1-s2.0-S0307904X15003832-main

Applied Mathematical Modelling xxx (2015) xxx–xxx

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Coordinating supplier retailer and carrier with price discountpolicy

http://dx.doi.org/10.1016/j.apm.2015.06.0060307-904X/� 2015 Published by Elsevier Inc.

⇑ Corresponding author. Tel.: +86 2363622951, mobile: +86 13527546605.E-mail addresses: [email protected] (L. Li), [email protected] (Y. Wang), [email protected] (W. Dai).

Please cite this article in press as: L. Li et al., Coordinating supplier retailer and carrier with price discount policy, Appl. Math.(2015), http://dx.doi.org/10.1016/j.apm.2015.06.006

Liying Li a, Yong Wang b,⇑, Wei Dai c

a Science College, Chongqing Jiaotong University, Chongqing, Chinab College of Economics and Business Administration, Chongqing University, Chongqing, Chinac School of Management and Information Systems, Victoria University, Melbourne, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 31 October 2012Received in revised form 30 April 2015Accepted 15 June 2015Available online xxxx

Keywords:CarrierPrice discountLead timeSupply chain coordination

Supply chain coordination with transport service providers (i.e., carriers) is seldomexplored. This paper adds a carrier to a supplier–retailer system and analyzes the effectof the two sources of double marginalization on pricing policies. We assume that lead timedemand is stochastic, and shortage during the lead time is permitted. In addition, weassume that the annual average demand rate is a decreasing function of the retail price.Our analysis shows that joint profit increases because demand increases, whereas unitoperating cost decreases as a result of joint coordination. The analytical results for thedecentralized and joint decision models are obtained for a specific annual average demandrate. Nonlinear transport-fee and wholesale-price discount schemes, which can facilitatesupply chain coordination, are then obtained using the profit sharing method. Finally,numerical examples are presented for illustrative and comparative purposes.

� 2015 Published by Elsevier Inc.

1. Introduction

Coordination among disparate partners within the supply chain and e-supply chain is widely discussed, whereas supplychain coordination with transport service providers is seldom explored. In this paper, we add a carrier to a supplier–retailersystem and analyze the effect of the two sources of double marginalization on pricing policies, which are profit margins ofthe retailer and carrier.

Most of the previous literature considered transport issues in supply chain decision took the assumption that the shippingcosts were exogenously given. Thus, the effect of carrier profit margins on supply chain pricing and inventory decisions is notadequately reflected. In this paper, the carrier’s shipping costs are assumed to be endogenous in supply chain decisions. Weassume that lead time demand is stochastic, and shortage during the lead time is permitted. In addition, we assume that theannual average demand rate is a decreasing function of the retail price. This paper primarily aims to analyze the effect of dou-ble marginalization on pricing policies and to identify an incentive scheme to coordinate the supply chain including a carrier.

The two research areas related to this research are: (1) lot-size coordination for product supply chains and (2) supplychain coordination incorporating transportation issues in decisions. There are available reviews in the literature on lot-sizecoordination in supply chains (see, e.g., reference lists in [1–3]). The body of the literature on coordinating order quantitiesbetween entities in a supply chain focused on a two-level supply chain for different assumptions. One of the first lot-sizemodels dealing with buyer–vendor coordination was proposed by Goyal [4], who analyzed a system consisting of a single

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2 L. Li et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

vendor and a single buyer. He implicitly assumed that the vendor provides an infinite replenishment rate, and that the pro-duction lot is transferred to the buyer in equal-sized shipments. The Goyal’s problem is referred to in the literature as theJoint Economic Lot-Sizing (JELS) problems. Among the works on the JELS problems, and of relevance to this paper, are thosewho considered all-unit discounts with joint buyer–seller perspective (see, e.g., reference lists in [5–7]). Other researchers(see, e.g., reference lists in [4,8,9]) investigated the JELS problems with quantity discounts by maximizing the profits of boththe seller-buyer with a profit sharing mechanism. Li and Huang [10] explored the cooperative JELS problems with a quantitydiscount scheme and profit sharing mechanism. They suggested that the party that has a more powerful bargaining positionwould gain a higher fraction of the profits.

Few works have investigated the JELS problems in a three-level supply chain. Munson and Rosenblatt [11] are believed tobe the first to investigate a three-level supply chain that consists of a supplier, a manufacturer and a retailer. In their model,the total demand was assumed to be constant and deterministic. Also, the manufacturer was assumed to be the most influ-ential channel player who is able to obtain a quantity discount from the supplier and pass some or all of this discount quan-tity to the retailer. The work of Munson and Rosenblatt [11] was extended by Khouja [12], who considered a systemconsisting of a supplier, multiple manufacturers and multiple buyers. He dealt with three inventory coordination mecha-nisms between chain members and developed closed-form expressions or simple algorithms for solving each of the coordi-nation mechanism models. Lee and Moon [13] developed inventory models for a three level supply chain with one supplier,one warehouse, and one retailer. They considered three types of individual models (independent model, retailer’s point ofview model, and supplier’s point of view model) and applied the compensation policy for the benefits and losses to the coor-dinated inventory model. Their study was based on the assumption of constant and deterministic demand. Jaber et al. [14]studied the coordination of a three-level (supplier–manufacturer–retailer) supply chain, where the retailer was faced withprice dependent demand. In their model, an all-unit price discounts scheme was used to coordinate the order quantitiesamong the supply chain levels, and a profit sharing mechanism is used to maximize the supply chain profit. In theabove-mentioned study, they assumed no shortages to occur and zero lead time.

Transport is a significant component of supply chain operations. Depending on the estimates used, upwards of 50% of thetotal annual logistics cost of a product can be attributed to transport. Therefore, any discussion on purchase quantities shouldconsider transport costs [15]. There is substantial literature on incorporating transport costs in the JELS problems. Kim and Ha[16] developed a buyer–supplier coordination model to determine optimal order quantity and the number of deliveries in asimple JIT scenario, assuming that transport cost is fixed per delivery regardless of shipping weight. Ertogral et al. [17] explic-itly incorporated the transport cost into the vendor–buyer lot-sizing problem and developed optimal solution procedures forsolving the integrated models. They considered all-unit-discount transport cost structures with and without over-declaration.In their models, the total demand and production rates are assumed to be constant and deterministic. Sajadieh et al. [18]developed an integrated production inventory marketing model to determine the relevant profit-maximizing decision vari-able values when demand is price sensitive. Lee and Wang [19] addressed an integrated three-level JELS problem with freightrate discounts. Madadi et al. [20] addressed specific inventory management decisions with transportation cost considerationin a multi-level environment consisting of one supplier, one warehouse and several retailers.

In all the above mentioned models with transport costs consideration, exogenously given shipping costs are used.However, the carrier’s decision-making has obvious effect on supply chain performance. Lei et al. [21] studied the optimalbusiness policies for a supply chain involving a third party transport partner, a supplier and a buyer who faces a price-sensitive demand. They assumed that the transporter charges supplier a shipping rate, be responsible for transporting theproduct from the supplier to the buyer. In their model, discount schemes are not provided to coordinate the order quantitiesand prices among the supply chain levels. Furthermore, they also assumed no shortages to occur and zero lead time. Unlikethe work of Lei et al. [21], this paper assumes all-unit price discounts, profit sharing mechanism, shortages to occur andstochastic demand during the lead time. This paper investigates the channel coordination issue of a supply chain withone supplier, one retailer, and one carrier. We assume that lead time demand is stochastic, and shortage during the lead timeis permitted. Furthermore, we assume that the annual average demand rate is a decreasing function of the retail price.Similar to the work of Jaber et al. [14], this paper uses an all-unit price discounts scheme to coordinate the supply chain.Price discounts are often suggested as incentives to facilitate coordination (or collaboration) owing to the negative demandeffect of double marginalization on supply chain management (see, e.g., reference lists in [22–24]). For a specific annualaverage demand rate, the present study demonstrates the effectiveness of transport-fee and wholesale-price discountschemes as a control mechanism in joint coordination among the carrier, supplier, and retailer.

This paper is organized as follows. After constructing the decentralized and joint decision models, we characterize theform of the optimal joint policy and discuss the impact of joint policy on the system’s expected profit in Section 2.Section 3 illustrates the optimal policy for a specific annual average demand rate (DðpÞ ¼ a � p�2). We develop a coordinationmechanism of the supply chain in Section 4. Section 5 presents the numerical examples for illustrative and comparative pur-poses. Section 6 concludes the paper.

2. Model development

We consider a multi-period supply chain with one supplier, one retailer, and one carrier. The retailer purchases an itemfrom the supplier, while the supplier orders integer multiple of the retailer’s order quantity of items from his/her supplier.The carrier is responsible for transporting this item from the supplier to the retailer.

Please cite this article in press as: L. Li et al., Coordinating supplier retailer and carrier with price discount policy, Appl. Math. Modell.(2015), http://dx.doi.org/10.1016/j.apm.2015.06.006


L. Li et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx 3

2.1. Model assumptions and formulation

To develop the proposed model, the following notations are adopted:

Pl(2

p

ease cite th015), http:/

unit retail price charged by the retailer
DðpÞ annual average demand rate faced by the retailer, which is a decreasing function of retail price p ~KS fixed (setup) cost per order placed by the supplier KR fixed cost per order incurred by the retailer KL fixed cost incurred by the carrier for each delivery from the supplier to the retailer ~hS
annual holding cost per unit of inventory at the supplier

hR
annual holding cost per unit of inventory at the retailer, hR >~hS
L
length of lead time m an integer representing the number of lots in which the items are delivered from the supply to the retailer AR unit shortage cost for the retailer k retailer’s safety factor for the stock rL standard deviation of forecast error over a replenishment lead time r standard deviation of forecast error of the annual demand c cost per unit ordered by the supplier t carrier’s transport cost per unit shipped from the supplier to the retailer w unit wholesale price charged by the supplier v unit transport fee charged by the carrier Q order size of the retailer k1 supplier’s sharing ratio of transport cost k2 retailer’s sharing ratio of transport cost, k2 ¼ 1� k1
The assumptions made in this paper are as follows.

(1) The carrier’s operation cost includes a fixed cost per shipment (e.g., value-added services per order processed, insur-ance per trip, and truck driver’s cost, etc) and a unit shipping variable cost (e.g., mileage cost and truck usage, etc)(similar assumption has been used by Lei et al. [21]).

(2) The retailer’s order size Q is less than the vehicle capacity, and so that each lot of size will be carried in a single vehicletrip.

(3) The shipping costs are shared by the supplier and retailer, with the sharing ratios set by the negotiation between thetwo parts. Moreover, the carrier charges the supplier and retailer a per unit shipping rate for each unit of itemtransported.

(4) The holding costs are charged per unit of inventory.(5) The annual average demand is dependent on the retail price but the standard deviation of forecast error of the annual

demand is not. Also, the demand X during the lead time L follows a normal distribution with mean DðpÞL and standarddeviation rL ¼ r

ffiffiffiLp

.(6) The ordering strategy of the retailer is ðs;QÞ, and the lead time L is a constant.(7) A stock out may occur during the replenishment lead time, and penalty cost for a lost sale occurs at the retailer.

Given that the retailer makes order decisions in terms of ðs;QÞ, and the predetermined safety factor is k, such that thesafety stock in the lead time L is krL. The expected yearly profit of the retailer is equal to the gross revenue minus the order-ing cost, shortage cost, and inventory holding cost:
Y
Rðp;QÞ ¼ ðp�w� k2vÞDðpÞ � ½KR þ ARrLguðkÞ�DðpÞ=Q � ðQ=2þ krLÞhR; ð1Þ

where

guðkÞ ¼1ffiffiffiffiffiffiffi2pp

Z 1

kðu� kÞ expð�u2=2Þdu: ð2Þ

Since the supplier’s order lot size is an integer multiple of the retailer’s order size Q , that is, mQ (m ¼ 1;2; . . .). It can be

shown that the supplier’s inventory holding cost is: hSQ=2, where hS ¼ ðm� 1Þ~hS [25,9]. It can also be shown that the sup-plier’s ordering cost is: KS � DðpÞ=Q , where KS ¼ ~KS=m. The supplier’s expected yearly profit is equal to the gross revenueminus the fixed ordering and inventory holding cost:
Y
SðwÞ ¼ ðw� c � k1vÞDðpÞ � KSDðpÞ=Q � hSQ=2: ð3Þ

is article in press as: L. Li et al., Coordinating supplier retailer and carrier with price discount policy, Appl. Math. Modell./dx.doi.org/10.1016/j.apm.2015.06.006



The carrier’s expected yearly profit is equal to the gross revenue minus the transport cost. Therefore, the carrier’sexpected yearly profit function is given by

Please(2015

YLðvÞ ¼ ðv � tÞDðpÞ � KLDðpÞ=Q : ð4Þ

2.2. The decentralized system

In the decentralized system, channel members (i.e., the carrier, supplier, or retailer) maximize their profit by optimizingthe decision variables within their control. The resulting maximum profits present lower bounds on the expected yearlyprofits if joint coordination prevails.

For any retailer’s selling price p, which determines the annual average demand rate DðpÞ, the retailer’s optimal order sizeis the EOQ order quantity, i.e.,

QR ¼ ½2DðpÞðKR þ ARrLguðkÞÞ=hR�1=2: ð5Þ

The resulting the ordering, inventory holding, and shortage cost is krLhR þ ½2hRDðpÞðKR þ ARrLguðkÞÞ�1=2. Therefore, with

the EOQ order quantity Q R, the retailer’s expected yearly profit function given by (1) can be written as
YRðp;Q RÞ ¼ ðp�w� k2vÞDðpÞ � krLhR � ½2hRDðpÞðKR þ ARrLguðkÞÞ�
1=2: ð6Þ

For any given unit transport fee v and unit wholesale price w, the retailer’s objective is to select a unit selling pricep ¼ pðw;vÞ that maximizes his/her expected yearly profit.

With the retailer’s unit selling price p ¼ pðw;vÞ and the EOQ order quantity QR, the supplier’s expected yearly profit func-tion given by (3) can be written as
Y
Sðw;mÞ ¼ ðw� c � k1vÞDðpÞ � f~KS=½mðKR þ ARrLguðkÞÞ� þ ðm� 1Þ~hS=hRg � ½hRDðpÞðKR þ ARrLguðkÞÞ=2�1=2

: ð7Þ

For any given unit transport fee v , the supplier’s objective is to select a unit wholesale price w ¼ wðvÞ and a lot size mul-tiplier m, which maximize his/her yearly profit.

For a fixed value of w, the optimal value of the supplier’s lot size multiplier, denoted by m�, can be obtained by maximiz-ing the supplier’s expected yearly profit.

Proposition 1. For a fixed value of w, the optimal lot size multiplier m� in the decentralized system must satisfy

m�ðm� � 1Þ 6 ð~KShRÞ=½ðKR þ ARrLguðkÞÞ~hS� 6 m�ðm� þ 1Þ: ð8Þ

Proof. See Appendix A.

Defining bxc as the nearest integer smaller than x, m� can be shown to be equal to [11]:

m� ¼ 12

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ð4hR

~KSÞ=½~hSðKR þ ARrLguðkÞÞ�q� ��

: ð9Þ

Finally, with the retailer’s unit retail price p ¼ pðwðvÞ;vÞ and the EOQ order quantity Q R, the carrier’s objective is to selecta unit transport fee v to maximize his/her expected yearly profit:
Y
LðvÞ ¼ ðv � tÞDðpÞ � ½KL=ðKR þ ARrLguðkÞÞ� � ½hRDðpÞðKR þ ARrLguðkÞÞ=2�1=2

: � ð10Þ

Proposition 2. With the retailer’s EOQ order quantity QR, the supplier’s expected yearly profit is never higher than the max-

imum, which can be achieved by the order quantity of the retailer: �QR ¼ ½2KSDðpÞ=hS�1=2, minimizing the supplier’s fixedordering and holding cost. The carrier’s expected yearly profit is also never higher than the maximum, which can be achieved

by the order quantity of the retailer: Q̂R ¼ DðpÞ, minimizing the carrier’s fixed transport cost.

Proof. See Appendix A.The above Proposition 2 shows the effect of the retailer’s order quantity on the expected yearly profits of the supplier and

the carrier. h

Property 1. For a fixed value of p, Q R decreases and m� increases as k increases.

cite this article in press as: L. Li et al., Coordinating supplier retailer and carrier with price discount policy, Appl. Math. Modell.), http://dx.doi.org/10.1016/j.apm.2015.06.006



Proof. See Appendix A.Property 1 shows that as the retailer’s safety factor for the stock increases, the retailer’s EOQ order quantity decreases, but

the supplier’s optimal lot size multiplier m� increases.

Property 2.

For a fixed value of p, QR increases and m� decreases as rL increases.

According to (5) and (9), Property 2 can be easily verified. Thus, the proof of Property 2 is omitted for simplicity. h

Property 2 shows that as the standard deviation of forecast error increases, the retailer’s EOQ order quantity increases, butthe supplier’s optimal lot size multiplier m� decreases.

Let v� denote the carrier’s unit transport fee that maximizesQ

LðvÞ as given in (10), and letQ�

L denote the correspondingexpected yearly profit. Accordingly, the supplier’s optimal wholesale price and maximized expected yearly profit are denotedby w� and

Q�S, respectively. The retailer’s optimal unit retail price and maximized expected yearly profit are denoted by p�

andQ�

R, respectively. We defineQ�

L þQ�

S þQ�

R

� �as the expected yearly profit of the decentralized system. In the following

section, we show the specific forms of the above results for a given annual average demand function.

2.3. The joint system

Now we analyze the joint policy if all channel members are willing to cooperate to maximize the joint expected profit ofthe whole supply chain. The joint expected profit is defined as the sum of the three parties’ expected profits:

Please(2015

YJðp;Q ; L;mÞ ¼

YLðvÞ þ

YSðw;mÞ þ

YRðp;Q ; LÞ ¼ ðp� t � cÞDðpÞ � KJDðpÞ=Q � hJQ=2� krLhR; ð11Þ

where

KJ ¼ KL þ ~KS=mþ KR þ ARrLguðkÞ; and hJ ¼ ðm� 1Þ~hS þ hR: ð12Þ

For fixed values of p and m, the joint ordering, shortage, and inventory holding cost is minimized by the joint EOQ orderquantity:

Q JðpÞ ¼ ½2KJDðpÞ=hJ�1=2: ð13Þ

Using the joint EOQ order quantity, we can rewrite (11) as
YJðp;Q JðpÞ; L;mÞ ¼ ðp� t � cÞDðpÞ � ½2KJhJDðpÞ�1=2 � krLhR: ð14Þ
For a fixed value of p, the optimal value of the lot size multiplier in the joint system, denoted by m��, is obtained by max-imizing the joint expected profit.

Proposition 3. For a fixed value of p, the optimal lot size multiplier in the joint system m�� must satisfy

m��ðm�� 1Þ 6 ~KSðhR � ~hSÞ=½~hSðKL þ KR þ ARrLguðkÞÞ� 6 m��ðm�� þ 1Þ: ð15Þ

The proof of Proposition 3 is similar to the proof of Proposition 1. Hence, the proof of Proposition 3 is omitted for simplicity.Further, the optimal lot size multiplier in the joint system can be written as:

m�� ¼ 12

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4~KSðhR � ~hSÞ=½~hSðKL þ KR þ ARrLguðkÞÞ�

q� �� : ð16Þ

Similarly, from (12), (13) and (16), we can easily verify that for a fixed value of p, the joint EOQ order quantity QJðpÞ decreasesas k increases, and increases as rL increases. In addition, the optimal lot size multiplier m�� in the joint system increases as kincreases, and decreases as rL increases.

Next, we examine the effect of using the joint EOQ order quantity, rather than the retailer’s EOQ order quantity, on the jointexpected profit.

Proposition 4. For the retailer’s unit retail price (p�), the joint expected yearly profit with the joint EOQ order quantity Q Jðp�Þ isQJðp�;QJðp�ÞÞ. The supply-chain-wide expected yearly profit with the retailer’s EOQ order quantity Q Rðp�Þ is

Q�L þ

Q�S þ

Q�R: We

have
YJðp�;QJðp�ÞÞP
Y�

LþY�

SþY�

R:

Proof. See Appendix A.




Proposition 4 shows that if the only joint effort is to use the joint EOQ order quantity, the joint expected yearly profit willnot be less than the expected yearly profit of the decentralized system. h

2.4. Impact of joint policy on system profit

Here, we characterize how a joint policy affects the system’s expected yearly profits. All channel members will becomewilling to accept a joint policy only if their profits are not decreased.

For a given joint policy ðp;QJðpÞÞ, the carrier’s expected yearly profit is

Please(2015

YJ

Lðv ;Q JðpÞÞ ¼ ðv � tÞDðpÞ � KLDðpÞ=Q JðpÞ:

Hence, the joint policy ðp;Q JðpÞÞ will appeal to the carrier only if the carrier can charge a unit transport fee v to the sup-plier and the retailer, leading to a expected yearly profit higher than the carrier’s expected yearly profit given by

Q�L . Letting

vmin denote the carrier’s smallest unit transport fee that satisfiesQJ

Lðv ;Q JðpÞÞPQ�

L and performing some algebra, we have

vminðpÞ ¼ t þY�

L=DðpÞ þ ðKL=KJÞ � ½hJKJ=2DðpÞ�1=2

: ð17Þ

Likewise, the supplier and the retailer will be interested in accepting the joint policy ðp;Q JðpÞÞ only when the unit trans-port fee v charged by the carrier leads to expected yearly profits higher than the expected yearly profits of the supplier andthe retailer given by

Q�S and

Q�R, respectively. That is,
YJ
Sðw;QJðpÞÞ ¼ ðw� c � k1vÞDðpÞ � KSDðpÞ=Q JðpÞ � hSQJðpÞ=2 P

Y�

S; ð18Þ

PJRðp;Q jðpÞÞ ¼ ðp�w� k2vÞDðpÞ � ½KR þ ARrLguðkÞ�DðpÞ=Q JðpÞ � ½Q JðpÞ=2þ krL�hR P

Y�

R: ð19Þ

Letting vmax denote the carrier’s largest unit transport fee that satisfies (18) and (19), we obtain

vmaxðpÞ ¼ p� c �Y�

RþY�

S

.DðpÞ � krLhR=DðpÞ � ½ðKS þ KR þ ARrLguðkÞÞ=KJ þ 1� � ½hJKJ=2DðpÞ�1=2

: ð20Þ

We can establish the relationship between the joint expected yearly profit with the coordination policy and the expectedyearly profit of the decentralized system. Applying (17) and (20), we can rewrite (11) as
Y
Jðp;Q JðpÞÞ ¼

Y�

LþY�

SþY�

Rþ ½vmaxðpÞ � vminðpÞ� � DðpÞ: ð21Þ

According to Proposition 4, the joint expected profit will increase if the joint unit retail price p satisfies p 6 p�, where p� isthe retailer’s unit retail price that maximizes the retailer’s expected yearly profit. The above result indicates that, with jointcoordination, if the joint unit retail price is chosen such that ½vmaxðpÞ � vminðpÞ� > 0, then ½vmaxðpÞ � vminðpÞ� represents theincreased unit profit due to employing the joint EOQ order quantity rather than the retailer’s EOQ order quantity.Simultaneously, the joint unit retail price p (p 6 p�) also leads to an increase in the annual average demand rate fromDðp�Þ to DðpÞ. Therefore, the term ½vmaxðpÞ � vminðpÞ� � DðpÞ reflects the impact of the joint policy on the system’s expectedyearly profit.

3. Decision analysis for a specific demand function

In this section, we apply the general results developed in the previous sections to a specific function for annual averagedemand. Here, we use the well-known constant price-sensitive demand function given by DðpÞ ¼ a � p�2l; ða > 0;l P 1Þ,where a is the scaling constant, and l is the constant price elasticity [26].

First, we develop the maximum profits individually achieved by the three parties and their optimal decision in the decen-tralized system.

Result 1. For DðpÞ ¼ a � p�2, in the decentralized system, the optimal unit transport fee and the carrier’s maximum expectedyearly profit are

v� ¼ ð2t þ cÞa1=2 � ½ð2t þ cÞM � 2cKL=P� � ð2hRPÞ1=2

a1=2 � ðM þ 2KL=PÞ � ð2hRPÞ1=2 ;

Y�

L¼ 1

64ðt þ cÞ ½a1=2 � ðM þ 2KL=PÞ � ð2hRPÞ1=2�

2;

where

P ¼ KR þ ARrLguðkÞ;M ¼ 1þ KS=P þ hS=hR:




The supplier’s optimal unit wholesale price and maximum expected yearly profit are

Please(2015

w� ¼ ½ð4� 2k2Þt þ ð4� k2Þc�a1=2 þ ½k2ð2t þ cÞM � 2k2cKL=P � 4ðt þ cÞ� � ð2hRPÞ1=2


Y�

S¼ 1

32ðt þ cÞ ½a1=2 �M � ð2hRPÞ1=2� � ½a1=2 � ðM þ 2KL=PÞ � ð2hRPÞ1=2�:

The retailer’s optimal unit retail price, optimal order quantity, and maximum expected yearly profit are

p� ¼ 8ðt þ cÞa1=2


Q � ¼ 14ðt þ cÞ ðP=2hRÞ1=2 � ½a1=2 � ðM þ 2KL=PÞ � ð2hRPÞ1=2�;

Y�

R¼ 1

16ðt þ cÞ ½a1=2 � ð2hRPÞ1=2� � ½a1=2 � ðM þ 2KL=PÞ � ð2hRPÞ1=2� � krLhR:

Note that the expected yearly profits of all channel members are not related to the sharing ratios of the transport costbetween the supplier and the retailer. The above formulae hold if

a1=2 > ðM þ 2KL=PÞ � ð2hRPÞ1=2:

Therefore,

@w�

@k2¼ �ð2t þ cÞ½a1=2 � ðM � 2cKL=PÞ � ð2hRPÞ1=2�

a1=2 � ðM þ 2KL=PÞ � ð2hRPÞ1=2 < 0:

It is shown that the supplier’s optimal unit wholesale price is a decreasing function of the retailer’s transport cost-sharingratio k2 [i.e., the supplier’s optimal unit wholesale price is an increasing function of the supplier’s transport cost-sharing ratiok1]. That is to say, the supplier expects to charge a markup if he/she incurs any transport cost.

Next, we provide the results when the joint policy is used in the joint system.

Result 2. For DðpÞ ¼ a � p�2 and with joint policy, the optimal joint unit retail price, optimal joint order quantity, andmaximum joint expected yearly profit are

p�� ¼ 2ðt þ cÞa1=2

a1=2 � ð2KJhJÞ1=2 ;

Q Jðp��Þ ¼1

t þ cðKJ=2hJÞ1=2 � ½a1=2 � ð2KJhJÞ1=2�;

YJðp��;Q Jðp��ÞÞ ¼

14ðt þ cÞ ½a

1=2 � ð2KJhJÞ1=2�2� krLhR:

The following relationships can be easily verified:

p�� < p�; Q Jðp��Þ > Qðp�Þ;

By Proposition 4 and Result 2, we obtain
YJðp��;Q Jðp��ÞÞ >
Y�

LþY�

SþY�

R:

The above example indicates that the optimal joint retail price with joint policy is less than the retailer’s optimal sellingprice; the optimal joint order quantity is greater than the retailer’s optimal order quantity; and the maximum joint yearlyprofit is greater than the maximum channel yearly profit in the decentralized system. Therefore, the joint policy results ingreater channel performance than the decentralized decision-making.




4. Coordination via transport-fee and wholesale-price discount schemes

Next, we design a method to divide the increased profit (½vmaxðpÞ � vminðpÞ� � DðpÞ), and develop a suitable incentivescheme to achieve supply chain coordination. According to Section 2.4, if the joint unit retail price p (p 6 p�) is chosen suchthat ½vmaxðpÞ � vminðpÞ� > 0, the joint expected profits increase. The retailer’s use of the joint policy will lead to a decrease inits own profit; therefore, its participation is subject to compensation for the decreased profit. In this section, price discountschemes for compensating for the decreased profit to the retailer are developed.

Depend on the market power of the players, if b1 percentage of the increased expected profit goes to the carrier, b2 per-centage of the increased expected profit goes to the supplier, and the other b3 ¼ 1� b1 � b2 percentage of the increasedexpected profit goes to the retailer. In this profit sharing arrangement, the coordinating unit transport fee charged by thecarrier, denoted as v J , and the coordinating unit wholesale price charged by the supplier, denoted as wJ , can be obtainedby the following formulae:

Please(2015

YJ

L¼Y�

Lþ b1 � ½vmaxðpÞ � vminðpÞ� � DðpÞ; ð22Þ

YJ

S¼Y�

Sþ b2 � ½vmaxðpÞ � vminðpÞ� � DðpÞ: ð23Þ

By applying some algebra to (22) and (23), we obtain

v J ¼ t þ b1½vmaxðpÞ � vminðpÞ� þ KL=Q JðpÞ þY�

L=DðpÞ; ð24Þ

wJ ¼ c þ k1t þ ðk1b1 þ b2Þ½vmaxðpÞ � vminðpÞ� þ ðk1KL þ KSÞ=QJðpÞ þ k1

Y�

LþY�

S

.DðpÞ þ hS½KJ=ð2hJDðpÞÞ�1=2

: ð25Þ

From (24) and (25), we can see that both the coordinating unit transport fee v J and the coordinating unit wholesale pricewJ are decreasing functions of the retailer’s order quantity and the annual sales volume. In the price discount scheme (25),the first component of the coordinating wholesale price offers a constant discount related to the transport cost-sharing ratiosand additional expected profits dividing ratios. The second component of the coordinating wholesale price offers a directincentive for the retailer to increase his/her order quantity. The last component of the coordinating wholesale price offersa direct incentive to increase the annual sales volume. The same construct can be applied to the coordinating transport fee.

Next, we examine the effectiveness of the transport-fee and wholesale-price discount schemes.

Proposition 5. For DðpÞ ¼ a � p�2; given that the retailer employs the joint EOQ order quantity QJðpÞ, the price discountschemes (24) and (25) generate a perfect coordination mechanism.

Proof. See Appendix A.In this price discount schemes, if the retailer is committed to the joint unit retail price and order quantity, then the carrier

charges the coordinating unit transport fee in (24), and the supplier charges the coordinating unit wholesale price in (25).

From (25), with the joint policy, given that the joint unit retail price p (p 6 p�) is chosen such that ½vmaxðpÞ � vminðpÞ� > 0,the following is easy to obtain:

@wJ

@k1¼ t þ b1½vmaxðpÞ � vminðpÞ� þ KL=Q JðpÞ þ

Y�

L=DðpÞ > 0:

This shows that, similar to the wholesale price in the decentralized system, the coordinating wholesale price wJ increasesas the supplier’s transport cost-sharing ratio k1 increases. That is to say, the transport costs borne by the supplier are passedon to the retailer under the price discount schemes. h

5. Numerical examples

In order to illustrate the model and properties provided in this paper, we implement a numerical study. First, we assumethat the annual average demand rate is given by DðpÞ ¼ a � p�2. The base parameters of the model are listed below:a ¼ 4� 106; t ¼ 5; c ¼ 10; KL ¼ 40; ~KS ¼ 100; KR ¼ 20; ~HS ¼ 15%; HR ¼ 30%ð~HS and HR represent the annual holding cost

percentage for the supplier and the retailer, respectively, thus ~hS ¼ ~HSc and hR ¼ HRc); Ar ¼ 1; rL ¼ 10; and k ¼ 1. The othereighteen examples are obtained by modifying the base case one parameter at a time using the following values: a ¼ 2� 106

and 6� 106;t ¼ 2 and10; c ¼ 5 and 20; KL ¼ 20 and 80; ~KS ¼ 50 and 200; KR ¼ 10 and50; ~HS ¼ 25%; HR ¼ 40%; Ar ¼ 0:5and 2; and rL ¼ 5 and 15.

The results are summarized in Table 1, whereQ

0 and Q S0 represent the decentralized system’s profit and the supplier’s lotsize, respectively;

Q1 and QS1 represent the system’s profit and the supplier’s lot size in joint coordination among the sup-

plier and retailer only, respectively;Q

2 and QS2 represent the system’s profit and the supplier’s lot size in joint coordination




including the carrier, respectively [We present the results for seven of these examples in Appendix B (see SupplementaryTables 5 and 6)]. Based on the experiments it can be found that, on average, including the carrier to the supplier-retailerchannel coordination problem can significantly increase the system’s profit and the supplier’s lot size versus only coordinat-ing through the retailer and the supplier. This is due to the negative demand effect of the two sources of double marginal-ization on three-echelon supply chain management.

Table 2 shows the effect of the three parties’ bargaining power on the allocation of the joint profit. From Table 2, it can beseen that the coordinating unit transport fee increases as the carrier’s bargaining power increases; the coordinating unitwholesale price decreases as the supplier’s bargaining power decreases. Additionally, the carrier (supplier, retailer) receivesmore extra profits when his/her bargaining power increases.

Table 1Summarized averages from the computational study.

Line Output measure Mean Standard deviation

1Q

0 27613.95 6696.582

Q1 48463.13 11671.46

3Q

2 64899.27 15605.204 QS0 177.05 42.845 QS1 367.42 84.696 QS2 719.53 172.227 ð

Q1 �

Q0Þ=Q

0 75.58% 0.99%8 ð

Q2 �

Q0Þ=Q

0 135.15% 1.68%9 ðQS1 � QS0Þ=QS0 108.98% 16.77%10 ðQS2 � QS0Þ=QS0 309.60% 51.44%

Table 2The allocation results of the joint profit.

ðb1; b2; b3Þ p�� QJ v J wJQJ

L

QJL

QJL

Q2

(0.20,0.50,0.30) 30.47 679.60 7.67 20.11 11242.14 26363.17 26976.48 64581.79(0.30,0.40,0.30) 30.47 679.60 8.53 19.68 14949.57 22655.74 26976.48 64581.79(0.20,0.40,0.40) 30.47 679.60 7.67 19.24 11242.14 22655.74 30683.91 64581.79

Table 3Effect of safety factor on lot size multipliers, order quantities, and expected profits of the whole supply chain in the decentralized and joint systems.

k m�� m� QJ QRQ

2Q

0 ðQ

2 �Q

0Þ=Q

0 (%)

0.0 1 3 686.13 63.96 64591.88 27570.33 134.280.2 1 3 684.23 62.70 64591.67 27556.02 134.400.4 1 3 682.65 61.64 64590.48 27542.37 134.510.6 1 3 681.37 60.76 64588.39 27529.64 134.610.8 1 3 680.36 60.06 64585.45 27518.00 134.701.0 1 3 679.60 59.53 64581.79 27507.48 134.781.2 1 3 679.03 59.13 64577.52 27498.01 134.841.4 1 3 678.63 58.84 64572.25 27489.46 134.901.6 1 3 678.35 58.65 64567.60 27481.67 134.951.8 1 3 678.12 58.52 64562.17 27474.46 134.992.0 1 3 678.04 58.43 64556.54 27467.65 135.03

Table 4Effect of standard deviation on lot size multipliers, order quantities, and expected profits of the whole supply chain in the decentralized and joint systems.

rL m�� m� QJ QRQ

2Q

0 ðQ

2 �Q

0Þ=Q

0 (%)

2 1 3 678.21 58.55 64610.02 27522.78 134.754 1 3 678.56 58.80 64602.96 27519.02 134.766 1 3 678.90 59.04 64595.90 27515.22 134.768 1 3 679.25 59.29 64588.85 27511.37 134.7710 1 3 679.60 59.53 64581.79 27507.48 134.7812 1 3 679.94 59.77 64574.74 27503.55 134.7914 1 3 680.29 60.01 64567.68 27499.57 134.8016 1 3 680.64 60.25 64560.63 27495.55 134.8018 1 3 680.98 60.49 64553.57 27491.50 134.8120 1 3 681.33 60.73 64546.52 27487.40 134.82




Table 3 reports the effect of the safety factor, and Table 4 reports the effect of the standard deviation of forecast error onthe lot size multipliers, order quantities, and expected profits of the whole supply chain in the decentralized and joint sys-tem, respectively. Both the safety factor and the standard deviation of forecast error have little effect on the lot size multi-pliers in the two-game setting, which is mainly due to the lot size multiplier being an integer. The order quantities tend todecrease in the safety factor, but a increase in the standard deviation of forecast error. These results are consistent withProperties 1 and 2 in Section 2.2. Furthermore, the expected profits of the whole supply chain tend to decrease in boththe safety factor and the standard deviation of forecast error. Nevertheless, the percentage increase in total channel profit(corresponding to ð

Q2 �

Q0Þ=Q

0 from Table 1) tends to increase in both the safety factor and the standard deviation of fore-cast error.

6. Conclusion

In this paper, we explicitly add a carrier to the supplier–retailer channel and analyze the effect of the two sources of dou-ble marginalization on pricing policies. The carrier profit margins are considered in supply chain decisions rather than theexogenous shipping cost, and the transport costs are assumed to be shared by the supplier and retailer. We assume that leadtime demand is stochastic, and shortage during the lead time is permitted. Also, the annual average demand rate is adecreasing function of the retail price. Our analysis shows that joint system profit is increased because the demand isincreased, but the unit operating cost is reduced as a result of joint coordination.

For a specific demand function, the current study demonstrates the effectiveness of transport-fee and wholesale-pricediscount schemes as a control mechanism in joint coordination among the carrier, supplier, and retailer. The coordinatingwholesale price and transport fee are decreasing functions of both the retailer’s order quantity and the annual sales volume.We also conclude that for this specific demand function, the sharing ratios of the unit transport cost between the supplierand retailer have no effect on the joint profit and all channel members’ profits in the decentralized system but have an effecton the value of wholesale prices in the joint and decentralized system. Therefore, the supplier expects to charge a markup ifhe/she incurs any transport cost. Numerical examples show that including the carrier to the supplier-retailer channel coor-dination can significantly improve the system’s performance versus only coordinating through the retailer and the supplier.The percentage increase in total channel profit tends to increase in both the safety factor and the standard deviation of fore-cast error. These observations provide managerial insights into the coordination pricing policy of supply chains with carriers.

For future studies, the following issues can be further investigated: (1) considering the lot-size coordination problemwhere demand is stochastic and the transport time as a part of the lead time affects the transport fee charged by the carrierin a three-level supply chain, and (2) formulating a coordination mechanism for a supply chain with multiple carriers, mul-tiple suppliers, or multiple retailers.

Acknowledgments

We sincerely thank the anonymous referees for their valuable suggestions and comments.

Appendix A

A.1 Proof of Proposition 1

Proof. For a fixed value of w, the maximum value ofQ

Sðw;mÞ in (7) can be obtained by minimizing the cost of the supplier.We let

Please(2015

HðmÞ ¼ f~KS=½mðKR þ ARrLguðkÞÞ� þ ðm� 1Þ~hS=hRg � ½hRDðpÞðKR þ ARrLguðkÞÞ=2�1=2; ðA1Þ

denote the cost of the supplier.After taking the first derivative of HðmÞ with respect to m, we obtain

dHðmÞdm

¼ �~KS

m2ðKR þ ARrLguðkÞÞþ

~hS

hR

!12ðhRDðpÞðKR þ ARrLguðkÞÞ

� �1=2

: ðA2Þ

Further, for a fixed value of w, HðmÞ is a convex function in m, because

d2HðmÞdm2 ¼ 2~KS

m3ðKR þ ARrLguðkÞÞ12ðhRDðpÞðKR þ ARrLguðkÞÞ

� �1=2

> 0:

By equating (A2) to zero, we obtain the following expression for the optimal lot size multiplier in the decentralizedsystem:




Please(2015

ðm�Þ2 ¼ ð~KShRÞ=½ðKR þ ARrLguðkÞÞ~hS�: ðA3Þ

From (A3), it is easy to verify that the optimal lot size multiplier m� satisfy (8). This completes the proof. h

A.2. Proof of Proposition 2

Proof. Let

h ¼ fKShR=½ðKR þ ARrLguðkÞÞhS�g1=2 þ fðKR þ ARrLguðkÞÞhS=ðKShRÞg1=2;

the fixed ordering and inventory holding cost incurred by the supplier are ½2KShSDðpÞ�1=2 and ðh=2Þ½2KShSDðpÞ�1=2, respec-

tively, when the order quantity of the retailer is �Q R ¼½2KSDðpÞ=hS�1=2 and Q R ¼ ½2DðpÞðKR þ ARrLguðkÞÞ=hR�1=2. It is easily ver-ified that h P 2.

With Q̂R ¼ DðpÞ and QR, the fixed transport cost incurred by the carrier is KL and ½hRDðpÞ=2ðKR þ ARrLguðkÞÞ�1=2KL,

respectively. Note that QR 6 DðpÞ, then

½hRDðpÞ=2ðKR þ ARrLguðkÞÞ�1=2KL P KL:

This completes the proof. h

A.3. Proof of Property 1

Proof. Taking the derivative of guðkÞ given by (2) with respect to k, we obtain

dguðkÞ=dk ¼ UðkÞ � 1 6 0; ðA4Þ

where UðkÞ is the cumulative distribution function of the standard normal distribution. From (A4), we can see that guðkÞdecreases as k increases. Therefore, from (5) and (9), Q R decreases and m� increases as k increases, which completes theproof. h


Proof. We let

d1 ¼ 1þ ½KL=ðKR þ ARrLguðkÞÞ þ KS=ðKR þ ARrLguðkÞÞ þ hS=hR�=2;

d2 ¼ f½1þ KL=ðKR þ ARrLguðkÞÞ þ KS=ðKR þ ARrLguðkÞÞ� � ð1þ hS=hRÞg=2:

Applying (6), (7), (10) and (14), we get
Y�
LþY�

SþY�

R¼ ðp� � t � cÞDðp�Þ � d1½2KRhRDðp�Þ�1=2 � krLhR;

YJðp�;QJðp�ÞÞ ¼ ðp� � t � cÞDðp�Þ � d2½2KRhRDðp�Þ�1=2 � krLhR:

It is easy to verify that d1 > d2. This completes the proof. h


Proof. Substituting DðpÞ ¼ a � p�2 in (14) and taking the derivative ofQ

Jðp;QJðpÞÞ with respect to p, we obtain

dY

Jðp;Q JðpÞÞ=dp ¼ ½ð2ahJKJÞ1=2 � a�=p2 þ 2aðt þ cÞ=p3: ðA5Þ

With the joint policy, the retailer’s expected yearly profit is
YJ
R¼ ðp�wJ � k2v JÞ � ða=p2Þ � krLhR � ½ðKR þ ARrLguðkÞÞ=KJ þ hR=hJ� � ½ahJKJ=2�1=2

=p: ðA6Þ

Substituting (24) and (25) in the retailer’s expected profit function in (A6) and taking the derivative ofQJ

Rðp;QJðpÞÞ withrespect to p; we obtain

dYJ

Rðp;Q JðpÞÞ=dp ¼ ð1� b1 � b2Þ � f½ð2ahJKJÞ1=2 � a�=p2 þ 2aðt þ cÞ=p3g ¼ ð1� b1 � b2Þ � ðd

YJðp;Q JðpÞÞ=dpÞ: ðA7Þ




From (A5) and (A7), we note that the retailer chooses the unit retail price that maximizes his/her expected profit as wellas the joint expected profit simultaneously. This completes the proof. h

Appendix B. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.apm.2015.06.006.

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