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Full returns policy and Pareto efficiency underdecentralized supply chainShu-Hui Chang a , Ai-Fen Tsai b & Lee-Mei Sheen aa Department of Business Administration , Takming University of Science andTechnology , No. 56, Sec. 1, Huanshan Road, Neihu District, Taipei City , 11451 , Taiwan,R.O.C.b Department of Applied Foreign Languages , Takming University of Science andTechnology , Taiwan, R.O.C.Published online: 14 Jun 2013.

To cite this article: Shu-Hui Chang , Ai-Fen Tsai & Lee-Mei Sheen (2010) Full returns policy and Pareto efficiencyunder decentralized supply chain, Journal of Statistics and Management Systems, 13:5, 1045-1053, DOI:10.1080/09720510.2010.10701519

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Full returns policy and Pareto efficiency under decentralized supplychain2

Shu-Hui Chang 1 ∗

Ai-Fen Tsai 24

Lee-Mei Sheen 1

1 Department of Business Administration6

Takming University of Science and TechnologyNo. 56, Sec. 1, Huanshan Road8

Neihu District, Taipei CityTaiwan 11451, R.O.C.10

2 Department of Applied Foreign LanguagesTakming University of Science and Technology12

Taiwan, R.O.C.

Abstract14

In this paper, we explore how manufacturer uses quantity discount and return policyto reach channel coordination and Pareto efficiency from the view of information symmetry.16

This paper can be modelized into a three-stage game. From the model we indicate the rangeof the retailer’s participation constraint that manufacturer is willing to give in order to reach18

channel coordination and Pareto efficiency under quantity discount strategy or together withreturn policy.20

Keywords and phrases : Quantity discount, return policy, channel coordination, Pareto efficiency.

1. Introduction22

The existing literature suggests that contract terms such as quantitydiscount and return policy may be used to coordinate profit distribu-24

tion among supply chain to improve the efficiency of supply chain.For example, Pasternack (1985) presents a model to overcome double26

marginalization by return policy, and suggests that in the case of a decen-tralized channel, channel coordination can be reached by the manufacturer28

∗E-mail: shchang@mail.takming.edu.tw——————————–Journal of Statistics & Management SystemsVol. 13 (2010), No. 5, pp. 1045–1053c© Taru Publications

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1046 S. H. CHANG, A. F. TSAI AND L. M. SHEEN

providing the retailer a returns policy with partial refund. Mantrala andRaman (1999) extended the model of Pasternack (1985) to the case of multi2

retailers. Pasternack (1985) shows in the case of a decentralized channelthe policy of a manufacturer allowing unlimited returns for full credit is4

system suboptimal. The theorem shows that if the manufacturer allowsthe retailer to make unlimited returns for full credit, it is impossible to6

achieve channel coordination. And Bose and Anand (2007) suggest thatin the case of a decentralized channel, if wholesale price is endogenously8

rather than exogenously provided, channel coordination can impossiblybe reached but Pareto efficiency may be attained when partial refund are10

allowed only. Su and Shi (2002), and Hsieh et al. (2008) discussed channelcoordination and Pareto efficiency under the combination of quantity12

discount strategy and return policy in the integrated channel, but theydid not discuss whether quantity discount strategy and return policy14

combined together can reach channel coordination or not.

In this paper, we also discuss whether manufacturer uses quantity16

discount and return policy allowing unlimited returns for full credit toreach channel coordination and Pareto efficiency or not from the view18

of information symmetry in the integrated channel. For the attempt toovercome double marginalization by contract terms of quantity discount20

and return policy, manufacturer must provide guaranty to retailer thata certain level of expected profit would be obtained1. Su and Shi (2002)22

and Hsieh et al. (2008) suggest that channel coordination can be reachedby quantity discount and return policy, but they did not derive definite24

range of participation condition of retailer for either case. Although Suand Shi (2002), and Hsieh et al. (2008) suggest that negative quantity26

discount might occur when two strategies of return policy and quantitydiscount are combined, neither of the two papers definitely indicates28

under what circumstance this situation will occur. From these modelswe explore the range of participation constraint that manufacturer can30

give to retailer in order to attain Pareto efficiency under the circumstancethat supply chain has reached channel coordination by quantity discount32

strategy or together with return policy. At what rage of buyback price, willmanufacturer give positive quantity discount when strategies of return34

policy and quantity discount strategy are combined? After participation

1Lau et al. (2000), Lau and Lau (1999), and Tsay (2001) point out that in the case when themanufacturer is the channel leader, a returns policy must divide the total channel profit insuch a way as to guarantee the retailer some exogenously given minimum expected profit;this is the retailer’s participation constraint.

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DECENTRALIZED SUPPLY CHAIN 1047

constraint is given to retailer by manufacturer, will the change in thebuyback price affect the expected profits of manufacturer and retailer? The2

outcomes mentioned in this paper show that the equilibrium solutionsunder channel coordination with a quantity discount strategy and a4

returns policy is Pareto-efficient with respect to the decentralized supplychain. But if retailer asks for positive purchase discount and also asks6

for return of good in full quantity and for full amount, then channelcoordination is not likely to achieve.8

The paper is organized as follows. In Section 2 we list the well-known results under decentralized and integrated supply chain without10

return policy and quantity discount strategy. In Section 3 we establishthe structure the model, and derive the equilibrium solutions with return12

policy and quantity discount strategy. The final Section is conclusion.

2. Basic model14

The notation used throughout this paper is summarized as follows:

Decision variables16

Q = inventory level; w = wholesale price per item

Parameters18

c = production cost per item; p = retail price per item;

u = buyback price per item; s = goodwill loss per item;20

f (·) = density function of the market demand, and f (·) is uniformlydistributed on the interval [1, 0] .222

F(·) = distribution function of the market demand

These assumptions mentioned below are made in this paper:24

(a) p > w > c > 0, (b) u ≤ w , (c) s > 0, (d) p is determined by themarket.26

3. Model analysis

3.1 The decentralized supply chain28

We now provide a model in which the manufacturer maximizes itsprofit by setting the wholesale price and the retailer maximizes its profit by30

ordering the stocking, respectively. The scenario is described as follows.

2See Bose and Ananad (2007).

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1048 S. H. CHANG, A. F. TSAI AND L. M. SHEEN

The expected profit for the retailer is

E(πUr ) = p

∫ QU

0x f (x)dx + pQU

∫ 1

QUf (x)dx2

− s∫ 1

QU(x − QU) f (x)dx − wUQU . (1)

And the manufacturer’s profit is4

πUm = (wU − c)QU . (2)

Since the manufacturer is the leader and the retailer is the follower,6

that is, the manufacturer decides its optimal wholesale price first, andgiven the wholesale price the retailer determines its optimal order quan-8

tity. The results are well known as

the optimal order quantity: QU∗=

(p + s − c)2(p + s)

, (3)10

the optimal wholesale price: wU∗=

12(p + s + c) , (4)

the expected profit for the retailer: E(πUr ) =

(p + s − c)2

8(p + s)− 1

2s , (5)12

the profit for the manufacturer: πUm =

(p + s − c)2

4(p + s). (6)

3.2 The integrated supply chain14

In the integrated supply chain, the manufacturer and the retailer arecombined as a single entity. The objective is to determine the optimal16

inventory level by maximizing the joint profits. Let πJ represent theexpected joint profits of the manufacturer and the retailer, which is18

E(πJ) = E(πr) + πm . Therefore, the expected joint profit is

E(π IJ ) = p

∫ QJ

0x f (x)dx + pQJ

∫ 1

Q j

f (x)dx20

− s∫ 1

QJ

(x − QJ) f (x)dx − cQJ . (7)

Maximizing (7) with respect to QJ to find the first-order condition,22

since f (·) is uniformly distributed on the interval [1, 0] , we can determinethe optimal quantity under the expected joint profit as24

Q∗J =

p + s − cp + s

. (8)

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DECENTRALIZED SUPPLY CHAIN 1049

Q∗J denotes the channel’s optimal inventory level. The profit of a vertically

integrated firm is the maximum attainable in the system.2

Now, we assume that the manufacturer decreases his wholesale pricefrom wU∗

= (p + s + c)/2 to4

wH∗=

12(p + s +αc) , (9)

where α ≤ 1.6

In this case the expected profit for the retailer in this case is

E[πHr (QH)] = p

∫ QH

0x f (x)dx + pQH

∫ 1

QHf (x)dx8

− s∫ 1

QH(x − QH) f (x)dx − wHQH . (10)

Following the line of maximization as in the previous case, the10

optimal order quantity is derived as

QH∗=

p + s − wH

p + s. (11)12

Using Eqs. (9), and from Eqs. (3) and (11), we get

QH∗ − QU∗=

c(1 −α)

2(p + s)≥ 0 .14

That is, QH∗is not smaller than QU∗

if the manufacturer reduces thewholesale price from wU∗

to wH∗. And the manufacturer’s profit in this16

case is

πHm = (WH − c)QH∗

. (12)18

Substituting Eqs. (9) and (11) into Eq. (10), then from Eqs. (5) and (10),we can obtain20

E(πHr )− E(πU

r ) =c(1 −α)

4(p + s)[2(p + s)− c(1 +α)] ≥ 0.

Obviously, because manufacturer did not set any additional condi-22

tion, if trading under this scenario, the expected profit of retailer willnot decrease, but the profit of manufacturer will decreased. Undoubtedly,24

manufacturer is not likely to trade under this scenario.

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1050 S. H. CHANG, A. F. TSAI AND L. M. SHEEN

3. Channel coordination

Subsequently, we deal with the effect of a return policy and quantity2

discount strategy on the manufacturer’s profit and the profit of the wholechannel. Under this situation, the stocking level ordered by the retailer4

from the manufacturer is still Q∗J . But now, to entice the retailer to the

volume, the manufacturer not only employs a quantity discount strategy,6

but also implements a return policy that allows the retailer to return anyunsold goods. Hence, the retailers profit π I

r , the manufacturer’s profit8

π Im , and the joint profit in this model π I

J = π Ir + π I

m , can be expressed,respectively, as10

E(π Ir ) = p

∫ QI

0x f (x)dx + pQI

∫ 1

QIf (x)dx

− s∫ 1

QI(x − QI) f (x)dx12

+ u∫ QI

0(QI − x) f (x)dx − wI QI , (13)

E(π Im) = (wI − c)QI − u

∫ QI

0(QI − x) f (x)dx , (14)14

E(π IJ ) = p

∫ QI

0x f (x)dx + pQI

∫ 1

QIf (x)dx

− s∫ 1

QI(x − QI) f (x)dx − cQI , (15)16

where

QI∗ = Q∗J =

p + s − cp + s

. (16)18

We set

E(π Ir ) = E(πH

r ). (17)20

By means of Eq. (17), and using wH∗, QH∗

and QI∗ defined inEq. (9), (11) and (16), respectively, the wholesale price in the scenario is22

determined as

wI∗ =c

8(θ− 1)[3θ2 + 2αθ− (4 +α2)] +

u(θ− 1)2θ

, (18)24

where θ = (p + s)/c > 1, because of p > c . That is, given the stockinglevel QI∗ and to keep the retailer’s profit equal to E(πH

r ) , the profit26

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DECENTRALIZED SUPPLY CHAIN 1051

for the manufacturer will increase in the case of providing quantitydiscount and returns policy compared to the case in which no policy2

is implemented. Although we know that manufacturer has motive toencourage retailer to participate in this channel coordination by quantity4

discount and return policy, but well still have to induce the most favorablecondition that manufacturer is willing to offer to retailer, i.e., the minimum6

value of α . From E(π Im) ≥ πU

m and by means of Eqs. (4) and (25), andQI∗ = 2QU∗

, we can get8

E(π Im)− πU

m = [2(wI∗ + uQU∗)− (2U∗

+ c)]QU∗ ≥ 0 . (19)

We obtain the following result,10

E(π I Im ) ≥ πU

m , if θ−√

2(θ− 1) ≤ α ≤ 1 . (20)

In this scenario the quantity discount can be expressed as12

∆w = wU∗−wI∗ =c2

8(p+s−c)[(p + s)(θ−α)2 − 4u(θ− 1)2] . (21)

From Eq. (21), we can derive the following condition:14

wU∗> wI∗ , if u ≤ (p + s)(θ−α)2

4(θ− 1)2 . (22)

That is, the quantity discount will be minus when u > (p + s)(θ −α)2/16

4(θ− 1)2 .Pasternack (1985) shows that if the manufacturer allows the retailer18

to make unlimited returns (i.e., all unsold stock can be returned) for fullrefund (i.e., u = wI∗ ), it is impossible to achieve channel coordination.20

But in this paper we demonstrate that as long as wI∗ ≤ (p + s)(θ −α)2/

4(θ − 1)2 , even manufacturer allows retailer to return goods for full22

quantity and full amount, channel coordination still can be achieved.Following the preceding assumption, i.e., let c = 50 , s = 0, p = 100 ,24

the under the circumstance of decentralized supply chain, wU∗= 75 ,

QU∗= 0.25 , E(πU

r ) = 3.125 , πUm = 6.25 and E(πU

J ) = 9.375 . For26

these parameters, let α = 1, if maintenance of positive purchase discountis desired, then return price u must not be larger than 25. Under such28

circumstance, the purchase quantity of retailer QI∗ = 0.5 . If suppose u =

20 , then the optimal wholesale price of manufacturer wI∗ = 73.75 , i.e.,30

∆w = wU∗ − wI∗ = 1.25 > 0, but if u = 25, then the optimal wholesaleprice of manufacturer wI∗ = 75 , i.e., ∆w = wU∗ − wI∗ = 1.25 > 0, but if32

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1052 S. H. CHANG, A. F. TSAI AND L. M. SHEEN

u = 25 , then the optimal wholesale price of manufacturer wI∗ = 75 , i.e.,∆w = wU∗ − wI∗ = 0. If α = 1, then no matter the u value is, we can get2

E(πUr ) = 3.125 , E(πU

m ) = 9.375 , E(πUJ ) = 12.5 . The outcomes mentioned

above show that the equilibrium solution under channel coordination with4

a quantity discount strategy and a returns policy is Pareto-efficient withrespect to the decentralized supply chain. But for the above parameters6

c = 50 , s = 0, p = 100 , if retailer asks for positive purchase discountand also asks for return of good in full quantity and for full amount, then8

channel coordination is not likely to achieve. That is because the wholesaleprice must be greater than unit cost. If unit cost is too great (i.e., c = 50),10

then the balanced wholesale price can not easily satisfy the conditionof wI∗ ≤ (p + s)(θ −α)2/4(θ − 1)2 . Therefore, we can understand the12

reason why Padmanabhan and Png (1997, 2004), and Jeon and Park (2002)suppose the unit cost c is small (or even set c → 0) when discussing return14

policy and return for full amount.

Conclusion16

This paper uses a game of three stages to explore how manufacturerto engage retailer to form a vertical integration unit so as to overcome18

the problem of double marginalization by quantity discount strategyand/or returns policy. In order to achieve this channel coordination,20

the contract between manufacturer and retailer must undoubtedly havePareto efficiency; otherwise, this contract cannot be established. This22

paper demonstrates that channel coordination can not be achieved un-der upstream-downstream integration by the contract with the terms of24

purchase discount and full return. We also find that, if the unit cost ofmanufacturer is small enough, and the manufacturer allows the larger26

unit profit attainable by retailer, then even under the return policy in fullquantity and for full amount, channel coordination can still be achieved.28

References

[1] I. Bose and P. Anand (2007), On returns policies with exogenous30

price, European Journal of Operational Research, Vol. 178, pp. 782–788.

[2] C. C. Hsieh, C. H. Wu and Y. J. Huang (2008), Ordering and32

pricing decisions in a two-echelon supply chain with asymmetric de-mand information, European Journal of Operational Research, Vol. 190,34

pp. 509–525.

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[7] V. Padmanabhan and I. P. L. Png (1997), Manufacturer’s returnpolicies and retail competition, Marketing Science, Vol. 16 (1), pp. 81–14

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Received March, 201026

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