back policy is used, the manufacturer offered a per-unit discount b to the retailer for each unit sold to consumers. The retailer profit in the scan-back case is

PRS ¼ ðPS þ bÞ

Z 0

QS

t dt þ ðPS þ bÞ 2ða  bP S Þ

Z

2ðabPS Þ QS

QS dt  W S Q S : 2ða  bP S Þ

ð6Þ

The manufacturer profit in the scan-back case is

PMS ¼ ðW S  cÞQ S  b

Z 0

QS

t dt  b 2ða  bPS Þ

Z

2ðabPS Þ

QS

QS dt: 2ða  bPS Þ

ð7Þ

For solving the optimization problem, the optimal retail price, order quantity, and wholesale price under the scan-back policy are defined as

PS

Q S

¼

¼

a  3bb þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ bb a þ bb þ 8bW S ; 4b

ð8Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  5ða þ bbÞ þ 4bW S  3 ða þ bbÞ þ 8ða þ bbÞbW S 2 

W S ¼

¼

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2  3 a þ bb  ða þ bbÞ þ 8bW S 4

;

ð9Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð5a þ 32bc þ 19bbÞða þ bbÞ þ 9b b2 þ 3ða þ 2bbÞ ða þ bbÞð17a þ 64bc þ 27bbÞ þ 9b b2 64bða þ bbÞ

ð10Þ

4. Unsold-discount policy 4.1. Model under unsold-discount policy In the unsold-discount policy, manufacturers offer discounts on the unsold quantity that retailers did not sell to end customers. We demonstrate that unsold-discount policy is more favorable than the off-invoice and scan-back policies under demand uncertainty. We use the following notations in this model: PB: retail price in the unsold-discount case WB: wholesale price in the unsold-discount case QB: retailer order quantity in the unsold-discount case B: per unsold unit discount in the unsold-discount case PMB : manufacturer’s profit in the unsold-discount case PRB : retailer’s profit in the unsold-discount case RQ R 2ðabPB Þ Q B t dt þ P B Q B dt and the expected income Under the unsold-discount policy, the expected revenue is P B 0 B 2ðabP 2ðabP B Þ BÞ R  QB Q B t from manufacturer’s discount for unsold products is B 0 2ðabPB Þ dt . The retailer profit in the unsold-discount case can be written as

PRB ¼ PB

Z 0

QB

t dt þ PB 2ða  bP B Þ

Z

2ðabPB Þ

QB

Z Q B

QB QB  t dt þ B dt  W B Q B : 2ða  bP B Þ 2ða  bP B Þ 0

ð11Þ

Taking the first-order condition with respect to order quantity and retail price, we obtain:

@ PRB 2ðP B  W B Þða  bP B Þ ¼ 0 ) QB ¼ ; PB  B @Q B

ð12Þ

2

@ PRB 4ða  bP B Þ ¼ 0 ) QB ¼ : @PB a  bB

ð13Þ

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

Next, equate Eqs. (12) and (13):

PB ¼

a þ 3bB þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a  bB a  9bB þ 8bW B ; 4b

ð14Þ

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a  bB  a  9bB þ 8bW B 5a  9bB þ 4bW  3 a  bB a  9bB þ 8bW B B : ¼ Q B ¼ 4 2

ð15Þ

The profit maximization problem encountered by the manufacturer in the unsold-discount case can be written as

 Z max PMB ¼ maxW B ðW B  cÞQ B  B WB

QB 0

 QB  t dt : 2ða  bP B Þ

ð16Þ

Subsequently, the optimal wholesale price is determined by



W B ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ða  bBÞð5a þ 45bB þ 32bcÞ þ 9b B2 þ 3ða  2bBÞ ða  bBÞð17a þ 64bc  27bBÞ þ 9b B2 :

64bða  bBÞ

ð17Þ

4.2. Comparison Based on the models, we compare the unsold-discount policy with the off-invoice and scan-back policies. In this section, the wholesale price, retail price and order quantity established when a scan-back promotion is offered are compared with that established when an off-invoice promotion is offered and when a scan-back is offered. Proposition 1 indicates that the wholesale price is lowest in the unsold-discount case and highest in the off-invoice case when the same trade promotion discount is provided in each case. Proposition 1. When a ¼ b ¼ B; W O P W S P W B are satisfied. Proof. (a) We first compare the wholesale prices W S and W B by taking the difference based on b ¼ B. We find that

n 2 W S  W B ¼ ða  bBÞ½ð5a þ 32bc þ 19bBÞða þ bBÞ þ 9b B2 þ 3ða þ 2bBÞwS   ða þ bBÞ½5a2 þ 8abð5B þ 4cÞ o 2 4b Bð9B þ 8cÞ þ 3ða  2bBÞwB  =½64bða þ bBÞða  bBÞ 2

2

¼ ½ð3a2  6b B2 ÞðwS  wB Þ þ 3abBðwS þ wB Þ  2bBð13a2  4b B2 Þ=½64bða þ bBÞða  bBÞ; where wS ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 17a2 þ 4b Bð9B þ 16cÞ þ abð64c þ 44BÞ and wB ¼ 17a2 þ 4b Bð9B  16cÞ þ abð64c  44BÞ: We then 2

2

compare ð3a2  6b B2 ÞðwS  wB Þ þ 3abBðwS þ wB Þ and 2bBð13a2  4b B2 Þ by squaring both sides and taking the difference. We obtain that 2

2

2

½ð3a2  6b B2 ÞðwS  wB Þ þ 3abBðwS þ wB Þ  ½2bBð13a2  4b B2 Þ 2

2

2

4

¼ 2ða2  b B2 Þ½153a4 þ 576a3 bc þ 472a2 b B2  1264b B4  9ða þ 2bBÞða  2bBÞwS wB : 2

Since a is much greater than b and B in general case, we know that a2  b B2 > 0 and a  2bB > 0: Then from the inequality of arithmetic and geometric means (AM–GM inequality), we have that 2

4

2

153a4 þ 576a3 bc þ 472a2 b B2  1264b B4  9ða þ 2bBÞða  2bBÞwS wB > 153a4 þ 576a3 bc þ 472a2 b B2 4 4

 1264b B  9ða þ 2bBÞða 

2bBÞðw2S

þ

w2B Þ=2

2 2

2 2

¼ 8b B ð95a2 þ 4b B þ 288abcÞ > 0:

Therefore, we can verify that W S  W B P 0 when b ¼ B. (b) We then compare W O and W S when a ¼ b. We compare the wholesale prices by taking the difference based on a ¼ b. We obtain that

W O  W S ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 ða þ bbÞ að17a þ 64bcÞ þ ð15a þ 12bbÞbb  ða þ 2bbÞ ½17a2 þ 4b bð16c þ 9bÞ þ abð64c þ 44bÞ 64bða þ bbÞ

:

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 We then compare ða þ bbÞ að17a þ 64bcÞ þ ð15a þ 12bbÞbb and ða þ 2bbÞ ½17a2 þ 4b bð16c þ 9bÞ þ abð64c þ 44bÞ by squaring both sides and taking the difference. We determine that

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ½ða þ bbÞ að17a þ 64bcÞ þ ð15a þ 12bbÞbb  ða þ 2bbÞ ½17a2 þ 4b bð16c þ 9bÞ þ abð64c þ 44bÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2bbða þ bbÞfð15a þ 12bbÞ að17a þ 64bcÞ  ½3að13a þ 32bcÞ þ 4bbð32bc  5aÞg: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Then we compare ð15a þ 12bbÞ að17a þ 64bcÞ and 3að13a þ 32bcÞ þ 4bbð32bc  5aÞ by squaring both sides and taking the difference. We have

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

X ¼ ½ð15a þ 12bbÞ að17a þ 64bcÞ  ½3að13a þ 32bcÞ þ 4bbð32bc  5aÞ

2

2

¼ 256ða  bcÞ½9a2 ða þ 4bcÞ þ 6abbð5a þ 16bcÞ þ 8b b2 ða þ 8bcÞ: According to the demand setting, a  bp should be greater than or equal to 0 (i.e., a P bp). The retail price p is larger than the constant marginal cost c. Thus, the value of a is much greater than the multiplication of b and c in general case. We can then verify that X P 0 in general case. This indicates that W O  W S P 0; therefore, we can conclude that W O P W S is satisfied when a ¼ b. From above, we can conclude that W O P W S P W B are satisfied when a ¼ b ¼ B. h Proposition 2 demonstrates that the retail price is highest in the unsold-discount case and lowest in the scan-back case when the same trade promotion discount is provided in each case. Proposition 2. When a ¼ b ¼ B; PB P PO P PS are satisfied. Proof. (a) We first compare PB and PO when a ¼ B. According to Eqs. (3) and (5), P O is not a function of the trade promotion discount a in the off-invoice case. pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In the unsold-discount case, PB ¼ ða þ 3bB þ a  bB a  9bB þ 8bW B Þ=4b ¼ ða  2bBþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a  bB a  9bB þ 8bW B Þ=4b þ 5B=4: Let x ¼ ða  bBÞð17a þ 64bc  27bBÞ þ 9b B2 and y ¼ a  2bB: The optimal wholesale price can be rewritten as

W B ¼

x2 þ 6xy  7y2 5B aB þ : þ 8 8ða  bBÞ 128bða  bBÞ

By substituting W B into P B , we determined that the optimal retail price is

PB ¼

y ðx þ 3yÞ 5B þ þ : 4b 16b 4

Subsequently, we take the derivative of the retail price with respect to the trade promotion discount for the scan-back case:

@PB 18bB  11a  16bc þ 3x 3x  ð11a þ 16bc  18bBÞ ¼ : ¼ 8x 8x @B Compare 3x and 11a þ 16bc  18bB by squaring both sides and taking the difference. We determine that 2

ð3xÞ2  ð11a þ 16bc  18bBÞ ¼ 32ða  bcÞða þ 8bcÞ. Because a is greater than b and c in general case (from the demand setting a  bp P 0), it is easy to see that @PB =@B > 0. Because @PB =@B > 0, PB reaches a minimum when B ¼ 0 and increases as B increases. Because W O ¼ W B ; when a ¼ B ¼ 0,  PO ¼ PB when a ¼ B ¼ 0. Therefore, P B P P O is satisfied when a ¼ B. and PS when a ¼ b. In the scan-back compare P O qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2  PS ¼ ða  3bb þ ða þ bbÞ þ 8bða þ bbÞW S Þ=4b ¼ ða þ 2bb þ ða þ bbÞ þ 8bða þ bbÞW S Þ=4b  5b=4: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x ¼ ða þ bbÞð17a þ 64bc þ 27bbÞ þ 9b b2 and y ¼ a þ 2bb, the optimal wholesale price can be rewritten as

(b) We

W S ¼

then

x2 þ 6xy  7y2 b bb2 þ þ : 128bða þ bbÞ 4 8ða þ bbÞ

By substituting W S into P S , we determine the optimal retail price:

case, Let

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

PS ¼

y ðx þ 3yÞ 5b þ  : 4b 16b 4

We subsequently take the derivative of the retail price with respect to the trade promotion discount in the scan-back case:

@P S @b

¼ 11aþ18bbþ16bc3x . 8x

We then compare 11a þ 18bb þ 16bc and 3x by squaring both sides and taking the difference. From qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 x ¼ ða þ bbÞð17a þ 64bc þ 27bbÞ þ 9b b2 , we determine that ð11a þ 18bb þ 16bcÞ  ð3xÞ2 ¼ 32ða  bcÞða þ 8bcÞ. Because a is much greater than b and c in the general case, it is easy to see that @P S =@b < 0. Because @P S =@b < 0, PS reaches a maximum when b ¼ 0 and decreases as b increases. Because W O ¼ W S is satisfied when a ¼ b ¼ 0, we know that PO ¼ PS is satisfied when a ¼ b ¼ 0. Therefore, PO P PS is satisfied when a ¼ b. From above, we can conclude that P B P PO P PS are satisfied when a ¼ b ¼ B. h Proposition 3 indicates that the retail order quantity is highest in the unsold-discount case and lowest in the scan-back case when the same trade promotion discount is provided in each case. Proposition 3. When a ¼ b ¼ B, Q B P Q O P Q S are satisfied. Proof. 2

y BÞ , P B ¼ 4b þ ðxþ3yÞ þ 5B and (a) We first compare Q B and Q O when a ¼ B. According to Q B ¼ 4ðabP 16b abB 4

have that

@Q B @B

¼

4bðabPB Þ 2 ðabBÞ



ðabBÞð44aþ64bc72bBÞx2 þ3xy 16x

@P B @B

¼ 18bB11a16bcþ3x , we 8x

:

4bðabPB Þ 2 ðabBÞ

þ3xy > 0, we focus on H ¼ ðabBÞð44aþ64bc72bBÞx . 16x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 and y ¼ a  2bB, we have From x ¼ ða  bBÞð17a þ 64bc  27bBÞ þ 9b B2  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 ða2bBÞð9a6bBÞða2bBÞ 17a2 þ4b Bð9B16cÞþabð64c44BÞ H¼ : 16x qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 We then compare ða  2bBÞð9a  6bBÞ and ða  2bBÞ 17a2 þ 4b Bð9B  16cÞ þ abð64c  44BÞ by squaring both sides and taking the difference. We determine that  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2 2 2 ½ða  2bBÞð9a  6bBÞ  ða  2bBÞ 17a2 þ 4b Bð9B  16cÞ þ abð64c  44BÞ ¼ 64ða  2bBÞ ða  bcÞða  bBÞ. This implies

Because

2

that H > 0 because a is much greater than b, c, and B in general case. Therefore, @Q B =@B > 0 is satisfied. From Eqs. (4) and (9), we know that Q O ¼ Q B when a ¼ B ¼ 0 and Q O is not a function of the trade promotion discount a. Because @Q B =@B > 0; Q B reaches a minimum when B ¼ 0 and increases as B increases. Therefore, Q B P Q O is satisfied when a ¼ B. dQ 

then compare Q O and Q S when a ¼ b. From Eqs. (4) and (5), we know that daO ¼ 0. Let ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 x ¼ ða þ bbÞð17a þ 64bc þ 27bbÞ þ 9b b2 and y ¼ a þ 2bb, the optimal order quantity in the scan-back case can be

(b) We

rewritten as

Q S ¼

2

5a b b2 x2 þ 6xy  7y2 3ðx þ yÞ þ 3bb þ : þ  2 8 4ða þ bbÞ 64ða þ bbÞ 2

þ6xy7y bb From W S ¼ x128bðaþbbÞ þ 4b þ 8ðaþbbÞ and 2

2

y þ ðxþ3yÞ  5b and P S ¼ 4b 16b 4

dP S db

dW S db

< 0 (Proposition 1), we have that   d 3ðxþ3yÞ < 0 (Proposition 2), we have that db . < 9b 2 8

d db



x2 þ6xy7y2 64ðaþbbÞ



2

bð2aþbbÞ <  8b  b16ðaþbbÞ From 2 < 0.

Then







 2 2 dQ S b bð2a þ bbÞ d x2 þ 6xy  7y2 d 3ðx þ 3yÞ b bð2a þ bbÞ d 3ðx þ 3yÞ < 3b þ < 3b  þ  ¼ 3b þ 2 2 db db 8 db 8 db 64ða þ bbÞ 4ða þ bbÞ 4ða þ bbÞ 2

þ

b bð2a þ bbÞ 4ða þ bbÞ

2

2



2

9b b ð6a2 þ 10abb þ 5b b2 Þ ¼ < 0: 2 2 4ða þ bbÞ

Because @Q S =@b < 0; Q S reaches a maximum when b ¼ 0 and decreases as b increases. Because Q O ¼ Q S is satisfied when a ¼ b ¼ 0, then we get that Q O P Q S is satisfied when a ¼ b. From above, we can conclude that Q B P Q O P Q S are satisfied when a ¼ b ¼ B. h Regarding the profits obtained under the three trade promotion policies, we offered the following two propositions. We demonstrate that there exists an unsold-discount price that results in the profits of the manufacturer and the profits of the retailer obtained under the unsold-discount policy being higher than those obtained under the off-invoice and scan-back policies. For the relationship between profit under off-invoice policy and profit under the scan-back policy, Yuan et al. (2013) has demonstrated that PMO P PMS and PRO P PRS .

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

Proposition 4 (a) There exists a value of B such that PMB P PMO ;

(b) There exists a value of B such that PMB P PMS .

Proof. Please see the proof in Appendix A. h Proposition 5 (a) There exists a value of B such that PRB P PRO ;

(b) There exists a value of B such that PRB P PRS .

Proof. Please see the proof in Appendix A. h According to Propositions 1 and 2, if the manufacturer realizes that he/she is not required to pay the scan-back discount for the entire quantity sold to the retailer, he/she obtains an incentive to lower the wholesale price; the lower wholesale price subsequently provides the retailer with an incentive to lower its retail price. Table 1 presents a summary of the theoretical predictions for manufacturer and retailer decisions and profits, given the specific trade promotion discounts used in the basic model. The table indicates that wholesale price, retail price and order quantity are high when the same trade promotion discount is allocated to an off-invoice policy, compared with that established under a scan-back policy. These results verify Propositions 1–3. Regarding the scan-back case, the retail price decreases as the trade promotion discount increases. In Proposition 2, we prove that @PS =@b < 0. Trade promotions are temporary price cuts that manufacturers offer retailers to encourage them to reduce retail prices (Kumar et al., 2001). The goal of trade promotion is for the retailer to offer the consumer a price discount (Blattberg and Levin, 1987). Intuitively, requiring the retailer to increase selling price while the manufacturer provides a high trade promotion discount is unreasonable. In practice, when the retailer receives a high trade promotion discount from the manufacturer, the retail price decreases and the discount is shifted to end customers, thereby increasing demand, or causing the demand to remain unchanged, thereby increasing the retailer’s per unit revenue. Table 1 indicates that the retail price decreases as the trade promotion discount increases in the scan-back case and remains unchanged as the trade promotion discount increases in the off-invoice case. In Table 1, the optimal retail price, order quantity, manufacturer profit and retailer profit for off-invoices case remain the same. For the manufacturer, the optimal wholesale price W O is derived as a linear function of off-invoice trade promotion pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5aþ32bcþ3 17a2 þ64abc þ a (see Eq. (5)). Then, the wholesale price after discount discount a, i.e. W O ¼ 64b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5aþ32bcþ3 17a2 þ64abc 5aþ32bcþ3 17a2 þ64abc  ) is free of a. Thus, the wholesale price after discount (W O  a ¼ ) is always (W O  a ¼ 64b 64b the same. The optimal retail price PO and order quantity Q O are determined based on the wholesale price after discount W O  a (see Eqs. (3) and (4)). Therefore, the retail price and order quantity keep unchanged because the wholesale price after discount is unchanged. Consequently, the manufacturer profit and the retailer profit remain the same. The scan-back promotion implies that the retailer receives a discount based on only the quantity it actually sells to customers. This provides the retailer with an incentive to lower its retail price to increase customer demand. However, Table 1 shows that the order quantity does not increase as the retail price decreases. This prompts the question as to why the order quantity decreases as the scan-back promotion discount increases. To answer this question, a more comprehensive discussion is required. When the scan-back promotion discount b ¼ 0, the optimal wholesale price is $6.28 and the optimal retail price is $14.37. This implies that the retailer’s marginal profit for selling a unit, $8.09, is higher than the marginal cost, $6.28. This explains why the optimal order quantity, 31.72, is higher than the mean of the demand, 28.15 (¼ 100  5  14:37). When the scan-back promotion discount b increases, the increase in the marginal cost incurred by the retailer is larger than the increase in their marginal profit. When b ¼ 3:5, the marginal profit, $8.7 (14:23  9:03 þ 3:5), is smaller than the marginal cost, $9.03. In this case, the optimal order quantity, 28.3, is smaller than the mean of the demand, 28.85 (¼ 100  5  14:23). This is why the order quantity decreases as the retail price decreases in the scan-back case. Table 2 presents the predictions of the basic model involving the unsold-discount policy. It indicates that the retail price and order quantity increase as the unsold-discount trade promotion discount increases. However, a high retail price implies that the retailer should purchase a small quantity. The retailer purchases a greater quantity as the unsold-discount trade promotion discount increases because ordering a large quantity reduces the likelihood of being out-of-stock. In this model, the retailer faces uncertain demand, which involves a high degree of risk. Therefore, the manufacturer provides an unsolddiscount trade promotion to share a portion of the risk with the retailer. Ordering a larger quantity may be profitable for the retailer, thereby reducing the probability of being out-of-stock and increasing profits for both the manufacturer and the retailer. Table 2 also shows that the retail price will increase as the unsold-discount increases. From the perspective

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

of goal, although unsold-discount policy increases both manufacturer’s and retailer’s profits, it cannot induce the retailer to offer a lower retail price, which violates the goal of trade promotion. This phenomenon can also be explained from another risk-taking perspective. When the unsold-discount trade promotion discount B is high, the decrease in marginal costs for the retailer is larger than the decrease in the marginal profit. For example, when B = 0 the retailer’s marginal profit is $8.09 and the marginal cost is $6.28; the difference is $1.81. When B = 3.5, the retailer’s marginal profit is $7.28 (14:58  7:3) and marginal cost is $3.8 (7:3  3:5); the difference is $3.48. The marginal profit of the retailer becomes much higher than the marginal cost as the unsold-discount price increases; therefore, the retailer orders a greater quantity. This explains why the retailer’s order quantity increases as the unsold-discount trade promotion discount increases. Figs. 2 and 3 verify Propositions 4 and 5. Fig. 2 shows that PMB P PMO when a ¼ B 6 11:60 and PMB P PMS when b ¼ B 6 12:34. Fig. 3 shows that PRB P PRO when a ¼ B 6 4:87 and PRB P PRS when b ¼ B 6 8:61. We determined that

PMB and PRB are concave functions of B, according to Figs. 2 and 3. In Fig. 2, when the manufacturer provides B = 10, the profit of the manufacturer PMB ¼ 204:56 is at a maximum. In this case, the wholesale price W B ¼ 10:50, the retail price PB ¼ 15:46, the order quantity of the retailer Q B ¼ 41:26, and the profit of the retailer PRB ¼ 102:28. In Fig. 3, when the manufacturer provides B = 2.67, the profit of the retailer PRB ¼ 179:56 is at a maximum. In this case, the wholesale price W B ¼ 7:02, the retail price PB ¼ 14:52, the order quantity of the retailer Q B ¼ 34:69, and the profit of the manufacturer PMB ¼ 130:00. However, when a ¼ B ¼ 10, the retailer prefers the off-invoice trade promotion because PRO > PRB . When the trade promotion discount is established within (0, 4.87), both the manufacturer and retailer profits obtained under the unsold-discount policy are higher than those obtained under the off-invoice and scan-back policies. Therefore, we concluded that manufacturers and retailers prefer the unsold-discount policy over the off-invoice and scan-back policies when profit is maximized and demand is uncertain. This provides a new research direction in which game theory can be used. Manufacturers and retailers can negotiate a discount for unsold products within (0, 4.87) to increase their profits according to their bargaining powers. This research proposal can be addressed in future research. 5. Target rebate policy Target rebate provides a direct incentive for the retailer to increase sales through a rebate paid by the manufacturer for any item sold above the target quantity. To understand the effects of target rebate on profits and decisions, we considered target rebate as another trade promotion in this study. We use the following notations in this section: PT : retail price in the target rebate case W T : wholesale price in the target rebate case Q T : retailer order quantity in the target rebate case X: target quantity T: rebate for any item sold above the target quantity In the target rebate policy, the rebate is paid to retailer for each unit sold beyond the target quantity X. When the retailer order quantity is less than the target quantity (Q T 6 X), the sales demand is not possibly beyond the target quantity. When R  R 2ðabPT Þ Q T X Q tX Q T > X, the expected rebate to the retailer is T X T 2ðabP dt þ Q T dt . Then the retailer profit in the target rebate 2ðabP T Þ TÞ case can be written as

PRT

8 RQ < PT 0 T ¼ : PT R Q T 0

t dt 2ðabP T Þ t dt 2ðabP T Þ

þ PT þ PT

R 2ðabPT Þ QT

QT dt 2ðabP T Þ

QT

QT dt 2ðabP T Þ

R 2ðabPT Þ

 WT QT ;  WT QT þ T

R QT X

tX dt 2ðabPT Þ

þ

Fig. 2. Manufacturer profit under difference case.

R 2ðabPT Þ QT

Q T X dt 2ðabP T Þ

if Q T 6 X;  ; if Q T > X:

ð18Þ

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

Fig. 3. Retailer profit under difference case.

When Q T > X, the expected rebate the manufacturer needs to pay to the retailer is T

R

QT tX dt X 2ðabPT Þ

þ

R 2ðabPT Þ QT

Q T X dt 2ðabP T Þ



. The

manufacturer’s profit in the target rebate case can be written as

(

PMT ¼

ðW T  cÞQ T ; ðW T  cÞQ T  T

R QT X

tX dt 2ðabPT Þ

þ

R 2ðabPT Þ QT

Q T X dt 2ðabP T Þ

if Q T 6 X;  ; if Q T > X:

ð19Þ

Because of the complexity of profit functions, we determine the optimal solution by numerical search methods. The optimal solution and its concave property are verified numerically in Figs. 4 and 5. Table 3 indicates that the wholesale price and order quantity increase and the retail price decrease as the rebate T increases. To obtain benefits from the target rebate policy, the retailer can reduce the selling price to raise the end demand. Table 3 also indicates that manufacturer profit increases

Fig. 4. Graphic illustration of PRT versus W T .

Fig. 5. Graphic illustration of PMT versus P T and Q T .

31

Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39 Table 3 Predictions of the theoretical model under target rebate (a = 100; b = 5; c = 1; X = 31.72). Trade promotion discount

Wholesale price Target rebate

Retail price Target rebate

Order quantity Target rebate

Manufacturer profits Target rebate

Retailer profits Target rebate

0.00

6.28 (0%)a 6.69 (15.65%) 7.07 (30.77%) 7.44 (45.32%) 7.80 (59.35%) 8.17 (72.78%)

14.37

31.72

167.35

128.34

14.36

31.85

181.21

115.16

14.32

32.19

195.09

103.05

14.26

32.65

209.03

91.49

14.19

33.17

223.09

80.07

14.11

33.71

237.31

68.51

1.00 2.00 3.00 4.00 5.00 a

Trade promotion as a percentage of wholesale price.

and retailer profit decreases under the target rebate policy when profit is maximized and demand is uncertain. However, the target rebate cannot benefit the retailer when the manufacturer determines the wholesale price to maximize their own profit. However, Fig. 6 shows that both manufacturer and retailer profits increases when the wholesale price is determine within an appropriate range. For examples, when the wholesale price is within (4.93, 6.29), (4.73, 6.34), and (4.70, 6.41), both manufacturer and retailer profits increase under the target rebate policy. The concern can be addressed through an agreement in which the wholesale price is determined within the appropriate range.

(a)

T=1 Manufacturer Profit under Target Rebate Retailer Profit under Target Rebate Manufacturer Profit without Target Rebate Retailer Profit without Target Rebate

Profit

210 160 110 4.70 4.90 5.10 5.30 5.50 5.70 5.90 6.10 6.30 6.50

Wholesale price

(b) T=2 Manufacturer Profit under Target Rebate Retailer Profit under Target Rebate Manufacturer Profit without Target Rebate Retailer Profit without Target Rebate

Profit

210 160 110 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00 6.20 6.40 6.6

Wholesale price

(c) T=3

Manufacturer Profit under Target Rebate Retailer Profit under Target Rebate Manufacturer Profit without Target Rebate Retailer Profit without Target Rebate

Profit

260 210 160 110 4.40 4.60 4.80 5.00 5.20 5.40 5.60 5.80 6.00 6.20 6.40 6.60

Wholesale price Fig. 6. Manufacturer and retailer profits under target rebate with different wholesale price.

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Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39 Table 4 Optimal decisions when trade promotion discount leads to market expansion. Optimal decision variable

Unsold-discount h

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffii

2

ð5AB þ45bBME þ32bcÞðAB bBME Þþ9b2 BME þ3ðAB 2bBME Þ

W ME

64bðAB bB

5AB 9bB

Q ME

AB þ3bB

P ME

ME

ME

ME

ME

ðAB bBME Þð17AB þ64bc27bBME Þþ9b2 BME

Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ME ME ME

þ4bW B 3

AB bB 2

AB 9bB

þ8bW B

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ME ME ME

þ

AB bB

AB 9bB

þ8bW B

4b

6. Model extension In this section, we consider two extensions of the basic models. First, we consider the market expansion model in which the trade promotion discount influences market expansion. Second, we consider price competition for various brands. In this case, the demand of a brand is dependent on its retail price and the price of competing brands. We also compared the trade promotions used in these extended models. 6.1. Market expansion According to the demand function provided by Yuan et al. (2013), new baseline demands can be determined for the offinvoice case and the scan-back case, which are AO and AS , respectively. These are functions of the trade promotion discount such that AO ¼ a þ g aME  haME and AS ¼ a þ gbME  hbME , where g P 0 and h P 0: Under the unsold-discount and target 2

2

ME2

ME2

rebate policies, the new baseline demand is AB ¼ a þ gBME  hB and AT ¼ a þ gT ME  hT , respectively, where g P 0 and h P 0. Thus, f ðpÞ ¼ A  bp, where A ¼ AO ; Ab ; AB ; or AT in the demand function. In the basic models, the trade promotion discounts were treated exogenously because the profits obtained were not concave. For market expansion, we decided that the trade promotion discounts should be endogenously chosen. Table 4 summarizes the optimal wholesale price, order quantity, and retail price under the unsold-discount policy. By substituting the closed form of these optimal decisions into manufacturer profit functions, we maximized manufacturer profits with respect ME ME to trade promotion discounts by solving dPME ¼ 0, dPME ¼ 0, and dPME M a =da Mb =db M B =dB

ME

¼ 0 for each case. For the target

rebate policy, we solved the problem numerically as well. Table 5 in this paper indicate that manufacturer and retailer profits obtained under the unsold-discount policy are higher than those obtained under the off-invoice and scan-back policies. Under the target rebate policy, the profit of the manufacturer is higher and the profit of the retailer is lower than those obtained under the off-invoice and scan-back policies. The new baseline demand caused the profit function of the manufacturer to be concave. Although the optimal trade promotion discount can be determined for the off-invoice and scan-back policies, the unsold-discount trade promotion still produces more favorable results than that of the other two trade promotions. Similar to the basic model, the wholesale price is lowest and the retail price and order quantity are highest under the unsold-discount policy. In addition, the manufacturer offers substantial discounts when trade promotion is allocated to unsold-discount and target rebate instead of to off-invoice and scan-back policies. 6.2. Price competition Retailers commonly sell products produced by competing manufacturers. Such firms wish to determine which trade promotion is most favorable under price competition. In this section, we consider competition that arises from various brands (manufacturers) that are uniquely priced. We considered that the demand of a brand is dependent on its retail price and the price of competing brands. f ðpi Þ in the demand function for channel i is assumed to be a  bP i þ mPj ; where a > 0 and b > m > 0. a is the market potential, b is the price elasticity for the brand, and m is the price elasticity for the competing brand. This section considers two manufacturers (manufacturer 1 and manufacturer 2) and compares four trade promotions (off-invoice, scan-back, unsold-discount and target rebate). Let a be the per-unit discount under the off-invoice policy, the retailer profit in the off-invoice case can be written as

" Z 2 X PRO ¼ PO;i i¼1

Q O;i 0

t 2ða  bP O;i þ mP O;j Þ

Z dt þ P O;i

2ðabPO;i þmPO;j Þ Q O;i

# Q O;i dt  ðW O;i  aÞQ O;i ; where j ¼ 3  i: 2ða  bP O;i þ mPO;j Þ

ð20Þ The profit of the manufacturer i in the off-invoice case is

PMO;i ¼ ðW O;i  a  cÞQ O;i :

ð21Þ

33

Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39 Table 5 Optimal decisions under unsold-discount and target rebate in market expansion (c = 1). Demand factor (a, b)

a b c

Expansion factors (g, h)

(80, 5)

(5, 1)

(80, 6)

(5, 2)

(80, 7)

(5, 3)

(100, 5)

(6, 1)

(100, 6)

(6, 2)

(100, 7)

(6, 3)

(120, 5)

(7, 1)

(120, 6)

(7, 2)

(120, 7)

(7, 3)

Trade promotion discount

Wholesale price

Retail price

Order quantity

Profits

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

3.08 (48.13%)a 1.59 (31.67%) 1.09 (25.41%) 3.61 (46.34%) 1.86 (30.69%) 1.27 (24.61%) 4.13 (44.84%) 2.12 (29.78%) 1.45 (24.05%)

4.22 (58.94%) 2.32 (42.41%) 1.69 (36.34%) 4.72 (54.69%) 2.56 (39.14%) 1.84 (33.33%) 5.22 (51.48%) 2.81 (36.83%) 2.00 (31.20%)

6.40

7.16

12.59

11.85

29.88

26.65

5.47

10.15

9.74

26.39

24.50

4.29

4.65

8.64

8.34

24.63

23.13

7.79

8.63

15.78

15.02

39.02

34.84

6.06

6.54

12.64

12.22

34.65

32.24

5.16

5.52

10.73

10.42

32.57

30.68

9.21

10.14

19.04

18.25

48.36

43.21

7.12

7.63

15.16

14.73

43.04

40.73

6.03

6.41

12.84

12.52

40.63

38.85

131.41b 92.50c 93.38 67.73 73.38 53.56 218.82 155.84 155.78 113.98 123.30 90.77 331.02 237.70 235.14 173.11 186.75 138.29

159.74 41.93 108.30 40.60 83.91 34.10 259.11 82.10 176.83 75.14 138.08 63.17 385.19 136.39 263.24 120.66 206.38 101.37

5.02

Trade promotion as a percentage of wholesale price. Manufacturer profit. Retailer profit.

Let b be the per-unit discount under the scan-back policy, the retailer profit in the scan-back case can be written as

PRS ¼

" Z 2 X ðPS;i þ bÞ

Q S;i

2ða  bP S;i þ mPS;j Þ

0

i¼1

Z

t

dt þ ðP S;i þ bÞ 

2ðabPS;i þmP S;j Þ

QS

i

# Q S;i dt  W S;i Q S;i ; 2ða  bPS;i þ mP S;j Þ

where j ¼ 3  i:

ð22Þ

The profit of the manufacturer i in the scan-back case is

PMS;i ¼ ðW S;i  cÞQ S;i  b

Z

Q S;i

t 2ða  bP S;i þ mPS;j Þ

0

Z

dt  b

2ðabP S;i þmPS;j Þ

Q S;i

Q S;i dt; 2ða  bP S;i þ mPS;j Þ

where j ¼ 3  i:

ð23Þ

The retailer profit in the unsold-discount case can be written as

PRB ¼

2 h X

PB;i

R Q B;i 0

i¼1

t dt 2ðabPB;i þmP B;j Þ

þ PBi 

R 2ðabPB;i þmPB;i Þ Q B;i

Q B;i dt 2ðabPB;i þmPB;j Þ

R Q þ B 0 B;i

Q B;i t dt 2ðabP B;i þmPB;j Þ



i  W B;i Q B;i ;

where j ¼ 3  i:

ð24Þ

The profit of the manufacturer i in the unsold-discount case is

PMB;i ¼ ðW B;i  cÞQ B;iB

Z

Q B;i

0

Q B;i  t dt 2ða  bP B;i þ mP B;j Þ

where j ¼ 3  i

ð25Þ

Let j = 3  i, the retailer profit in the target rebate case can be written as

PRT

8 2 3 RQ > 2 PT;i 0 T;i 2ðabPT;it þmPT;j Þ dtþ > X > > 4 5; > > R 2ðabPT;i þmPT;i Þ > Q T;i > > P dt  W Q i¼1 T;i T;i T;i > Q T;i 2ðabP T;i þmPT;j Þ > > > 3 2 RQ > > < PT;i 0 T;i 2ðabP t þmP Þ dtþ T;i T;j 7 6 ¼ 1 7 6 0 R Q T;i tX > > 2 6 7 dtþ X > X 2ðabP T;i þmPT;j Þ > 7 6 @ > A þ T > 7; 6 R > 2ðabP þmP Þ Q X T;i T;j > T;i 7 6 > dt i¼1 > 7 6 Q T;i 2ðabPT;i þmPT;j Þ > > 5 4 > > R 2ðabP T;i þmP T;j Þ Q T;i > : PT i Q T;i dt  W Q T;i T;i 2ðabPT;i þmPT;j Þ

if Q T;i 6 X;

ð26Þ if Q T;i > X;

Let j = 3  i, the profit of the manufacturer i in the target rebate case can be written as

PMT;i ¼

8 < ðW T;i  cÞQ T;i ;

: ðW T;i  cÞQ T;i  T

R Q T;i X

tX dt 2ðabPT;i þmPT;j Þ

þ

R 2ðabPT;i þmPT;j Þ Q T;i

9 if Q T;i 6 X; =  Q T;i X dt ; if Q T;i > X: ; 2ðabP T;i þmPT;j Þ

ð27Þ

34

Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

To determine the optimal wholesale price, we first selected an initial guess of wholesale price. According to this wholesale price, the retailer determines optimal retail prices and order quantities to maximize their profit for various brands. The profit of the manufacturer can also be calculated. We then chose an error tolerance and set a new wholesale price as the sum of the original wholesale price and the error tolerance. Based on the new wholesale price, the optimal retail prices and order quantities for various brands could be determined. We also computed the profit of the manufacturer based on the new wholesale price. If the profit of the manufacturer obtained using the new wholesale price was larger than that of the original wholesale price, we set another new wholesale price (the sum of the new wholesale price and the error tolerance) and determined the retailer decisions and profits of the manufacturer. We repeated these processes until the manufacturer profit decreased at the first time. Thus, we can determine the optimal wholesale prices, retail prices, and order quantities under price competition. Figs. 7–11 present the manufacturer and retailer behaviors based on various trade promotions. The wholesale price will increase as the trade promotion discount increases. The manufacturer reasonably increases his selling price as the discount increases. The wholesale price is highest under the off-invoice policy because the manufacturer provides discounts on all of the products the retailer orders in the off-invoice case. Regarding the off-invoice policy, when the trade promotion discount increases, the retail price, order quantity, manufacturer profit, and retailer profit remain unchanged. Regarding the scan-back policy, when the trade promotion discount increases, the retail price, order quantity, manufacturer profit, and retailer profit decrease. Regarding the unsold-discount policy, when the trade promotion discount increases, the retail price, order quantity, manufacturer profit, and retailer profit increases. Regarding the target rebate policy, when the trade promotion discount increases, the retail price and manufacturer profit increase but the order quantity and retailer profit decrease. This indicates that the manufacturer and retailer receive benefits from only the unsold-discount policy when profits are maximized. The manufacturer and retailer profit functions are concave according to the unsold-discount. When the unsold-discount is determined within an appropriate range, both manufacturer and retailer’s profits are higher than those obtained under the offinvoice and scan-back policies. Thus, the appropriate range becomes wider as the price elasticity for the competing brand m increases. 6.3. Other demand functions In above sections, we consider the linear price sensitive and uncertain demand with uniform distribution. This demand setting makes the model analytically tractable. To test our results under a different demand function, we consider a demand with another common distribution, i.e. the price sensitive and uncertain demand with normal distribution (D  N½l ¼ a  bp; r). Table 6 indicates that both manufacturer and retailer benefit from the unsold-discount policy; only manufacturer benefits from the target rebate policy; neither manufacturer nor retailer benefits from the off-invoice and scan-back policies. Similar to the linear price sensitive and uncertain demand with uniform distribution, the total profit

(b) m=1 Off-invoice

Scan-back

Off-invoice

Scan-back

Buy-back

Target rebate

Buy-back

Target rebate

Wholesale price

Wholesale price

(a) m=0

11 10 9 8 7 6 0

1

2

3

12 11 10 9 8 7 0

4

Trade promoon discount

1

2

Wholesale price

(c) m=2 Off-invoice

Scan-back

Buy-back

Target rebate

15 14 13 12 11 10 9 0

1

2

3

3

Trade promoon discount

4

Trade promoon discount Fig. 7. Wholesale price under different trade promotion and competition level.

4

35

Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

(b) m=1 Off-invoice

Scan-back

Off-invoice

Scan-back

Buy-back

Target rebate

Buy-back

Target rebate

14.7

Retail price

Retail price

(a) m=0

14.5 14.3 14.1

18.2 18 17.8 17.6

0

1

2

3

4

0

Trade promoon discount

1

2

3

4

Trade promoon discount

Retail price

(c) m=2 Off-invoice

Scan-back

Buy-back

Target rebate

23.9 23.7 23.5 23.3 0

1

2

3

4

Trade promoon discount Fig. 8. Retail price under different trade promotion and competition level.

(b) m=1 Off-invoice

Scan-back

Buy-back

Target rebate

37 35 33 31 29 27

Order quanty

Order quanty

(a) m=0

0

1

2

3

4

Off-invoice

Scan-back

Buy-back

Target rebate

37 35 33 31 29 0

1

2

3

4

Trade promoon discount

Trade promoon discount

Order quanty

(c) m=2 Off-invoice

Scan-back

Buy-back

Target rebate

38 36 34 32 30 0

1

2

3

4

Trade promoon discount Fig. 9. Order quantity under different trade promotion and competition level.

of manufacturer and retailer increases in the target-rebate case. Thus, the target rebate policy can benefit both manufacturers and retailers when the wholesale price is determined within an appropriate range. To the end of this section we consider a non-linear price-elasticity demand, D ¼ f ðpÞ þ U, where f ðpÞ ¼ gph is a function of unit retail price p with g > 0, h > 1, and U is a continuous random variable following a uniform distribution on ½gph ; gph . The results from this non-linear price-elasticity are different with those from linear price demand function. These support that performances of promotional strategies are influenced by demand function (Saha, 2013). However, as shown in Table 7, target rebate policy still provides benefits to both manufacturer and retailer. But, neither manufacturer nor retailer benefits from the off-invoice, scan-back and unsold discount policies.

36

Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

(b) m=1 Off-invoice

Scan-back

Off-invoice

Scan-back

Buy-back

Target rebate

Buy-back

Target rebate

270 220 170 120 0 1 2 3 4 5 6 7 8 9 10 11 12

Manufacturer profit

Manufacturer profit

(a) m=0

290 240 190 140 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Trade promoon discount

Trade promoon discount

Manufacturer profit

(c) m=2 Off-invoice

Scan-back

Buy-back

Target rebate

420 370 320 270 220 0

2

4

6

8

10 12 14 16 18

Trade promoon discount Fig. 10. Manufacturer profit under different trade promotion and competition level.

(a) m=0

(b) m=1 Off-invoice

Scan-back

Off-invoice

Scan-back

Buy-back

Target rebate

Buy-back

Target rebate

345

260

Retailer profit

Retailer profit

280

240 220 200 180

325 305 285 265

0

1

2

3

4

5

6

0

1

2

3

4

(c) m=2 Off-invoice

Scan-back

Buy-back

Target rebate

Retailer profit

480 460 440 420 400 380 0

1

2

3

4

5

6

5

6

Trade promoon discount

Trade promoon discount

7

Trade promoon discount Fig. 11. Retailer profit under different trade promotion and competition level.

37

Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39 Table 6 Predictions of the theoretical model under linear price sensitive and normal distribution demand (a = 100; b = 15; c = 1;

r ¼ 2).

Trade promotion discount

Wholesale price

Retail price

Order quantity

Manufacturer profits

Retailer profits

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

0.00 0.10 0.20 0.30 0.40 0.50

3.72 3.82 3.92 4.02 4.12 4.22

3.72 3.82 3.92 4.01 4.11 4.21

5.1424 5.1424 5.1424 5.1424 5.1424 5.1424

5.1424 5.1417 5.1409 5.1353 5.1346 5.1334

21.68 21.68 21.68 21.68 21.68 21.68

21.68 21.65 21.64 21.70 21.68 21.66

58.96 58.96 58.96 58.96 58.96 58.96

58.96 58.94 58.92 58.90 58.87 58.85

29.08 29.08 29.08 29.08 29.08 29.08

29.08 29.04 29.01 29.20 29.17 29.14

0.00 0.10 0.20 0.30 0.40 0.50

Wholesale price

Retail price

Unsolddiscount

Target rebate

Unsolddiscount

3.72 3.72 3.72 3.72 3.72 3.72

3.72 3.76 3.83 3.90 3.98 4.07

5.1424 5.1432 5.1440 5.1449 5.1457 5.1466

Order quantity

Manufacturer profits

Retailer profits

Target rebate

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

5.14 5.12 5.10 5.09 5.08 5.07

21.68 21.70 21.72 21.74 21.77 21.79

21.68 22.10 22.29 22.50 22.63 22.68

58.96 58.99 59.01 59.03 59.06 59.08

58.96 60.81 62.70 64.60 66.52 68.45

29.08 29.11 29.15 29.19 29.22 29.26

29.08 28.36 27.00 25.64 24.06 22.26

Table 7 Predictions of the theoretical model under price-elasticity and uncertain demand. Trade promotion discount

Wholesale price

Retail price

Order quantity

Manufacturer profits

Retailer profits

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

Offinvoices

Scanbacks

0.00 0.10 0.20 0.30 0.40 0.50

1.50 1.60 1.70 1.80 1.90 2.00

1.50 1.61 1.72 1.83 1.94 2.04

3.00 3.00 3.00 3.00 3.00 3.00

3.00 3.04 3.08 3.12 3.16 3.20

185.19 185.19 185.19 185.19 185.19 185.19

185.19 173.11 162.42 152.90 144.37 136.68

92.59 92.59 92.59 92.59 92.59 92.59

92.59 88.35 84.49 80.97 77.75 74.78

138.89 138.89 138.89 138.89 138.89 138.89

138.89 132.77 127.20 122.11 117.44 113.14

0.00 0.10 0.20 0.30 0.40 0.50

Wholesale price

Retail price

Order quantity

Manufacturer profits

Retailer profits

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

Unsolddiscount

Target rebate

1.50 1.53 1.55 1.58 1.61 1.65

1.50 1.57 1.65 1.72 1.80 1.88

3.00 3.03 3.05 3.09 3.11 3.15

3.00 2.97 2.95 2.92 2.90 2.88

185.19 185.03 184.54 183.69 182.46 180.85

185.19 186.97 185.32 187.72 186.64 185.83

92.59 92.55 92.43 92.22 91.92 91.53

92.59 95.32 97.99 100.60 103.17 105.69

138.89 138.79 138.48 137.96 137.19 136.19

138.89 137.00 133.35 131.66 128.22 124.87

7. Conclusion In this paper, we consider unsold-discount and target rebate as new trade promotion strategies and compared them with off-invoice and scan-back trade promotions. In the models the manufacturer determines the wholesale price and trade promotion policy, and the retailer determines the retail price and order quantity to maximize his/her profits when demand is uncertain. We also discuss the behaviors of channel members and the effectiveness of these trade promotions. Several managerial insights are provided based on propositions and numerical analysis. Consider the linear price sensitive and uncertain demand, the results indicate that manufacturers and retailers prefer unsold-discount and target rebate trade promotions over off-invoice and scan-back trade promotions under demand uncertainty. We demonstrated that wholesale price is lowest and retail price and order quantity are highest under the unsold-discount policy; retail price and order quantity increase as the unsold-discount increases; and retail price decreases and order quantity increases as the target rebate increases. Fig. 12 summarizes the results for the profits of both manufacturers and retailers based on the use of the four trade promotions. We determine the existence of an unsold-discount that causes manufacturer and retailer profits obtained under the unsold-discount policy to be higher than those obtained under the off-invoice and scan-back policies. The manufacturer profit is highest and retailer profit is lowest under the target rebate policy. However, the target rebate policy can benefit both manufacturers and retailers when the wholesale price is determined within an appropriate range. This means that the target-rebate shifts to a location where both manufacturer and retailer profits obtained under the unsold-discount policy are higher than those obtained under the off-invoice and scan-back policies after a wholesale price negotiation. However,

38

Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

Manufacturer profit Targetrebate

Wholesale price negotiation Unsolddiscount

Off-invoice

Scan-back Retailer profit Fig. 12. Manufacturer and retailer profits under different trade promotions.

when consider the price-elasticity and uncertain demand, both manufacturer and retailer can only benefit from the target rebate policy. The performances of promotional strategies are strongly influenced by demand function. In addition, we consider some extensions of the model: when the trade promotion discount influences market expansion and when price is competitive for various brands. We conclude that the aforementioned results are valid when applied to these extended models. Under the market expansion condition, the manufacturer offers substantial discounts when trade promotion is allocated to unsold-discount and target rebate instead of to off-invoice and scan-back policies. In the case of price competition, the appropriate unsold-discount range is wider because the price elasticity of the competing brand increases. The results of this study can be used as a reference by business managers and administrators. Several opportunities for further research are provided. First, this paper considers the demand with uniform distribution. One may consider a more general demand form to see the new predictions. Secondly, we can be extended to consider a game problem in which the unsold-discount under the unsold-discount policy or the wholesale price under the target rebate are negotiated, based on the bargaining powers of channel members. Third, we compared the unsold-discount and target rebate policies with the off-invoice and scan-back policies and demonstrated that the unsold-discount and target rebate policies produce more favorable results than that produced by the off-invoice and scan-back policies. Therefore, providing other trade promotions that can benefit channel members is worthwhile. In addition, discussing the decision behaviors of companies according to the four trade promotions by conducting lab experiments can be undertaken in future studies. The experimental results can subsequently be compared with the results of the theoretical model used in this study. If we consider the centralized decision-making (i.e. benchmark model that channel members make decisions to maximize the total channel profit), the profit sharing between channel members depends on the discount negotiated. In this case, negotiation (discount) between the channel members under the decentralized scenario is an interesting research topic. Also, design coordinating contracts based on trade promotions is worthy to research. Acknowledgement This paper is supported in part by the Ministry of Science and Technology – Taiwan under Grant MOST 104-2221-E-011171-MY3. Appendix A

Proof of Proposition 4 (a) PMO ¼ ðW O  c  aÞQ O and PMB ¼ ðW B  cÞQ B  B

0

Q B t dt 2ðabP B Þ

BQ 2

B ¼ ðW B  cÞQ B  4ðabP  . Þ B

i From Q B ¼ abB , we have PMB ¼ ðW B  cÞ  abB Q B . Consider the slope when B = 0 and take the first derivative of h i.  pffiffiffi pffiffiffi pffiffiffi BðabP  Þ BðabP Þ ðW B  cÞ  abB B , dBjB¼0 ¼ 45 abw  ð117a þ 288bcÞ ab  4ð4 aþ we have d ðW B  cÞ  abBB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 26a þ 64bc þ 6wÞw=64 abw; where w ¼ að17a þ 64bcÞ. Because a is much greater than b and c in general case, we h  i.  BðabP Þ know that d ðW B  cÞ  abBB dBjB¼0 > 0 in general case. Then from dQ B =dB > 0 in Proposition 8, we know that 2 4ðabPB Þ

h

R Q B

BðabP B Þ

dPMB =dB > 0. This means when B = 0, PMB increases as B increases in general case. Moreover, from PMB jB¼0 ¼ PMO , we know that there exists a value of B > 0 such tat PMB P PMO in general case. Therefore, there exists a value of unsold-discount such

39

Y.-C. Tsao, J.-C. Lu / Transportation Research Part E 96 (2016) 20–39

that the manufacturer’s profit under the unsold-discount policy is higher than the manufacturer’s profit under the offinvoice policy. (b) Yuan et al. (2013) showed that PMO P PMS . Then from Proposition 4(a), we can know that there exists a value of B

such tat PMB P PMS in general case. This indicates that there exists a value of unsold-discount such that the manufacturer’s profit under the unsold-discount policy is higher than the manufacturer’s profit under the scan-back policy. h

Proof of Proposition 5

Z Q B Q B Q B  t    dt þ B  dt  W B Q B 2ða  bP B Þ 2ða  bPB Þ 0 0 Qs ! 2 ðPB  BÞQ 2 bP O      B and PRO ¼ ¼ ðPB  W B ÞQ B   W O Q O :  a 4ða  bP B Þ

(a) PR ¼ P B B

Z

Q B

t   dt þ P B 2ða  bP B Þ

we

2ðabPB Þ





 4ðabPB Þ bP2 B þaB2bBP B , we have PRB ¼  W B Q B . Consider the slope when B = 0 and take the first derivative abB abB h pffiffiffi have dPRB =dBjB¼0 ¼ 15759a4 þ 93600a3 bc þ 23616abcf þ 4608b2 c2 f þ 63a2 ð2048b2 c2 þ 119fÞ  2112 abcf 2

From Q B ¼

PRB ,

Z

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f ¼ a3 ð17a þ 64bcÞ

of

-

3196a -13120a bc-  4a ð1024b c þ 201fÞ-=1024a ð17a þ 64bcÞ; where and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 26a þ 64bc þ 6f=a. Because a is much larger than b and c in general case, we know that dPRB =dBjB¼0 > 0 in general case. This means when B = 0, PRB increases as B increases in general case. Furthermore, from PRB jB¼0 ¼ PRO , we know that there exists a value of B > 0 such tat PRB P PRO in general case. Therefore, there exists a value of unsold-discount such that the retailer’s profit under the unsold-discount policy is higher than the retailer’s profit is under the off-invoice policy. 7=2

5=2

3=2

2 2

2

-¼

(b) Yuan et al. (2013) showed that PRO P PRS . Then from Proposition 5(a), we can know that there exists a value of B such tat PRB P PRS in general case. This indicates that there exists a value of unsold-discount such that the retailer’s profit under the unsold-discount policy is higher than the retailer’s profit is under the scan-back policy. h

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