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Physics Procedía 24 (2012) 1500 - 1506

2012 International Conference on Applied Physics and Industrial Engineering

Distribution Free Approach for Coordination of a Supply Chain with Consumer Return

1 • • 2

Jinsong Hu ,Yuanji Xu

'Department of Management Science and Engineering Qingdao University Qingdao, China 2Department of Management Science and Engineering Zhejiang Sci-Tech University Hangzhou, China

Abstract

Consumer return is considered in a coordination of a supply chain consisting of one manufacturer and one retailer. A distribution free approach is employed to deal with a centralized decision model and a decentralized model which are constructed under the situation with only knowing the demand function's mean and variance, respectively. A markdown money contract is designed to coordinate the supply chain, and it is also proved that the contract can make the supply chain perfectly coordinated. Several numerical examples are given at the end of this paper. © 2011 Published by Elsevier B.V Selection and/or peer-review under responsibility of ICAPIE Organization Committee.

Keywords:distribution free, consumer return, markdown money contract, supply chain coordination.

1. Introduction

With the intensification of market competition, retailers provide more after sale services to attract consumers. A returns policy is prevalent in the retail industries, as reported that the value of returned items exceeds $100 billion per year in the United States [1]. But only 5% of the items are returned just because of the imperfect quality [2], the reason for the rest returned items lies in the uncertain evaluation when consumers buy the specific items. Different types of consumer returns policy have been studied in recent years. Moorthy and Srinivasan (1995) [3] pointed out that generous returns policy helps to signal high quality. Hess et al. (1996) [4] showed that inappropriate returns could be controlled in a profitable way by imposing nonrefundable charges and that these charges increased with the value of the merchandise ordered. Heiman et al. (2002) [5] modelled money-back guarantees as an option. Yabalik et al. (2005) [6] developed an integrated approach for analyzing logistics and marketing decisions within the context of designing an optimal returns system for a retailer servicing two distinct market segments. Su (2007) [7] and Aviv and Pazgal (2008) [8] studied dynamic pricing problems where consumers made purchase decisions based on expected future price.

1875-3892 © 2011 Published by Elsevier B.V. Selection and/or peer-review under responsibility of ICAPIE Organization Committee. doi:10.1016/j.phpro.2012.02.222

Our work is closely related to the literature on markdown money contract with supply chain. As pointed out by Cachon (2003) [9] , the name of returns policy or buyback contract is somewhat misleading since it implies physical returns of overstock at the end of the selling season, which only happens when the manufacturer's salvage value is higher than the retailer's. If the retailer has the liquidation advantage, the markdown money contract rather than the buyback policy can coordinate the channel [10]. Xiao et al. (2010) [11] integrated consumer returns policy and manufacturer buyback/markdown money policy.

Foregoing literatures always assume that the market demand is a stochastic with a known distribution. However, in most cases, the distributional information of the demand is very limited. Sometimes, all that is available is an educated guess of the mean and of the variances. There is a tendency to use normal distribution under these conditions. Nevertheless, the normal distribution does not provide the best protection against the occurrence of other distributions with the same mean and the same variance. Scarf (1958) [12] took a conservative approach to model a newsboy problem where only the mean and the variance of the demand are know without any further assumptions about the form of the distribution of the demand. Through a lengthy mathematical argument, he obtained a closed form expression for the optimal order quantity. Gallego and Moon (1993) [13] considerably simplified the proof of Scart's ordering rule and made it easier to understand and easier to remember. In this paper, we use the conservative approach to deal with the outlined above when the distributional information is limited to the mean and the variance.

In this paper, we develop our model on the base of Xiao's one (Xiao et al., 2010) by limiting the demand distribution to the mean and the variance and differing the aptitude for the recovering value from the surplus product between the manufacture and the retailer. The rest of this paper is organized as follows. In the next section, we give relative notation and assumptions. Section 3 introduces the basic model under the markdown money contract. Section 4 studies the coordination mechanism via markdown money. In Section 5 numerical examples are given to illustrate explicitly the effects of the consumer return behavior on the supply chain performance. Finally, the paper concludes in Section 8 with some suggestions for future work in this area.

2. Notation and Assumptions

The following notation is used: c the unit production cost; w(> c) the unit wholesale price; p(> w) the unit retail price;

v the random valuation for the consumer's preference, which has an increasing cumulative distribution

G() over the interval [v, v]; r the refund amount paid to the consumers when there are products returned, and r e [0, p]; X a random demand consisting of a stochastic, E , and non-stochastic, D(r), i.e., x = D(r) + E, where D(r) is an increasing function of the refund amount r, and E is a random variable with only the mean n and the deviation a known. Therefore, the mean and the standard deviation of X denotes D(r) + n and a, respectively; si the salvage value per unit when liquidated by party i where i = M refers to the manufacturer and

i = R refersto the retailer; si the salvage value per unit sold but returned of party i, i e {M, R} ; l, the party is inspection and disposition cost per unit of the returned products , i e {M,R} ; l the total inspection and disposition cost per returned products;

m the manufacturer's markdown money per unit leftover of the retailer; c0 the consumer's returns cost per unit return of consumers; G1 the probability of returning the product for consumers, and G1 = G(r - c0). The mathematical models presented in this study have the following assumptions:

1. The supply chain which we considered is composed of one manufacturer, one retailer and the end consumers.

2. The consumers did not fully know their valuation for the products until they have some relevant experiences.

3. The end consumers could return products if they want.

4. The error s is independent of the consumer's valuation v .

5. The leftovers include the products that were unsold as well as that were sold but returned.

6. The manufacture and the retailer incur the inspection and disposition of the returned products by consumers.

7. The salvage for the unsold products si are greater than si, the residual for the sold but returned ones.

8. There is a relationship used to constrain the parameters: si < c < w. The first inequality is to prevent infinite order by retailer, and the second one is necessary for the manufacturer's participation.

3. The Basic Model

3.1 The retailer's profit model

According to the external demand, the retailer decides a proper order quantity Q to maximize his own profit. Note that min{X, Q} units are sold, where Gj min{X, Q} are sold and kept by consumers and Gj min{X, Q} are sold but returned; Q - min{X, Q}are unsold products quantity. Thereby, the revenue from the products that are sold and kept by consumers is pGj min{X, Q}, the revenue from the products that are unsold but subsidized by the manufacture is (m + s)[Q - min{X, Q}], and the revenue from the products that are bought but returned by consumers is (p -r + m + s -lR)Gjmin{X,Q} . The retailer's procurement cost is wQ . Then his profit function can be expressed as

"R (Q) = pGt min{X, Q} + (m + s) (Q - min{X, Q})

+(p - r + m + s- lR )G1min^_X, Q} - wQ . (1)

Further, we can rewrite Eq. (1) as

nR (Q) = [p - m - s - (r + Ir - m - sOGJminjX ,Q}

-(w - m - s)Q . (2)

Note that min{X,Q} = X - max{X - Q,0} , and insert it into Eq. (2) and after simply geometry operations , then we can get the retailer's expected profit as follows

E[nR(Q)] = [p - m - s - (r + Ir - m - s')Gj(E(X) - E[max{X - Q,0}])

-(w - m - s)Q . (3)

3.2 The manufacturer's profit model

The manufacturer's profit depends on Q , the retailer ordered, and the markdown money m he provides. Then the profit function for the manufacturer can be showed as

*m(Q) = (w - c - m)Q + m(l - G1)mm{X,Q} . (4)

Hence, the expected profit of the manufacture is as follows

E[*M(Q)] = (w - c - m)Q + m(1 - Gi)(E(X) - E[max{X - Q,0}]) . (5)

3.3 The supply chain's profit model

The profit of the supply chain is composed by the retailer's and the manufacturer's ones the channel expected profit's expression as follows

E[nc (Q)] = [p - s - (r + lR - s')Gi ](E(X) - E[max{X - Q, 0}])

-(c - s)Q.

4. Coordination Mechanism via Markdown Money Contract

4.1 The centralized decision model

In this subsection, the optimal order quantity will be considered in a centralized decision model as a benchmark.

Rather than derive the optimality condition of the channel, we first digress by discussing the expected value of max{X - Q,0} . From the notion of X , we know that the mean and the standard deviation of X are D(r) + /u and u, respectively. SoE(X) = D(r) + /u . However, since the distribution of X is unknown, it is difficult for us to get the above expected profits directly. Taking the conservative approach proposed by Gallego and Moon (1993), we estimate E[max{X - Q,0}] against the worst possible distribution as follows

E[max{X - Q,0}] < i [V D(X) + (Q - E(X ))2 - (Q - E(X))] = 2 [ju2 + (Q - D(r) - M)2 - (Q - D(r). (7)

Now we recover to consider the optimality order quantity. Using first and second-order derivative in Eq. (6), it can be easily find E[nc (Q)] is a concave function in Q. By solving the first-order condition dE[nc (Q)]/dQ = 0 for Q, we have the following proposition.

Proposition 1 The channel profit is a concave function in Q, and the equilibrium quantity is

Q'C = D(r) + M + p - c - (lR+ r - sR)Gi - . J^R = .

2 I Vc - sr \jp - c - (Ir + r - sR)Gi )

4.2 The decentralized decision model

The time sequence of this game is as follows: the manufacturer sets a pair of markdown money contract parameters (w, m); and then the retailer reacts to determine order quantity according to the

markdown money policy provided by the manufacturer; finally, the manufacturer decides the optimal contract parameters depending on the retailer's reaction.

Then we have

For the given markdown money contract parameters ( w, m ), the retailer determines his optimal order quantity to get the maximum profit of his own, which means he faces the problem to solve maxE[^ (Q)].

Similar to the method of deriving Proposition 1, we can get the optimal order quantity in the decentralized model, which is presented in Proposition 2.

Proposition 2 The retailer's profit is a concave function in Q, and the optimal order quantity is

Qr = D(r) + M + -

■y/p - w - (r + lR - m - s'R)Oi

-Jw — m — s

w — m — sD

tjp - w - (r + lR - m — sR)G1

4.3 The decentralized decision model with coordination

A coordination mechanism utilized by the manufacturer in a decentralized is to induce the retailer to order the quantity Q*C1, which means the retailer's order quantity in the decentralized model equals to that in the centralized one (i.e. Q'r = Q'C ) . Proposition 3 summarizes the coordination mechanism. Proposition 3 The supply chain can be coordinated by the markdown money contract (w(m), m) with

0 < m < mR , where mR = P Sr—(r+iR—Sr)Gl , and

w(m) =

1 — Gi

C[p — m — SR — (r + lR — m — SR)G1] + m[P — (lR + r + SR — SR)G1]

p - sR - (lR + r - sR )G1

Inserting the expression of w(m) into Eq. (6), we have the following Corollary 1. Corollary 1 When the supply chain is coordinated, we have

EK (Q*)] = -

'SR - (r + lR - m - SR)G1

E[KC2 (QC2 )] .

P - SR - (lR + r - SR )G1

Corollary 1 explicitly means that the retailer's expected profit is a decreasing function of m . Note that E[kc (QC)] = E[kr (QC)] + E[km (Q'c )] and E[kc(QC)] is independent of m . Thus, E[km (QC)] is an increasing function of m . Since the manufacture's expected profit is an increasing function of the markdown money, he will set a unit markdown money as high as possible to maximize his own profit. But there is an upper bound for the markdown money as pointed in Proposition 3. The retailer will make a negative profit when the markdown money exceeds a certain one (i.e. mR ), in that situation, the retailer will refuse the contract.

Corollary 2 The w(m) is an increasing function of m .

c[p - m - Sr - (r + Ir - m - sR)GJ

Proof dw(m) = _d_ dm dm

P - Sr - (IR + r - SRG

m[P — (lR + r + SR — SR )G1]

P — SR — (lR + r — SR )G1

R ) G1

-> 0 .

= P - C - (lR + r - SR)G1 + _

P - Sr - (Ir + r - SrG P - Sr - (Ir + r - Sr)G1

From corollary 2, we find that when the manufacturer offers the retailer a high unit markdown price, the manufacture will charge a high unit wholesale price at first.

5. Numerical Illustration

In this section, we illustrate the effects of some factors on the equilibrium outcome and expected profits of the players. Relative parameters are given as follows:

m = 3.0, c = 2.0, c0 = 2.0, D(r) = 5 + 0.5r , lM = lR = 0.2, p = 6, r = 4, sM = 1.2, s'M = 0.8, sR = 1.0, = 0.6 and v ~ N(6,1), n = 5, a = 1.

From table 1, given the markdown money, we derive following observations. When the unit returns cost (c0) of consumer increases, the optimal quantity of the supply chain increases due to a lower returning probability, and the manufacturer will decrease the unit wholesale price. And both of the manufacturer and the retailer profit from the higher return cost.

TABLE I. The Effects Of Return Cost

c0 w* QC nM nc

0.10 4.4030 12.7399 27.0091 18.2353 45.2444

0.15 4.4027 12.7411 27.0781 18.2546 45.3327

0.20 4.4024 12.7421 27.1400 18.2719 45.4119

0.25 4.4021 12.7431 27.1955 18.2874 45.4829

0.30 4.4018 12.7439 27.2451 18.3013 45.5464

From table 2, with the increase of return price ( r ), the optimal order quantity goes up together with the wholesale price. An interesting thing is observed that the channel profit increases first but decreases when the return price exceeds a certain one.

TABLE II. The Eccect Of The Return Price

r w* QC nM nC

3.00 4.4002 12.2497 26.3792 17.6007 43.9799

3.50 4.4008 12.4983 26.8903 17.9861 44.8764

4.00 4.4024 12.7421 27.1400 18.2719 45.4119

4.50 4.4050 12.9710 26.6609 18.1470 44.8079

5.00 4.4062 13.1631 24.6558 16.8647 41.5205

As a detail discussion about the optimal return price under different return costs, we give table 3 to explicitly illustrate the effects of return cost. The optimal return price adds when the return cost becomes higher.

TABLE III. The Effect of The Return cost On The Optiaml Return Price

C0 r* w* QC nM nc

0.10 3.9787 4.4029 12.7298 27.0149 18.2306 45.2454

0.15 4.0226 4.4028 12.7519 27.0738 18.2603 45.3340

0.20 4.0669 4.4027 12.7740 27.1326 18.2901 45.4227

0.25 4.1111 4.4025 12.7961 27.1914 18.3202 45.5116

0.30 4.1153 4.4024 12.8128 27.2501 18.3504 45.6005

6. Conclusion

In this paper, we study coordination of a supply chain integrating consumer return under a decision environment with little information of demand distribution. Differing from Xiao's model, we limit the demand distribution to the mean and the variance. By the means of distribution free approach, we establish a centralized model and a decentralized one, respectively. We design a mechanism to make the decentralized model perfectly coordinated. At the end of this paper, we provide several numerical illustrations to examine the effects of the consumer return behavior on the supply chain's performance.

Future research can be done in following directions. First, return time can be considered in the supply chain. Second, consumer's valuation can be described as a fuzzy function, which may be more proper.

Acknowledgment

This research was supported by the National Nature Science Foundation of China Grand No.70671056 and No.71071082 and Shandong Province Nature Science Foundation Grand No.Y2008H07.

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