Research ArticleService and Price Decisions of a Supply Chain withOptional After-Sale Service

Xiaochen Sun1 Qingshuai Zhang1 and Yancong Zhou2

1Department of Mathematics Tianjin University Tianjin 300072 China2School of Information Engineering Tianjin University of Commerce Tianjin 300134 China

Correspondence should be addressed to Xiaochen Sun sxc722163com

Received 4 December 2015 Revised 22 February 2016 Accepted 10 March 2016

Academic Editor Zhimin Huang

Copyright copy 2016 Xiaochen Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For durable products the high quality after-sales service has been playing an increasingly important role in consumersrsquo purchasebehaviors We mainly study a supply chain composed of a manufacturer and a retailer In a process of products sales themanufacturer will provide a basic free quality assurance service On this basis the retailer provides paid optional quality assuranceservice to consumers to promote sales Users are divided into two categories in this paper users with no optional service andusers with optional services We derive the equilibrium decisions between the manufacturer and the retailer under the followingtwo cases (i) the optional after-sales service level and the wholesale price determined by the manufacturer and the retail pricedetermined by the retailer (ii) the wholesale price determined by the manufacturer and the optional after-sales service level andthe retail price determined by the retailer

1 Introduction

In recent years as the living standard of people is improvingwith the rapid development of economy people becomemoreandmore sensitive to nonprice factors they could enjoy ratherthan a price attribute In 2007 the global consumer electronicsproducts consumer research showed that service has becomethe second important factor that affects consumerrsquos purchasebehaviorsMore andmore enterprises have realized this pointin time Relying solely on price advantage it is difficult tomaintain a lasting competitive edge such as Lenovo WAL-MART SONY General Electric andDELLThese enterprisesalso have to provide better services to customers so that theycan maintain their market shares

For the same product providing different services indaily life is also becoming more common When we buya computer we will encounter this kind of situation Basicquality assurance services are to be purchased but not allconsumers are satisfied with this basic service there is a partof the consumers who want to get more and better serviceTherefore for retailers and manufacturers it is necessary to

set up a reasonable and optional service policy to meet moreconsumers

Kameshwaran et al [1] considered the pricing problemin the following three cases (i) only selling product notproviding service (ii) offering product and service indepen-dently and (iii) offering product and service bundled Lu etal [2] studied the supply chain decision-making problem oftwo competing manufacturers and one retailer It emphasizesthe importance of service provided by the manufacturerwhen customers are more sensitive to the price and servicelevel of the product Xiao and Yang [3] constructed a modelthat considers both price and service competition to studythe optimal decision of each member when the demand isuncertain

In the model of Cohen and Whang [4] a customer canobtain after-sales service only from either the manufactureror an independent service shop Li et al [5] considered asupply chain comprising a manufacturer and a retailer Themanufacturer supplies a product to the retailer while theretailer sells the product bundled with after-sales serviceto consumers in a fully competitive market He finds that

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 1505817 11 pageshttpdxdoiorg10115520161505817

2 Mathematical Problems in Engineering

the manufacturerrsquos sharing of the cost with the retailer tobuild service capacity improves the profits of both partiesIn the model of Kurata and Nam [6] a customer can obtainsimultaneously two after-sales services one from the manu-facturer and the other from the retailer And formulating fiveanalytical models they find that the service level maximizingprofits is not the service level that can satisfy consumersthe most As an extension of Kurata and Nam [6] Kurataand Nam [7] explored the effect of uncertainty on after-sales service decisions by comparing several informationstructures in a two-stage supply chain

In fact the optional service to a certain extent to meet theneeds of customers is a stimulating factor so we assume thatthe demand is subject to the impact of optional services Thisis consistent with the study of Xu et al [8] In this case allusers are divided into two categories users with no optionalservice and users with an optional service We will focus onthe equilibrium decisions between the manufacturer and theretailer with two categories of users in which a basic serviceis provided by the manufacturer and an optional service isprovided by the retailer or the manufacturer

Our problem is most relevant to Kurata and Nam [6]However we consider a model with an optional after-saleservice in which the customers need to pay for selectingthe optional service and the price of the optional serviceis a decision variable And we study the decision problemsunder the following two cases (i) the optional after-salesservice level and the wholesale price determined by themanufacturer and the retail price determined by the retailer(ii) the wholesale price determined by the manufacturerand the optional after-sales service level and the retail pricedetermined by the retailer

The remainder of this paper is organized as follows Thesecond part is the problem description and modeling Inthe third part we establish the model where the optionalservice level is decided by the manufacturer The fourth partis the model analysis and the solution In the fifth part weestablish themodel where the optional service level is decidedby the retailer The sixth part is the model analysis and thesolution The seventh part is the comparisons of the twokinds of equilibrium decisions by numerical results Finallyconclusions and the future research directions are addressedin Section 8

2 Problem Description and Models

We consider a supply chain composed of a manufacturer anda retailer in which the manufacturer sells the product to theretailer at the wholesale price 119908 and the retailer sells theproduct to the customer at the price 119901

There are two categories of users in the market The firstcategory of users only use basic service items such as the basicquality assurance services the second category of users arenot satisfied with the basic services but also need enterprisesto provide more optional services such as extended warrantyserviceWe use 119886

1to represent the base-case potential market

size for the first category of users We use 1198862to represent

the base-case potential market size for the second category

of users We assume that the basic service and the retail priceare bound that is after the payment of the cost of the salesthe basic service does not require an additional paymentso the two categories of users are similarly sensitive to thebasic service But the optional service will attract a part ofthe first category of users and other product users to transfertherefore the optional service is negative for the first categoryof usersrsquo demand and is positive for the second category ofusersrsquo demand and the impact of the second category of usersis more Let 119889

1and 119889

2 respectively represent the first and

second categories of userrsquos demand function for the productAs popularly used in other literatures such as [9ndash11] on priceand service competition here 119889

1and 119889

2are thought of as

linear functions about retail price and optional service levelThus the demand functions could be described by price andoptional service as follows

1198891(119901 119910) = 119886

1minus 120573119901 minus 120583

1119910

1198892(119901 119910) = 119886

2minus 120573119901 + 120583

2119910

(1)

where 119910 indicates the optional service level and 120573 indicatesthe price sensitive coefficient 120583

1measures the responsiveness

of market demand for the first category of users to theoptional service level When the optional service is highenough the customer insensitive to the service level willchange the attitude so the first category of customers will beless Therefore the optional service level has a negative effecton 1198891 where 120583

1gt 0 Corresponding to the first category

of customers the second category of customers are sensitiveto the optional service level so the optional service levelhas a positive effect on 119889

2 1205832measures the responsiveness

of market demand for the second category of users to theoptional service level so 120583

2gt 0 With the optional service

level increasing the total demand of two categories shouldbe increased so there should be 120583

1lt 1205832 meaning that with

the optional service level improving the overall demand forthe product will increase (120583

2minus 1205831)119910

If one user chooses the optional service shehe needs topay a certain service charge whichwe call the optional serviceprice Of course the optional service price changes with theoptional service level Let119910 indicate the optional service levelthen the optional service price is 119903(119910) and it is a concavefunction For convenience make 119903(119910) = 120572119910

In order to maximize their own profits the manufacturerand the retailer need to make decisions based on their ownpieces of information We assume that the basic servicelevel is known as well as the optional service price givenby the retailer We consider the decision-making problem ofsupply chain members under the condition of determinedenvironment

3 The Optional Service Level Decidedby the Manufacturer

In this part the optional service level 119910 is determined bythe manufacturer the manufacturer also determines thewholesale price119908 and the retail price 119901 is determined by theretailer

Mathematical Problems in Engineering 3

Hence the manufacturerrsquos profit function is

Π119872(119908 119910) = (119908 minus 119888) (119889

1(119901 119910) + 119889

2(119901 119910)) minus

1205781199102

2

(2)

In the above equation the first item is the sales revenueand the second is the cost of the optional service as Tsay andAgrawal [10] does

Then the retailerrsquos profit function can be formulated bythe following

Π119877(119901) = (119901 minus 119908) 119889

1(119901 119910) + (119901 + 120572119910 minus 119908) 119889

2(119901 119910) (3)

Based on (1)ndash(3) the general optimization model can bedescribed as followsmaxΠ

119872(119908 119910) = (119908 minus 119888) (119886

1+ 1198862minus 2120573119901 minus 120583

1119910 + 1205832119910)

minus

1205781199102

2

(4)

subject to

maxΠ119877(119901) = (119901 minus 119908) (119886

1minus 120573119901 minus 120583

1119910)

+ (119901 + 120572119910 minus 119908) (1198862minus 120573119901 + 120583

2119910)

(5)

We can see that based on the analysis stated above inthis model the manufacturer determines the wholesale price119908 and the optional service level 119910 and the retailer determinesthe retail price 119901 respectively

4 Results and Discussions forthe Manufacturer Case

We now derive chain membersrsquo optimal strategies by maxi-mizing their profits in the above general optimization model

Theorem 1 For any given wholesale price 119908 and optionalservice level 119910 the profit functionΠ

119877(119901) is concave with respect

to the retail price 119901

Proof Consider the following derivatives

120597Π119877(119901)

120597119901

= 1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 4120573119901 minus 120572120573119910

1205972Π119877(119901)

1205971199012

= minus4120573

(6)

Because 1205972Π119877(119901)120597119901

2= minus4120573 lt 0 Π

119877(119901) is concave with

respect to the retail price119901 that is to say there does exist retailprice 1199011015840 maximizing Π

119877(119901)

In the following all best response functions are denotedby the superscript ldquo1015840rdquo and let the superscript ldquolowastrdquo denote theequilibrium decisions

Proposition 2 For any given 119908 and 119910 the retailer bestresponse function satisfies the following

1199011015840=

1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 120572120573119910

4120573

(7)

Proof It can be easily proved by solving the first condition120597Π119877(119901)120597119901 = 0

Paying attention to the impact of wholesale price andoptional service level on the retailerrsquos best response functionwe derive the first derivatives of 1199011015840 with respect to wholesaleprice and optional service level that is 120597119901120597119908 = 05 gt 0 and120597119901120597119908 = (minus120583

1+1205832minus120572120573)4120573 We can easily see that the higher

the wholesale price the higher the retail price However theeffect of optional service level on the retail price is not veryevident

Obviously if the manufacturer increases the wholesaleprice 119908 of the product the retailer tends to increase thesales price 119901 of the product which is very consistent withour intuition And for the optional service level 119910 becausethe optional service price is linear with optional service level119910 when the correlation coefficient 120572 is relatively large theretailer tending to decrease retail price increases sales toincrease his profit

Lemma 3 The inequality 8120573(120578minus21205721205832) gt (120572120573minus120583

1+1205832)2 holds

for any practical problem

Proof As the retailerrsquos operation problem its profitmust havea maximum value in any practical case

Bringing the formulation of 1199011015840 into the manufacturerprofit function Π

119872(119908 119910) we can get

Π119872= (119908 minus 119888) (119886

1+ 1198862

minus 2120573(

1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 120572120573119910

4120573

) minus 1205831119910

+ 1205832119910) minus

1205781199102

2

(8)

Consider the following derivatives

120597Π119872

120597119908

=

1198861+ 1198862minus 1205831119910 + 1205832119910 minus 4120573119908 + 120572120573119910

2

+ 120573119888

120597Π119872

120597119910

= (119908 minus 119888)

minus1205831+ 1205832+ 120572120573

2

minus 120578119910

(

1205972Π119872

1205971199082

1205972Π119872

120597119908120597119910

1205972Π119872

120597119910120597119908

1205972Π119872

1205971199102

)

=(

minus2120573

120572120573 minus 1205831+ 1205832

2

120572120573 minus 1205831+ 1205832

2

minus120578

)

(9)

From the existence condition of extreme function formultivariable function there must be

Δ2= 2120573120578 minus (

120572120573 minus 1205831+ 1205832

2

)

2

gt 0 (10)

Therefore the lemma holds

4 Mathematical Problems in Engineering

Proposition 4 The manufacturer equilibrium decisions sat-isfy

119908lowast=

2 (1198861+ 1198862) 120578 + 4120573120578119888 minus 119888 (minus120583

1+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

119910lowast=

(1198861+ 1198862minus 2120573119888) (minus120583

1+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

(11)

Proof With the best response function for retail price 1199011015840 in(7) themanufacturerrsquos best response function for the productcan be achieved by solving 119908lowast 119910lowast isin argmaxΠ

119872(119908 119910)

Let 120597Π119872(119909 119910)120597119910 = (119908 minus 119888)((minus120583

1+ 1205832+ 120572120573)2) minus 120578119910 = 0

we can get

119910 =

(119908 minus 119888) (minus1205831+ 1205832+ 120572120573)

2120578

(12)

Bring it into 120597Π119872120597119908 = (119886

1+ 1198862minus 1205831119910 + 120583

2119910 minus 4120573119908 +

120572120573119910)2 + 120573119888 Making it zero we can get the manufacturerequilibrium decisions

119908lowast=

2 (1198861+ 1198862) 120578 + 4120573120578119888 minus 119888 (minus120583

1+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (13)

119910lowast=

(1198861+ 1198862minus 2120573119888) (minus120583

1+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (14)

Bringing 119908lowast 119910lowast into 1199011015840 = (1198861+ 1198862minus 1205831119910 + 120583

2119910 +

2120573119908 minus 120572120573119910)4120573 we can get the retailer equilibrium deci-sions

119901lowast=

6120578 (1198861+ 1198862) minus 2119888 (minus120583

1+ 1205832+ 120572120573) (minus120583

1+ 1205832) minus 120572 (119886

1+ 1198862) (minus1205831+ 1205832+ 120572120573) + 4120573120578119888

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(15)

Proposition 5 The equilibrium wholesale price119908lowast is increas-ing in 119886

1 1198862 1205832 and 120572 but decreasing in 120583

1

Proof From (13) we get the following derivatives

120597119908lowast

1205971198861

=

2120578

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2gt 0

120597119908lowast

1205971198862

=

2120578

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2gt 0

(16)

From (16) we can see that the equilibriumwholesale price119908lowastis increasing in 119886

1 1198862

From (13) it is easily got that

119908lowast=

2 (1198861+ 1198862) 120578 minus 4120573120578119888

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2+ 119888 (17)

From (17) we can see that when 1205831is increasing the

equilibrium wholesale price 119908lowast is decreasing in 1205831 And the

equilibriumwholesale price119908lowast is increasing in 1205832and 120572

Proposition 6 If 119888 gt 4(1198861+ 1198862) the equilibrium wholesale

price119908lowast is increasing in 120578 otherwise the equilibriumwholesaleprice 119908lowast is decreasing in 120578

Proof From (13) we get the following derivative

120597119908lowast

120597120578

=

(119888 minus 4 (1198861+ 1198862)) (minus120583

1+ 1205832+ 120572120573)

2

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

2 (18)

It is easy to know that if 119888 gt 4(1198861+ 1198862) the equilibrium

wholesale price 119908lowast is increasing in 120578 otherwise the equilib-rium wholesale price 119908lowast is decreasing in 120578

Proposition 7 If 4120573120578 gt (minus1205831+ 1205832+ 120572120573)

2 the equilibriumwholesale price119908lowast is increasing in 119888 otherwise the equilibriumwholesale price 119908lowast is decreasing in 119888

Proof From (13) we get the following derivative

120597119908lowast

120597119888

=

4120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (19)

It is easy to know that if 4120573120578 gt (minus1205831+ 1205832+ 120572120573)

2 theequilibrium wholesale price 119908lowast is increasing in 119888 otherwisethe equilibrium wholesale price 119908lowast is decreasing in 119888

Proposition 8 The equilibrium optional service level 119910lowast isincreasing in 119886

1 1198862 1205832 and 120572 but decreasing in 120583

1 120578 and 119888

Proof From (12) it is easily got that

119910lowast=

(119908lowastminus 119888) (minus120583

1+ 1205832+ 120572120573)

2120578

(20)

From (20) we can see that when 1205831is increasing minus120583

1+

1205832+120572120573 is decreasing and is greater than zero And because119908lowast

is decreasing in 1205831 when 120583

1is increasing119908lowastminus119888 is decreasing

and is greater than zero Hence the equilibrium optionalservice level 119910lowast is decreasing in 120583

1 And the equilibrium

wholesale price 119908lowast is increasing in 1205832and 120572

Mathematical Problems in Engineering 5

From (20) we can see that when 1198861is increasing 119908lowast is

increasing so 119908lowast minus 119888 is increasing and is greater than zeroHence the equilibrium optional service level 119910lowast is decreasingin 1198861 And the equilibriumwholesale price119908lowast is increasing in

1198862From (14) we can see that when 120578 is increasing the

equilibrium optional service level 119910lowast is decreasing in 120578

Proposition 9 (i) If 6120578 gt 120572(minus1205831+ 1205832+ 120572120573) the equilibrium

retail price 119901lowast is increasing in 1198861and 119886

2 otherwise the

equilibrium retail price 119901lowast is decreasing in 1198861and 119886

2 (ii) If

2120573120578 gt (minus1205831+ 1205832)(minus1205831+ 1205832+ 120572120573) the equilibrium retail price

119901lowast is increasing in 119888 otherwise the equilibrium retail price 119901lowast

is decreasing in 119888

Proof From (15) it is easily got that

120597119901lowast

1205971198861

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

120597119901lowast

1205971198862

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(21)

It is easy to know that if 6120578 gt 120572(minus1205831+ 1205832+ 120572120573)

the equilibrium retail price 119901lowast is increasing in 1198861and 119886

2

otherwise the equilibrium retail price 119901lowast is decreasing in 1198861

and 1198862

From (15) it is easily got that

120597119901lowast

120597119888

=

2120573120578 minus (minus1205831+ 1205832) (minus1205831+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (22)

It is easy to know that if 2120573120578 gt (minus1205831+1205832)(minus1205831+1205832+120572120573)

the equilibrium retail price 119901lowast is increasing in 119888 otherwisethe equilibrium retail price 119901lowast is decreasing in 119888

5 The Optional Service Level Decidedby the Retailer

In this part the optional service level 119910 is determined by theretailer the retailer also determines the retail price 119901 and thewholesale price 119908 is determined by the manufacturer

Hence the manufacturer profit function is

Π119872= (119908 minus 119888) (119889

1(119901 119910) + 119889

2(119901 119910)) (23)

Then the profit function of retailer can be formulated bythe following

Π119877= (119901 minus 119908) 119889

1(119901 119910) + (119901 + 120572119910 minus 119908) 119889

2(119901 119910)

minus

1205781199102

2

(24)

In the above equation the first item is sales revenue andthe second is the cost of the optional service as Tsay andAgrawal [10] does

Based on (1) (23) and (24) the general optimizationmodel can be described as follows

maxΠ119872= (119908 minus 119888) (119886

1+ 1198862minus 2120573119901 minus 120583

1119910 + 1205832119910) (25)

subject to

maxΠ119877= (119901 minus 119908) (119886

1minus 120573119901 minus 120583

1119910)

+ (119901 + 120572119910 minus 119908) (1198862minus 120573119901 + 120583

2119910) minus

1205781199102

2

(26)

We can see that based on the analysis stated above in thismodel the manufacturer determines the wholesale price 119908and the retailer determines the retail price 119901 and the optionalservice level 119910 respectively

6 Results and Discussions forthe Retailer Case

We now derive chain membersrsquo optimal strategies by maxi-mizing their profits in the above general optimization model

Firstly the following parameters hold

Lemma 10 The inequalities 4120573(120578 minus 21205721205832) gt (minus120572120573 minus 120583

1+ 1205832)2

and 21205721205832lt 120578 hold for any practical problem

Proof As the retailerrsquos operation problem its profit musthave a maximum value in any practical case Consider thefollowing derivatives

120597Π119877

120597119901

= 1198861+ 1198862minus 4120573119901 + 2120573119908 minus 120572120573119910 minus 120583

1119910 + 1205832119910

120597Π119877

120597119910

= 1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 + 2120572120583

2119910

minus 120578119910

(

1205972Π119877

1205971199012

1205972Π119877

120597119901120597119910

1205972Π119877

120597119910120597119901

1205972Π119877

1205971199102

)

= (

minus4120573 minus120572120573 minus 1205831+ 1205832

minus120572120573 minus 1205831+ 1205832

21205721205832minus 120578

)

(27)

From the existence condition of extreme function formultivariable function there must be

Δ3= 2120572120583

2minus 120578 lt 0

Δ4= 4120573 (120578 minus 2120572120583

2) minus (minus120572120573 minus 120583

1+ 1205832)2

gt 0

(28)

Therefore the lemma holds

6 Mathematical Problems in Engineering

Proposition 11 For any given 119908 the retailer best responsefunctions satisfy

1199011015840=

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

1199101015840=

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(29)

Proof Let 120597Π119877120597119910 = 120572119886

2minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 +

21205721205832119910 minus 120578119910 = 0 we can get

119910 =

1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901

120578 minus 21205721205832

(30)

Bring it into 120597Π119877120597119901 = 119886

1+1198862minus4120573119901+2120573119908minus120572120573119910minus120583

1119910+

1205832119910 Making it zero we can get the retailerrsquos best response

decisions119901 =

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

(31)

It is easy to get that

119910 =

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(32)

Among them119860 = 1198861+1198862 120590 = 120583

1minus1205832minus120572120573 120585 = minus120583

1+1205832

andΔ4= 4120573120577minus120582

2 that is 120577 = 120578minus21205721205832 120582 = minus120572120573minus120583

1+1205832

Paying attention to the impact of wholesale price on theretailerrsquos best response function we derive the first derivativesof 1199011015840 with respect to wholesale price and the first derivativesof 1199101015840 with respect to wholesale price that is 120597119901120597119908 = (2120573120577 +120582(1205831minus 1205832))Δ4and 120597119910120597119908 = 2120573120577(120583

1minus 1205832minus 120572120573)120577Δ

4lt 0 We

can easily see that the higher the wholesale price the higherthe optional service level However the effect of the wholesaleprice on the retail price is not very evident

Theorem 12 The profit function Π119872(119908) is concave with

respect to the wholesale price 119908

Proof Bringing the retailer best response functions into themanufacturer profit function Π

119872(119908) we can get

Π119872= (119908 minus 119888)

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

(33)

Consider the following derivatives

120597Π119872

120597119908

=

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

+ (119908 minus 119888)

2120573 (120582120577 (minus1205831+ 1205832) minus 2120573120577

2) + (120583

1minus 1205832) (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582)

120577Δ4

(34)

1205972Π119872

1205971199082=

minus812057321205772minus 8120572120577120573

2(minus1205831+ 1205832)

120577Δ4

lt 0 (35)

Because 1205972Π1198721205971199082= (minus8120573

21205772minus8120572120577120573

2(minus1205831+1205832))120577Δ

4lt

0 Π119872(119908) is concave with respect to the wholesale price 119908

that is to say there does exist wholesale price119908lowast maximizingΠ119872(119908)

Proposition 13 The manufacturer equilibrium decisions sat-isfy

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(36)

Proof Let (34) be equal to zero we can get

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(37)

Bringing 119908lowast into (31) and (32) we can get the retailerequilibrium decisions

119901lowast=

81205732120577 (120577 + 120572120585) (120577119860 + 120582120572119886

2) + (2120573120577 minus 120582120585) (119860Δ

4120577 + (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888))

8Δ4(12057321205772+ 120572120577120573

2120585)

119910lowast=

812057321205772(120577 + 120572120585) (120582119860 + 4120572120573119886

2) + 2120573119860Δ

41205772120590 + 2120573120577120590 (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 4120573

212057721205851205902(1205721198862+ 120585119888)

8120577Δ4(12057321205772+ 120572120577120573

2120585)

(38)

Mathematical Problems in Engineering 7

95

100

105

110

115

120

125

130

80 10060 80 1006080 1006080 1006080 10060122

124

126

128

130

132

134

136

138

0

05

1

15

2

25

3

35

200

400

600

800

1000

1200

1400

100

200

300

400

500

600

700

800

Figure 1 Effects of unit production cost

7 Numerical Simulation

In this section we will analyze the changes of differentparameters of supply chain members by numerical simu-lation The objective of numerical analyses is to examinehow the equilibrium profit functions change when the valuesof parameters change And we compare the equilibriumdecisions and the equilibrium profits under two models Thebaseline parameters are set as follows 119888 = 80 119886

1= 100

1198862= 50 120572 = 02 120583

1= 01 120583

2= 04 120578 = 3 and

120573 = 05 In order to compare directly we will use a diagramshowing the curves of two models The red curve representsthe equilibrium decisions and the equilibrium profits wherethe optional service level is determined by the manufacturerand the blue curve represents the equilibrium decisions andthe equilibrium profits where the optional service level isdetermined by the retailer

The ordinate axes in Figures 1ndash8 from left to rightrespectively represent the equilibrium wholesale price theequilibrium retail price the equilibrium optional servicelevel the manufacturerrsquos profit and the retailerrsquos profit

Figure 1 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119888It is easy to see that as 119888 increases the retail price andthe wholesale price are linearly increasing but the optimaloptional service level and the profits of all chain membersare linearly decreasing In Figure 1 119888 is from 50 to 100 Wecan see that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer

Figure 2 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120578 Itis easy to see that as 120578 increases in the model where theoptional service level is determined by manufacturer theretail price and the wholesale price and the optional servicelevel are decreasing but in the model where the optionalservice level is determined by the retailer the retail price andthe wholesale price are increasing and the optional servicelevel is decreasing In Figure 2 120578 is from 2 to 6 We cansee that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer For 120583

1 there are similar results

It is represented in Figure 3 The only difference is thatthe retailerrsquos profit in the model where the optional servicelevel is determined by the retailer firstly decreases and thenincreases

Figure 4 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119886

1

It is easy to see that as 1198861increases all of the equilibrium

decisions and the profits of all chain members are increasingIn Figure 4 119886

1is from 50 to 200 We can see that the

optimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by manufacturer are greater than theoptimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by the retailer But the retailerrsquos profitin the model where the optional service level is determinedby the manufacturer is less than the retailerrsquos profit in themodel where the optional service level is determined by the

8 Mathematical Problems in Engineering

50 1000 50 1000 50 1000 50 1000 50 1000106

107

108

109

110

111

112

113

114

115

116

128

129

130

131

132

133

134

0

05

1

15

2

25

3

35

4

580

585

590

595

600

605

610

615

620

625

300

320

340

360

380

400

420

440

460

480

500

Figure 2 Effects of optional service cost coefficient

02 040 02 040 02 040 02 040 02 040109

110

111

112

113

114

115

116

1295

130

1305

131

1315

132

1325

133

minus1

minus05

0

05

1

15

2

25

602

604

606

608

610

612

614

616

618

620

622

300

320

340

360

380

400

420

Figure 3 Effects of service elastic coefficient of the first category of customers

retailer Moreover the optimal optional service level in themodel where the optional service level is determined by themanufacturer is firstly less and then greater than the optimaloptional service level in the model where the optional servicelevel is determined by the retailer For 119886

2 there are similar

results It is represented in Figure 5 The only difference isthat the optimal optional service level in the model wherethe optional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer

Figure 6 indicates how the equilibrium decisions and theprofits of all chainmembers changewith respect to120573 It is easyto see that as 120573 increases all of the equilibrium decisions andthe profits of all chain members are decreasing In Figure 6120573 is from 01 to 05 We can see that the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model where the

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the manufacturerrsquos sharing of the cost with the retailer tobuild service capacity improves the profits of both partiesIn the model of Kurata and Nam [6] a customer can obtainsimultaneously two after-sales services one from the manu-facturer and the other from the retailer And formulating fiveanalytical models they find that the service level maximizingprofits is not the service level that can satisfy consumersthe most As an extension of Kurata and Nam [6] Kurataand Nam [7] explored the effect of uncertainty on after-sales service decisions by comparing several informationstructures in a two-stage supply chain

In fact the optional service to a certain extent to meet theneeds of customers is a stimulating factor so we assume thatthe demand is subject to the impact of optional services Thisis consistent with the study of Xu et al [8] In this case allusers are divided into two categories users with no optionalservice and users with an optional service We will focus onthe equilibrium decisions between the manufacturer and theretailer with two categories of users in which a basic serviceis provided by the manufacturer and an optional service isprovided by the retailer or the manufacturer

Our problem is most relevant to Kurata and Nam [6]However we consider a model with an optional after-saleservice in which the customers need to pay for selectingthe optional service and the price of the optional serviceis a decision variable And we study the decision problemsunder the following two cases (i) the optional after-salesservice level and the wholesale price determined by themanufacturer and the retail price determined by the retailer(ii) the wholesale price determined by the manufacturerand the optional after-sales service level and the retail pricedetermined by the retailer

The remainder of this paper is organized as follows Thesecond part is the problem description and modeling Inthe third part we establish the model where the optionalservice level is decided by the manufacturer The fourth partis the model analysis and the solution In the fifth part weestablish themodel where the optional service level is decidedby the retailer The sixth part is the model analysis and thesolution The seventh part is the comparisons of the twokinds of equilibrium decisions by numerical results Finallyconclusions and the future research directions are addressedin Section 8

2 Problem Description and Models

We consider a supply chain composed of a manufacturer anda retailer in which the manufacturer sells the product to theretailer at the wholesale price 119908 and the retailer sells theproduct to the customer at the price 119901

There are two categories of users in the market The firstcategory of users only use basic service items such as the basicquality assurance services the second category of users arenot satisfied with the basic services but also need enterprisesto provide more optional services such as extended warrantyserviceWe use 119886

1to represent the base-case potential market

size for the first category of users We use 1198862to represent

the base-case potential market size for the second category

of users We assume that the basic service and the retail priceare bound that is after the payment of the cost of the salesthe basic service does not require an additional paymentso the two categories of users are similarly sensitive to thebasic service But the optional service will attract a part ofthe first category of users and other product users to transfertherefore the optional service is negative for the first categoryof usersrsquo demand and is positive for the second category ofusersrsquo demand and the impact of the second category of usersis more Let 119889

1and 119889

2 respectively represent the first and

second categories of userrsquos demand function for the productAs popularly used in other literatures such as [9ndash11] on priceand service competition here 119889

1and 119889

2are thought of as

linear functions about retail price and optional service levelThus the demand functions could be described by price andoptional service as follows

1198891(119901 119910) = 119886

1minus 120573119901 minus 120583

1119910

1198892(119901 119910) = 119886

2minus 120573119901 + 120583

2119910

(1)

where 119910 indicates the optional service level and 120573 indicatesthe price sensitive coefficient 120583

1measures the responsiveness

of market demand for the first category of users to theoptional service level When the optional service is highenough the customer insensitive to the service level willchange the attitude so the first category of customers will beless Therefore the optional service level has a negative effecton 1198891 where 120583

1gt 0 Corresponding to the first category

of customers the second category of customers are sensitiveto the optional service level so the optional service levelhas a positive effect on 119889

2 1205832measures the responsiveness

of market demand for the second category of users to theoptional service level so 120583

2gt 0 With the optional service

level increasing the total demand of two categories shouldbe increased so there should be 120583

1lt 1205832 meaning that with

the optional service level improving the overall demand forthe product will increase (120583

2minus 1205831)119910

If one user chooses the optional service shehe needs topay a certain service charge whichwe call the optional serviceprice Of course the optional service price changes with theoptional service level Let119910 indicate the optional service levelthen the optional service price is 119903(119910) and it is a concavefunction For convenience make 119903(119910) = 120572119910

In order to maximize their own profits the manufacturerand the retailer need to make decisions based on their ownpieces of information We assume that the basic servicelevel is known as well as the optional service price givenby the retailer We consider the decision-making problem ofsupply chain members under the condition of determinedenvironment

3 The Optional Service Level Decidedby the Manufacturer

In this part the optional service level 119910 is determined bythe manufacturer the manufacturer also determines thewholesale price119908 and the retail price 119901 is determined by theretailer

Mathematical Problems in Engineering 3

Hence the manufacturerrsquos profit function is

Π119872(119908 119910) = (119908 minus 119888) (119889

1(119901 119910) + 119889

2(119901 119910)) minus

1205781199102

2

(2)

In the above equation the first item is the sales revenueand the second is the cost of the optional service as Tsay andAgrawal [10] does

Then the retailerrsquos profit function can be formulated bythe following

Π119877(119901) = (119901 minus 119908) 119889

1(119901 119910) + (119901 + 120572119910 minus 119908) 119889

2(119901 119910) (3)

Based on (1)ndash(3) the general optimization model can bedescribed as followsmaxΠ

119872(119908 119910) = (119908 minus 119888) (119886

1+ 1198862minus 2120573119901 minus 120583

1119910 + 1205832119910)

minus

1205781199102

2

(4)

subject to

maxΠ119877(119901) = (119901 minus 119908) (119886

1minus 120573119901 minus 120583

1119910)

+ (119901 + 120572119910 minus 119908) (1198862minus 120573119901 + 120583

2119910)

(5)

We can see that based on the analysis stated above inthis model the manufacturer determines the wholesale price119908 and the optional service level 119910 and the retailer determinesthe retail price 119901 respectively

4 Results and Discussions forthe Manufacturer Case

We now derive chain membersrsquo optimal strategies by maxi-mizing their profits in the above general optimization model

Theorem 1 For any given wholesale price 119908 and optionalservice level 119910 the profit functionΠ

119877(119901) is concave with respect

to the retail price 119901

Proof Consider the following derivatives

120597Π119877(119901)

120597119901

= 1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 4120573119901 minus 120572120573119910

1205972Π119877(119901)

1205971199012

= minus4120573

(6)

Because 1205972Π119877(119901)120597119901

2= minus4120573 lt 0 Π

119877(119901) is concave with

respect to the retail price119901 that is to say there does exist retailprice 1199011015840 maximizing Π

119877(119901)

In the following all best response functions are denotedby the superscript ldquo1015840rdquo and let the superscript ldquolowastrdquo denote theequilibrium decisions

Proposition 2 For any given 119908 and 119910 the retailer bestresponse function satisfies the following

1199011015840=

1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 120572120573119910

4120573

(7)

Proof It can be easily proved by solving the first condition120597Π119877(119901)120597119901 = 0

Paying attention to the impact of wholesale price andoptional service level on the retailerrsquos best response functionwe derive the first derivatives of 1199011015840 with respect to wholesaleprice and optional service level that is 120597119901120597119908 = 05 gt 0 and120597119901120597119908 = (minus120583

1+1205832minus120572120573)4120573 We can easily see that the higher

the wholesale price the higher the retail price However theeffect of optional service level on the retail price is not veryevident

Obviously if the manufacturer increases the wholesaleprice 119908 of the product the retailer tends to increase thesales price 119901 of the product which is very consistent withour intuition And for the optional service level 119910 becausethe optional service price is linear with optional service level119910 when the correlation coefficient 120572 is relatively large theretailer tending to decrease retail price increases sales toincrease his profit

Lemma 3 The inequality 8120573(120578minus21205721205832) gt (120572120573minus120583

1+1205832)2 holds

for any practical problem

Proof As the retailerrsquos operation problem its profitmust havea maximum value in any practical case

Bringing the formulation of 1199011015840 into the manufacturerprofit function Π

119872(119908 119910) we can get

Π119872= (119908 minus 119888) (119886

1+ 1198862

minus 2120573(

1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 120572120573119910

4120573

) minus 1205831119910

+ 1205832119910) minus

1205781199102

2

(8)

Consider the following derivatives

120597Π119872

120597119908

=

1198861+ 1198862minus 1205831119910 + 1205832119910 minus 4120573119908 + 120572120573119910

2

+ 120573119888

120597Π119872

120597119910

= (119908 minus 119888)

minus1205831+ 1205832+ 120572120573

2

minus 120578119910

(

1205972Π119872

1205971199082

1205972Π119872

120597119908120597119910

1205972Π119872

120597119910120597119908

1205972Π119872

1205971199102

)

=(

minus2120573

120572120573 minus 1205831+ 1205832

2

120572120573 minus 1205831+ 1205832

2

minus120578

)

(9)

From the existence condition of extreme function formultivariable function there must be

Δ2= 2120573120578 minus (

120572120573 minus 1205831+ 1205832

2

)

2

gt 0 (10)

Therefore the lemma holds

4 Mathematical Problems in Engineering

Proposition 4 The manufacturer equilibrium decisions sat-isfy

119908lowast=

2 (1198861+ 1198862) 120578 + 4120573120578119888 minus 119888 (minus120583

1+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

119910lowast=

(1198861+ 1198862minus 2120573119888) (minus120583

1+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

(11)

Proof With the best response function for retail price 1199011015840 in(7) themanufacturerrsquos best response function for the productcan be achieved by solving 119908lowast 119910lowast isin argmaxΠ

119872(119908 119910)

Let 120597Π119872(119909 119910)120597119910 = (119908 minus 119888)((minus120583

1+ 1205832+ 120572120573)2) minus 120578119910 = 0

we can get

119910 =

(119908 minus 119888) (minus1205831+ 1205832+ 120572120573)

2120578

(12)

Bring it into 120597Π119872120597119908 = (119886

1+ 1198862minus 1205831119910 + 120583

2119910 minus 4120573119908 +

120572120573119910)2 + 120573119888 Making it zero we can get the manufacturerequilibrium decisions

119908lowast=

2 (1198861+ 1198862) 120578 + 4120573120578119888 minus 119888 (minus120583

1+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (13)

119910lowast=

(1198861+ 1198862minus 2120573119888) (minus120583

1+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (14)

Bringing 119908lowast 119910lowast into 1199011015840 = (1198861+ 1198862minus 1205831119910 + 120583

2119910 +

2120573119908 minus 120572120573119910)4120573 we can get the retailer equilibrium deci-sions

119901lowast=

6120578 (1198861+ 1198862) minus 2119888 (minus120583

1+ 1205832+ 120572120573) (minus120583

1+ 1205832) minus 120572 (119886

1+ 1198862) (minus1205831+ 1205832+ 120572120573) + 4120573120578119888

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(15)

Proposition 5 The equilibrium wholesale price119908lowast is increas-ing in 119886

1 1198862 1205832 and 120572 but decreasing in 120583

1

Proof From (13) we get the following derivatives

120597119908lowast

1205971198861

=

2120578

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2gt 0

120597119908lowast

1205971198862

=

2120578

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2gt 0

(16)

From (16) we can see that the equilibriumwholesale price119908lowastis increasing in 119886

1 1198862

From (13) it is easily got that

119908lowast=

2 (1198861+ 1198862) 120578 minus 4120573120578119888

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2+ 119888 (17)

From (17) we can see that when 1205831is increasing the

equilibrium wholesale price 119908lowast is decreasing in 1205831 And the

equilibriumwholesale price119908lowast is increasing in 1205832and 120572

Proposition 6 If 119888 gt 4(1198861+ 1198862) the equilibrium wholesale

price119908lowast is increasing in 120578 otherwise the equilibriumwholesaleprice 119908lowast is decreasing in 120578

Proof From (13) we get the following derivative

120597119908lowast

120597120578

=

(119888 minus 4 (1198861+ 1198862)) (minus120583

1+ 1205832+ 120572120573)

2

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

2 (18)

It is easy to know that if 119888 gt 4(1198861+ 1198862) the equilibrium

wholesale price 119908lowast is increasing in 120578 otherwise the equilib-rium wholesale price 119908lowast is decreasing in 120578

Proposition 7 If 4120573120578 gt (minus1205831+ 1205832+ 120572120573)

2 the equilibriumwholesale price119908lowast is increasing in 119888 otherwise the equilibriumwholesale price 119908lowast is decreasing in 119888

Proof From (13) we get the following derivative

120597119908lowast

120597119888

=

4120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (19)

It is easy to know that if 4120573120578 gt (minus1205831+ 1205832+ 120572120573)

2 theequilibrium wholesale price 119908lowast is increasing in 119888 otherwisethe equilibrium wholesale price 119908lowast is decreasing in 119888

Proposition 8 The equilibrium optional service level 119910lowast isincreasing in 119886

1 1198862 1205832 and 120572 but decreasing in 120583

1 120578 and 119888

Proof From (12) it is easily got that

119910lowast=

(119908lowastminus 119888) (minus120583

1+ 1205832+ 120572120573)

2120578

(20)

From (20) we can see that when 1205831is increasing minus120583

1+

1205832+120572120573 is decreasing and is greater than zero And because119908lowast

is decreasing in 1205831 when 120583

1is increasing119908lowastminus119888 is decreasing

and is greater than zero Hence the equilibrium optionalservice level 119910lowast is decreasing in 120583

1 And the equilibrium

wholesale price 119908lowast is increasing in 1205832and 120572

Mathematical Problems in Engineering 5

From (20) we can see that when 1198861is increasing 119908lowast is

increasing so 119908lowast minus 119888 is increasing and is greater than zeroHence the equilibrium optional service level 119910lowast is decreasingin 1198861 And the equilibriumwholesale price119908lowast is increasing in

1198862From (14) we can see that when 120578 is increasing the

equilibrium optional service level 119910lowast is decreasing in 120578

Proposition 9 (i) If 6120578 gt 120572(minus1205831+ 1205832+ 120572120573) the equilibrium

retail price 119901lowast is increasing in 1198861and 119886

2 otherwise the

equilibrium retail price 119901lowast is decreasing in 1198861and 119886

2 (ii) If

2120573120578 gt (minus1205831+ 1205832)(minus1205831+ 1205832+ 120572120573) the equilibrium retail price

119901lowast is increasing in 119888 otherwise the equilibrium retail price 119901lowast

is decreasing in 119888

Proof From (15) it is easily got that

120597119901lowast

1205971198861

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

120597119901lowast

1205971198862

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(21)

It is easy to know that if 6120578 gt 120572(minus1205831+ 1205832+ 120572120573)

the equilibrium retail price 119901lowast is increasing in 1198861and 119886

2

otherwise the equilibrium retail price 119901lowast is decreasing in 1198861

and 1198862

From (15) it is easily got that

120597119901lowast

120597119888

=

2120573120578 minus (minus1205831+ 1205832) (minus1205831+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (22)

It is easy to know that if 2120573120578 gt (minus1205831+1205832)(minus1205831+1205832+120572120573)

the equilibrium retail price 119901lowast is increasing in 119888 otherwisethe equilibrium retail price 119901lowast is decreasing in 119888

5 The Optional Service Level Decidedby the Retailer

In this part the optional service level 119910 is determined by theretailer the retailer also determines the retail price 119901 and thewholesale price 119908 is determined by the manufacturer

Hence the manufacturer profit function is

Π119872= (119908 minus 119888) (119889

1(119901 119910) + 119889

2(119901 119910)) (23)

Then the profit function of retailer can be formulated bythe following

Π119877= (119901 minus 119908) 119889

1(119901 119910) + (119901 + 120572119910 minus 119908) 119889

2(119901 119910)

minus

1205781199102

2

(24)

In the above equation the first item is sales revenue andthe second is the cost of the optional service as Tsay andAgrawal [10] does

Based on (1) (23) and (24) the general optimizationmodel can be described as follows

maxΠ119872= (119908 minus 119888) (119886

1+ 1198862minus 2120573119901 minus 120583

1119910 + 1205832119910) (25)

subject to

maxΠ119877= (119901 minus 119908) (119886

1minus 120573119901 minus 120583

1119910)

+ (119901 + 120572119910 minus 119908) (1198862minus 120573119901 + 120583

2119910) minus

1205781199102

2

(26)

We can see that based on the analysis stated above in thismodel the manufacturer determines the wholesale price 119908and the retailer determines the retail price 119901 and the optionalservice level 119910 respectively

6 Results and Discussions forthe Retailer Case

We now derive chain membersrsquo optimal strategies by maxi-mizing their profits in the above general optimization model

Firstly the following parameters hold

Lemma 10 The inequalities 4120573(120578 minus 21205721205832) gt (minus120572120573 minus 120583

1+ 1205832)2

and 21205721205832lt 120578 hold for any practical problem

Proof As the retailerrsquos operation problem its profit musthave a maximum value in any practical case Consider thefollowing derivatives

120597Π119877

120597119901

= 1198861+ 1198862minus 4120573119901 + 2120573119908 minus 120572120573119910 minus 120583

1119910 + 1205832119910

120597Π119877

120597119910

= 1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 + 2120572120583

2119910

minus 120578119910

(

1205972Π119877

1205971199012

1205972Π119877

120597119901120597119910

1205972Π119877

120597119910120597119901

1205972Π119877

1205971199102

)

= (

minus4120573 minus120572120573 minus 1205831+ 1205832

minus120572120573 minus 1205831+ 1205832

21205721205832minus 120578

)

(27)

From the existence condition of extreme function formultivariable function there must be

Δ3= 2120572120583

2minus 120578 lt 0

Δ4= 4120573 (120578 minus 2120572120583

2) minus (minus120572120573 minus 120583

1+ 1205832)2

gt 0

(28)

Therefore the lemma holds

6 Mathematical Problems in Engineering

Proposition 11 For any given 119908 the retailer best responsefunctions satisfy

1199011015840=

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

1199101015840=

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(29)

Proof Let 120597Π119877120597119910 = 120572119886

2minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 +

21205721205832119910 minus 120578119910 = 0 we can get

119910 =

1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901

120578 minus 21205721205832

(30)

Bring it into 120597Π119877120597119901 = 119886

1+1198862minus4120573119901+2120573119908minus120572120573119910minus120583

1119910+

1205832119910 Making it zero we can get the retailerrsquos best response

decisions119901 =

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

(31)

It is easy to get that

119910 =

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(32)

Among them119860 = 1198861+1198862 120590 = 120583

1minus1205832minus120572120573 120585 = minus120583

1+1205832

andΔ4= 4120573120577minus120582

2 that is 120577 = 120578minus21205721205832 120582 = minus120572120573minus120583

1+1205832

Paying attention to the impact of wholesale price on theretailerrsquos best response function we derive the first derivativesof 1199011015840 with respect to wholesale price and the first derivativesof 1199101015840 with respect to wholesale price that is 120597119901120597119908 = (2120573120577 +120582(1205831minus 1205832))Δ4and 120597119910120597119908 = 2120573120577(120583

1minus 1205832minus 120572120573)120577Δ

4lt 0 We

can easily see that the higher the wholesale price the higherthe optional service level However the effect of the wholesaleprice on the retail price is not very evident

Theorem 12 The profit function Π119872(119908) is concave with

respect to the wholesale price 119908

Proof Bringing the retailer best response functions into themanufacturer profit function Π

119872(119908) we can get

Π119872= (119908 minus 119888)

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

(33)

Consider the following derivatives

120597Π119872

120597119908

=

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

+ (119908 minus 119888)

2120573 (120582120577 (minus1205831+ 1205832) minus 2120573120577

2) + (120583

1minus 1205832) (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582)

120577Δ4

(34)

1205972Π119872

1205971199082=

minus812057321205772minus 8120572120577120573

2(minus1205831+ 1205832)

120577Δ4

lt 0 (35)

Because 1205972Π1198721205971199082= (minus8120573

21205772minus8120572120577120573

2(minus1205831+1205832))120577Δ

4lt

0 Π119872(119908) is concave with respect to the wholesale price 119908

that is to say there does exist wholesale price119908lowast maximizingΠ119872(119908)

Proposition 13 The manufacturer equilibrium decisions sat-isfy

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(36)

Proof Let (34) be equal to zero we can get

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(37)

Bringing 119908lowast into (31) and (32) we can get the retailerequilibrium decisions

119901lowast=

81205732120577 (120577 + 120572120585) (120577119860 + 120582120572119886

2) + (2120573120577 minus 120582120585) (119860Δ

4120577 + (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888))

8Δ4(12057321205772+ 120572120577120573

2120585)

119910lowast=

812057321205772(120577 + 120572120585) (120582119860 + 4120572120573119886

2) + 2120573119860Δ

41205772120590 + 2120573120577120590 (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 4120573

212057721205851205902(1205721198862+ 120585119888)

8120577Δ4(12057321205772+ 120572120577120573

2120585)

(38)

Mathematical Problems in Engineering 7

95

100

105

110

115

120

125

130

80 10060 80 1006080 1006080 1006080 10060122

124

126

128

130

132

134

136

138

0

05

1

15

2

25

3

35

200

400

600

800

1000

1200

1400

100

200

300

400

500

600

700

800

Figure 1 Effects of unit production cost

7 Numerical Simulation

In this section we will analyze the changes of differentparameters of supply chain members by numerical simu-lation The objective of numerical analyses is to examinehow the equilibrium profit functions change when the valuesof parameters change And we compare the equilibriumdecisions and the equilibrium profits under two models Thebaseline parameters are set as follows 119888 = 80 119886

1= 100

1198862= 50 120572 = 02 120583

1= 01 120583

2= 04 120578 = 3 and

120573 = 05 In order to compare directly we will use a diagramshowing the curves of two models The red curve representsthe equilibrium decisions and the equilibrium profits wherethe optional service level is determined by the manufacturerand the blue curve represents the equilibrium decisions andthe equilibrium profits where the optional service level isdetermined by the retailer

The ordinate axes in Figures 1ndash8 from left to rightrespectively represent the equilibrium wholesale price theequilibrium retail price the equilibrium optional servicelevel the manufacturerrsquos profit and the retailerrsquos profit

Figure 1 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119888It is easy to see that as 119888 increases the retail price andthe wholesale price are linearly increasing but the optimaloptional service level and the profits of all chain membersare linearly decreasing In Figure 1 119888 is from 50 to 100 Wecan see that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer

Figure 2 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120578 Itis easy to see that as 120578 increases in the model where theoptional service level is determined by manufacturer theretail price and the wholesale price and the optional servicelevel are decreasing but in the model where the optionalservice level is determined by the retailer the retail price andthe wholesale price are increasing and the optional servicelevel is decreasing In Figure 2 120578 is from 2 to 6 We cansee that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer For 120583

1 there are similar results

It is represented in Figure 3 The only difference is thatthe retailerrsquos profit in the model where the optional servicelevel is determined by the retailer firstly decreases and thenincreases

Figure 4 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119886

1

It is easy to see that as 1198861increases all of the equilibrium

decisions and the profits of all chain members are increasingIn Figure 4 119886

1is from 50 to 200 We can see that the

optimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by manufacturer are greater than theoptimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by the retailer But the retailerrsquos profitin the model where the optional service level is determinedby the manufacturer is less than the retailerrsquos profit in themodel where the optional service level is determined by the

8 Mathematical Problems in Engineering

50 1000 50 1000 50 1000 50 1000 50 1000106

107

108

109

110

111

112

113

114

115

116

128

129

130

131

132

133

134

0

05

1

15

2

25

3

35

4

580

585

590

595

600

605

610

615

620

625

300

320

340

360

380

400

420

440

460

480

500

Figure 2 Effects of optional service cost coefficient

02 040 02 040 02 040 02 040 02 040109

110

111

112

113

114

115

116

1295

130

1305

131

1315

132

1325

133

minus1

minus05

0

05

1

15

2

25

602

604

606

608

610

612

614

616

618

620

622

300

320

340

360

380

400

420

Figure 3 Effects of service elastic coefficient of the first category of customers

retailer Moreover the optimal optional service level in themodel where the optional service level is determined by themanufacturer is firstly less and then greater than the optimaloptional service level in the model where the optional servicelevel is determined by the retailer For 119886

2 there are similar

results It is represented in Figure 5 The only difference isthat the optimal optional service level in the model wherethe optional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer

Figure 6 indicates how the equilibrium decisions and theprofits of all chainmembers changewith respect to120573 It is easyto see that as 120573 increases all of the equilibrium decisions andthe profits of all chain members are decreasing In Figure 6120573 is from 01 to 05 We can see that the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model where the

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Hence the manufacturerrsquos profit function is

Π119872(119908 119910) = (119908 minus 119888) (119889

1(119901 119910) + 119889

2(119901 119910)) minus

1205781199102

2

(2)

In the above equation the first item is the sales revenueand the second is the cost of the optional service as Tsay andAgrawal [10] does

Then the retailerrsquos profit function can be formulated bythe following

Π119877(119901) = (119901 minus 119908) 119889

1(119901 119910) + (119901 + 120572119910 minus 119908) 119889

2(119901 119910) (3)

Based on (1)ndash(3) the general optimization model can bedescribed as followsmaxΠ

119872(119908 119910) = (119908 minus 119888) (119886

1+ 1198862minus 2120573119901 minus 120583

1119910 + 1205832119910)

minus

1205781199102

2

(4)

subject to

maxΠ119877(119901) = (119901 minus 119908) (119886

1minus 120573119901 minus 120583

1119910)

+ (119901 + 120572119910 minus 119908) (1198862minus 120573119901 + 120583

2119910)

(5)

We can see that based on the analysis stated above inthis model the manufacturer determines the wholesale price119908 and the optional service level 119910 and the retailer determinesthe retail price 119901 respectively

4 Results and Discussions forthe Manufacturer Case

We now derive chain membersrsquo optimal strategies by maxi-mizing their profits in the above general optimization model

Theorem 1 For any given wholesale price 119908 and optionalservice level 119910 the profit functionΠ

119877(119901) is concave with respect

to the retail price 119901

Proof Consider the following derivatives

120597Π119877(119901)

120597119901

= 1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 4120573119901 minus 120572120573119910

1205972Π119877(119901)

1205971199012

= minus4120573

(6)

Because 1205972Π119877(119901)120597119901

2= minus4120573 lt 0 Π

119877(119901) is concave with

respect to the retail price119901 that is to say there does exist retailprice 1199011015840 maximizing Π

119877(119901)

In the following all best response functions are denotedby the superscript ldquo1015840rdquo and let the superscript ldquolowastrdquo denote theequilibrium decisions

Proposition 2 For any given 119908 and 119910 the retailer bestresponse function satisfies the following

1199011015840=

1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 120572120573119910

4120573

(7)

Proof It can be easily proved by solving the first condition120597Π119877(119901)120597119901 = 0

Paying attention to the impact of wholesale price andoptional service level on the retailerrsquos best response functionwe derive the first derivatives of 1199011015840 with respect to wholesaleprice and optional service level that is 120597119901120597119908 = 05 gt 0 and120597119901120597119908 = (minus120583

1+1205832minus120572120573)4120573 We can easily see that the higher

the wholesale price the higher the retail price However theeffect of optional service level on the retail price is not veryevident

Obviously if the manufacturer increases the wholesaleprice 119908 of the product the retailer tends to increase thesales price 119901 of the product which is very consistent withour intuition And for the optional service level 119910 becausethe optional service price is linear with optional service level119910 when the correlation coefficient 120572 is relatively large theretailer tending to decrease retail price increases sales toincrease his profit

Lemma 3 The inequality 8120573(120578minus21205721205832) gt (120572120573minus120583

1+1205832)2 holds

for any practical problem

Proof As the retailerrsquos operation problem its profitmust havea maximum value in any practical case

Bringing the formulation of 1199011015840 into the manufacturerprofit function Π

119872(119908 119910) we can get

Π119872= (119908 minus 119888) (119886

1+ 1198862

minus 2120573(

1198861+ 1198862minus 1205831119910 + 1205832119910 + 2120573119908 minus 120572120573119910

4120573

) minus 1205831119910

+ 1205832119910) minus

1205781199102

2

(8)

Consider the following derivatives

120597Π119872

120597119908

=

1198861+ 1198862minus 1205831119910 + 1205832119910 minus 4120573119908 + 120572120573119910

2

+ 120573119888

120597Π119872

120597119910

= (119908 minus 119888)

minus1205831+ 1205832+ 120572120573

2

minus 120578119910

(

1205972Π119872

1205971199082

1205972Π119872

120597119908120597119910

1205972Π119872

120597119910120597119908

1205972Π119872

1205971199102

)

=(

minus2120573

120572120573 minus 1205831+ 1205832

2

120572120573 minus 1205831+ 1205832

2

minus120578

)

(9)

From the existence condition of extreme function formultivariable function there must be

Δ2= 2120573120578 minus (

120572120573 minus 1205831+ 1205832

2

)

2

gt 0 (10)

Therefore the lemma holds

4 Mathematical Problems in Engineering

Proposition 4 The manufacturer equilibrium decisions sat-isfy

119908lowast=

2 (1198861+ 1198862) 120578 + 4120573120578119888 minus 119888 (minus120583

1+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

119910lowast=

(1198861+ 1198862minus 2120573119888) (minus120583

1+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

(11)

Proof With the best response function for retail price 1199011015840 in(7) themanufacturerrsquos best response function for the productcan be achieved by solving 119908lowast 119910lowast isin argmaxΠ

119872(119908 119910)

Let 120597Π119872(119909 119910)120597119910 = (119908 minus 119888)((minus120583

1+ 1205832+ 120572120573)2) minus 120578119910 = 0

we can get

119910 =

(119908 minus 119888) (minus1205831+ 1205832+ 120572120573)

2120578

(12)

Bring it into 120597Π119872120597119908 = (119886

1+ 1198862minus 1205831119910 + 120583

2119910 minus 4120573119908 +

120572120573119910)2 + 120573119888 Making it zero we can get the manufacturerequilibrium decisions

119908lowast=

2 (1198861+ 1198862) 120578 + 4120573120578119888 minus 119888 (minus120583

1+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (13)

119910lowast=

(1198861+ 1198862minus 2120573119888) (minus120583

1+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (14)

Bringing 119908lowast 119910lowast into 1199011015840 = (1198861+ 1198862minus 1205831119910 + 120583

2119910 +

2120573119908 minus 120572120573119910)4120573 we can get the retailer equilibrium deci-sions

119901lowast=

6120578 (1198861+ 1198862) minus 2119888 (minus120583

1+ 1205832+ 120572120573) (minus120583

1+ 1205832) minus 120572 (119886

1+ 1198862) (minus1205831+ 1205832+ 120572120573) + 4120573120578119888

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(15)

Proposition 5 The equilibrium wholesale price119908lowast is increas-ing in 119886

1 1198862 1205832 and 120572 but decreasing in 120583

1

Proof From (13) we get the following derivatives

120597119908lowast

1205971198861

=

2120578

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2gt 0

120597119908lowast

1205971198862

=

2120578

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2gt 0

(16)

From (16) we can see that the equilibriumwholesale price119908lowastis increasing in 119886

1 1198862

From (13) it is easily got that

119908lowast=

2 (1198861+ 1198862) 120578 minus 4120573120578119888

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2+ 119888 (17)

From (17) we can see that when 1205831is increasing the

equilibrium wholesale price 119908lowast is decreasing in 1205831 And the

equilibriumwholesale price119908lowast is increasing in 1205832and 120572

Proposition 6 If 119888 gt 4(1198861+ 1198862) the equilibrium wholesale

price119908lowast is increasing in 120578 otherwise the equilibriumwholesaleprice 119908lowast is decreasing in 120578

Proof From (13) we get the following derivative

120597119908lowast

120597120578

=

(119888 minus 4 (1198861+ 1198862)) (minus120583

1+ 1205832+ 120572120573)

2

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

2 (18)

It is easy to know that if 119888 gt 4(1198861+ 1198862) the equilibrium

wholesale price 119908lowast is increasing in 120578 otherwise the equilib-rium wholesale price 119908lowast is decreasing in 120578

Proposition 7 If 4120573120578 gt (minus1205831+ 1205832+ 120572120573)

2 the equilibriumwholesale price119908lowast is increasing in 119888 otherwise the equilibriumwholesale price 119908lowast is decreasing in 119888

Proof From (13) we get the following derivative

120597119908lowast

120597119888

=

4120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (19)

It is easy to know that if 4120573120578 gt (minus1205831+ 1205832+ 120572120573)

2 theequilibrium wholesale price 119908lowast is increasing in 119888 otherwisethe equilibrium wholesale price 119908lowast is decreasing in 119888

Proposition 8 The equilibrium optional service level 119910lowast isincreasing in 119886

1 1198862 1205832 and 120572 but decreasing in 120583

1 120578 and 119888

Proof From (12) it is easily got that

119910lowast=

(119908lowastminus 119888) (minus120583

1+ 1205832+ 120572120573)

2120578

(20)

From (20) we can see that when 1205831is increasing minus120583

1+

1205832+120572120573 is decreasing and is greater than zero And because119908lowast

is decreasing in 1205831 when 120583

1is increasing119908lowastminus119888 is decreasing

and is greater than zero Hence the equilibrium optionalservice level 119910lowast is decreasing in 120583

1 And the equilibrium

wholesale price 119908lowast is increasing in 1205832and 120572

Mathematical Problems in Engineering 5

From (20) we can see that when 1198861is increasing 119908lowast is

increasing so 119908lowast minus 119888 is increasing and is greater than zeroHence the equilibrium optional service level 119910lowast is decreasingin 1198861 And the equilibriumwholesale price119908lowast is increasing in

1198862From (14) we can see that when 120578 is increasing the

equilibrium optional service level 119910lowast is decreasing in 120578

Proposition 9 (i) If 6120578 gt 120572(minus1205831+ 1205832+ 120572120573) the equilibrium

retail price 119901lowast is increasing in 1198861and 119886

2 otherwise the

equilibrium retail price 119901lowast is decreasing in 1198861and 119886

2 (ii) If

2120573120578 gt (minus1205831+ 1205832)(minus1205831+ 1205832+ 120572120573) the equilibrium retail price

119901lowast is increasing in 119888 otherwise the equilibrium retail price 119901lowast

is decreasing in 119888

Proof From (15) it is easily got that

120597119901lowast

1205971198861

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

120597119901lowast

1205971198862

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(21)

It is easy to know that if 6120578 gt 120572(minus1205831+ 1205832+ 120572120573)

the equilibrium retail price 119901lowast is increasing in 1198861and 119886

2

otherwise the equilibrium retail price 119901lowast is decreasing in 1198861

and 1198862

From (15) it is easily got that

120597119901lowast

120597119888

=

2120573120578 minus (minus1205831+ 1205832) (minus1205831+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (22)

It is easy to know that if 2120573120578 gt (minus1205831+1205832)(minus1205831+1205832+120572120573)

the equilibrium retail price 119901lowast is increasing in 119888 otherwisethe equilibrium retail price 119901lowast is decreasing in 119888

5 The Optional Service Level Decidedby the Retailer

In this part the optional service level 119910 is determined by theretailer the retailer also determines the retail price 119901 and thewholesale price 119908 is determined by the manufacturer

Hence the manufacturer profit function is

Π119872= (119908 minus 119888) (119889

1(119901 119910) + 119889

2(119901 119910)) (23)

Then the profit function of retailer can be formulated bythe following

Π119877= (119901 minus 119908) 119889

1(119901 119910) + (119901 + 120572119910 minus 119908) 119889

2(119901 119910)

minus

1205781199102

2

(24)

In the above equation the first item is sales revenue andthe second is the cost of the optional service as Tsay andAgrawal [10] does

Based on (1) (23) and (24) the general optimizationmodel can be described as follows

maxΠ119872= (119908 minus 119888) (119886

1+ 1198862minus 2120573119901 minus 120583

1119910 + 1205832119910) (25)

subject to

maxΠ119877= (119901 minus 119908) (119886

1minus 120573119901 minus 120583

1119910)

+ (119901 + 120572119910 minus 119908) (1198862minus 120573119901 + 120583

2119910) minus

1205781199102

2

(26)

We can see that based on the analysis stated above in thismodel the manufacturer determines the wholesale price 119908and the retailer determines the retail price 119901 and the optionalservice level 119910 respectively

6 Results and Discussions forthe Retailer Case

We now derive chain membersrsquo optimal strategies by maxi-mizing their profits in the above general optimization model

Firstly the following parameters hold

Lemma 10 The inequalities 4120573(120578 minus 21205721205832) gt (minus120572120573 minus 120583

1+ 1205832)2

and 21205721205832lt 120578 hold for any practical problem

Proof As the retailerrsquos operation problem its profit musthave a maximum value in any practical case Consider thefollowing derivatives

120597Π119877

120597119901

= 1198861+ 1198862minus 4120573119901 + 2120573119908 minus 120572120573119910 minus 120583

1119910 + 1205832119910

120597Π119877

120597119910

= 1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 + 2120572120583

2119910

minus 120578119910

(

1205972Π119877

1205971199012

1205972Π119877

120597119901120597119910

1205972Π119877

120597119910120597119901

1205972Π119877

1205971199102

)

= (

minus4120573 minus120572120573 minus 1205831+ 1205832

minus120572120573 minus 1205831+ 1205832

21205721205832minus 120578

)

(27)

From the existence condition of extreme function formultivariable function there must be

Δ3= 2120572120583

2minus 120578 lt 0

Δ4= 4120573 (120578 minus 2120572120583

2) minus (minus120572120573 minus 120583

1+ 1205832)2

gt 0

(28)

Therefore the lemma holds

6 Mathematical Problems in Engineering

Proposition 11 For any given 119908 the retailer best responsefunctions satisfy

1199011015840=

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

1199101015840=

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(29)

Proof Let 120597Π119877120597119910 = 120572119886

2minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 +

21205721205832119910 minus 120578119910 = 0 we can get

119910 =

1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901

120578 minus 21205721205832

(30)

Bring it into 120597Π119877120597119901 = 119886

1+1198862minus4120573119901+2120573119908minus120572120573119910minus120583

1119910+

1205832119910 Making it zero we can get the retailerrsquos best response

decisions119901 =

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

(31)

It is easy to get that

119910 =

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(32)

Among them119860 = 1198861+1198862 120590 = 120583

1minus1205832minus120572120573 120585 = minus120583

1+1205832

andΔ4= 4120573120577minus120582

2 that is 120577 = 120578minus21205721205832 120582 = minus120572120573minus120583

1+1205832

Paying attention to the impact of wholesale price on theretailerrsquos best response function we derive the first derivativesof 1199011015840 with respect to wholesale price and the first derivativesof 1199101015840 with respect to wholesale price that is 120597119901120597119908 = (2120573120577 +120582(1205831minus 1205832))Δ4and 120597119910120597119908 = 2120573120577(120583

1minus 1205832minus 120572120573)120577Δ

4lt 0 We

can easily see that the higher the wholesale price the higherthe optional service level However the effect of the wholesaleprice on the retail price is not very evident

Theorem 12 The profit function Π119872(119908) is concave with

respect to the wholesale price 119908

Proof Bringing the retailer best response functions into themanufacturer profit function Π

119872(119908) we can get

Π119872= (119908 minus 119888)

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

(33)

Consider the following derivatives

120597Π119872

120597119908

=

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

+ (119908 minus 119888)

2120573 (120582120577 (minus1205831+ 1205832) minus 2120573120577

2) + (120583

1minus 1205832) (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582)

120577Δ4

(34)

1205972Π119872

1205971199082=

minus812057321205772minus 8120572120577120573

2(minus1205831+ 1205832)

120577Δ4

lt 0 (35)

Because 1205972Π1198721205971199082= (minus8120573

21205772minus8120572120577120573

2(minus1205831+1205832))120577Δ

4lt

0 Π119872(119908) is concave with respect to the wholesale price 119908

that is to say there does exist wholesale price119908lowast maximizingΠ119872(119908)

Proposition 13 The manufacturer equilibrium decisions sat-isfy

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(36)

Proof Let (34) be equal to zero we can get

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(37)

Bringing 119908lowast into (31) and (32) we can get the retailerequilibrium decisions

119901lowast=

81205732120577 (120577 + 120572120585) (120577119860 + 120582120572119886

2) + (2120573120577 minus 120582120585) (119860Δ

4120577 + (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888))

8Δ4(12057321205772+ 120572120577120573

2120585)

119910lowast=

812057321205772(120577 + 120572120585) (120582119860 + 4120572120573119886

2) + 2120573119860Δ

41205772120590 + 2120573120577120590 (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 4120573

212057721205851205902(1205721198862+ 120585119888)

8120577Δ4(12057321205772+ 120572120577120573

2120585)

(38)

Mathematical Problems in Engineering 7

95

100

105

110

115

120

125

130

80 10060 80 1006080 1006080 1006080 10060122

124

126

128

130

132

134

136

138

0

05

1

15

2

25

3

35

200

400

600

800

1000

1200

1400

100

200

300

400

500

600

700

800

Figure 1 Effects of unit production cost

7 Numerical Simulation

In this section we will analyze the changes of differentparameters of supply chain members by numerical simu-lation The objective of numerical analyses is to examinehow the equilibrium profit functions change when the valuesof parameters change And we compare the equilibriumdecisions and the equilibrium profits under two models Thebaseline parameters are set as follows 119888 = 80 119886

1= 100

1198862= 50 120572 = 02 120583

1= 01 120583

2= 04 120578 = 3 and

120573 = 05 In order to compare directly we will use a diagramshowing the curves of two models The red curve representsthe equilibrium decisions and the equilibrium profits wherethe optional service level is determined by the manufacturerand the blue curve represents the equilibrium decisions andthe equilibrium profits where the optional service level isdetermined by the retailer

The ordinate axes in Figures 1ndash8 from left to rightrespectively represent the equilibrium wholesale price theequilibrium retail price the equilibrium optional servicelevel the manufacturerrsquos profit and the retailerrsquos profit

Figure 1 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119888It is easy to see that as 119888 increases the retail price andthe wholesale price are linearly increasing but the optimaloptional service level and the profits of all chain membersare linearly decreasing In Figure 1 119888 is from 50 to 100 Wecan see that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer

Figure 2 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120578 Itis easy to see that as 120578 increases in the model where theoptional service level is determined by manufacturer theretail price and the wholesale price and the optional servicelevel are decreasing but in the model where the optionalservice level is determined by the retailer the retail price andthe wholesale price are increasing and the optional servicelevel is decreasing In Figure 2 120578 is from 2 to 6 We cansee that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer For 120583

1 there are similar results

It is represented in Figure 3 The only difference is thatthe retailerrsquos profit in the model where the optional servicelevel is determined by the retailer firstly decreases and thenincreases

Figure 4 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119886

1

It is easy to see that as 1198861increases all of the equilibrium

decisions and the profits of all chain members are increasingIn Figure 4 119886

1is from 50 to 200 We can see that the

optimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by manufacturer are greater than theoptimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by the retailer But the retailerrsquos profitin the model where the optional service level is determinedby the manufacturer is less than the retailerrsquos profit in themodel where the optional service level is determined by the

8 Mathematical Problems in Engineering

50 1000 50 1000 50 1000 50 1000 50 1000106

107

108

109

110

111

112

113

114

115

116

128

129

130

131

132

133

134

0

05

1

15

2

25

3

35

4

580

585

590

595

600

605

610

615

620

625

300

320

340

360

380

400

420

440

460

480

500

Figure 2 Effects of optional service cost coefficient

02 040 02 040 02 040 02 040 02 040109

110

111

112

113

114

115

116

1295

130

1305

131

1315

132

1325

133

minus1

minus05

0

05

1

15

2

25

602

604

606

608

610

612

614

616

618

620

622

300

320

340

360

380

400

420

Figure 3 Effects of service elastic coefficient of the first category of customers

retailer Moreover the optimal optional service level in themodel where the optional service level is determined by themanufacturer is firstly less and then greater than the optimaloptional service level in the model where the optional servicelevel is determined by the retailer For 119886

2 there are similar

results It is represented in Figure 5 The only difference isthat the optimal optional service level in the model wherethe optional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer

Figure 6 indicates how the equilibrium decisions and theprofits of all chainmembers changewith respect to120573 It is easyto see that as 120573 increases all of the equilibrium decisions andthe profits of all chain members are decreasing In Figure 6120573 is from 01 to 05 We can see that the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model where the

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Proposition 4 The manufacturer equilibrium decisions sat-isfy

119908lowast=

2 (1198861+ 1198862) 120578 + 4120573120578119888 minus 119888 (minus120583

1+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

119910lowast=

(1198861+ 1198862minus 2120573119888) (minus120583

1+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

(11)

Proof With the best response function for retail price 1199011015840 in(7) themanufacturerrsquos best response function for the productcan be achieved by solving 119908lowast 119910lowast isin argmaxΠ

119872(119908 119910)

Let 120597Π119872(119909 119910)120597119910 = (119908 minus 119888)((minus120583

1+ 1205832+ 120572120573)2) minus 120578119910 = 0

we can get

119910 =

(119908 minus 119888) (minus1205831+ 1205832+ 120572120573)

2120578

(12)

Bring it into 120597Π119872120597119908 = (119886

1+ 1198862minus 1205831119910 + 120583

2119910 minus 4120573119908 +

120572120573119910)2 + 120573119888 Making it zero we can get the manufacturerequilibrium decisions

119908lowast=

2 (1198861+ 1198862) 120578 + 4120573120578119888 minus 119888 (minus120583

1+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (13)

119910lowast=

(1198861+ 1198862minus 2120573119888) (minus120583

1+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (14)

Bringing 119908lowast 119910lowast into 1199011015840 = (1198861+ 1198862minus 1205831119910 + 120583

2119910 +

2120573119908 minus 120572120573119910)4120573 we can get the retailer equilibrium deci-sions

119901lowast=

6120578 (1198861+ 1198862) minus 2119888 (minus120583

1+ 1205832+ 120572120573) (minus120583

1+ 1205832) minus 120572 (119886

1+ 1198862) (minus1205831+ 1205832+ 120572120573) + 4120573120578119888

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(15)

Proposition 5 The equilibrium wholesale price119908lowast is increas-ing in 119886

1 1198862 1205832 and 120572 but decreasing in 120583

1

Proof From (13) we get the following derivatives

120597119908lowast

1205971198861

=

2120578

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2gt 0

120597119908lowast

1205971198862

=

2120578

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2gt 0

(16)

From (16) we can see that the equilibriumwholesale price119908lowastis increasing in 119886

1 1198862

From (13) it is easily got that

119908lowast=

2 (1198861+ 1198862) 120578 minus 4120573120578119888

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2+ 119888 (17)

From (17) we can see that when 1205831is increasing the

equilibrium wholesale price 119908lowast is decreasing in 1205831 And the

equilibriumwholesale price119908lowast is increasing in 1205832and 120572

Proposition 6 If 119888 gt 4(1198861+ 1198862) the equilibrium wholesale

price119908lowast is increasing in 120578 otherwise the equilibriumwholesaleprice 119908lowast is decreasing in 120578

Proof From (13) we get the following derivative

120597119908lowast

120597120578

=

(119888 minus 4 (1198861+ 1198862)) (minus120583

1+ 1205832+ 120572120573)

2

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

2 (18)

It is easy to know that if 119888 gt 4(1198861+ 1198862) the equilibrium

wholesale price 119908lowast is increasing in 120578 otherwise the equilib-rium wholesale price 119908lowast is decreasing in 120578

Proposition 7 If 4120573120578 gt (minus1205831+ 1205832+ 120572120573)

2 the equilibriumwholesale price119908lowast is increasing in 119888 otherwise the equilibriumwholesale price 119908lowast is decreasing in 119888

Proof From (13) we get the following derivative

120597119908lowast

120597119888

=

4120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (19)

It is easy to know that if 4120573120578 gt (minus1205831+ 1205832+ 120572120573)

2 theequilibrium wholesale price 119908lowast is increasing in 119888 otherwisethe equilibrium wholesale price 119908lowast is decreasing in 119888

Proposition 8 The equilibrium optional service level 119910lowast isincreasing in 119886

1 1198862 1205832 and 120572 but decreasing in 120583

1 120578 and 119888

Proof From (12) it is easily got that

119910lowast=

(119908lowastminus 119888) (minus120583

1+ 1205832+ 120572120573)

2120578

(20)

From (20) we can see that when 1205831is increasing minus120583

1+

1205832+120572120573 is decreasing and is greater than zero And because119908lowast

is decreasing in 1205831 when 120583

1is increasing119908lowastminus119888 is decreasing

and is greater than zero Hence the equilibrium optionalservice level 119910lowast is decreasing in 120583

1 And the equilibrium

wholesale price 119908lowast is increasing in 1205832and 120572

Mathematical Problems in Engineering 5

From (20) we can see that when 1198861is increasing 119908lowast is

increasing so 119908lowast minus 119888 is increasing and is greater than zeroHence the equilibrium optional service level 119910lowast is decreasingin 1198861 And the equilibriumwholesale price119908lowast is increasing in

1198862From (14) we can see that when 120578 is increasing the

equilibrium optional service level 119910lowast is decreasing in 120578

Proposition 9 (i) If 6120578 gt 120572(minus1205831+ 1205832+ 120572120573) the equilibrium

retail price 119901lowast is increasing in 1198861and 119886

2 otherwise the

equilibrium retail price 119901lowast is decreasing in 1198861and 119886

2 (ii) If

2120573120578 gt (minus1205831+ 1205832)(minus1205831+ 1205832+ 120572120573) the equilibrium retail price

119901lowast is increasing in 119888 otherwise the equilibrium retail price 119901lowast

is decreasing in 119888

Proof From (15) it is easily got that

120597119901lowast

1205971198861

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

120597119901lowast

1205971198862

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(21)

It is easy to know that if 6120578 gt 120572(minus1205831+ 1205832+ 120572120573)

the equilibrium retail price 119901lowast is increasing in 1198861and 119886

2

otherwise the equilibrium retail price 119901lowast is decreasing in 1198861

and 1198862

From (15) it is easily got that

120597119901lowast

120597119888

=

2120573120578 minus (minus1205831+ 1205832) (minus1205831+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (22)

It is easy to know that if 2120573120578 gt (minus1205831+1205832)(minus1205831+1205832+120572120573)

the equilibrium retail price 119901lowast is increasing in 119888 otherwisethe equilibrium retail price 119901lowast is decreasing in 119888

5 The Optional Service Level Decidedby the Retailer

In this part the optional service level 119910 is determined by theretailer the retailer also determines the retail price 119901 and thewholesale price 119908 is determined by the manufacturer

Hence the manufacturer profit function is

Π119872= (119908 minus 119888) (119889

1(119901 119910) + 119889

2(119901 119910)) (23)

Then the profit function of retailer can be formulated bythe following

Π119877= (119901 minus 119908) 119889

1(119901 119910) + (119901 + 120572119910 minus 119908) 119889

2(119901 119910)

minus

1205781199102

2

(24)

In the above equation the first item is sales revenue andthe second is the cost of the optional service as Tsay andAgrawal [10] does

Based on (1) (23) and (24) the general optimizationmodel can be described as follows

maxΠ119872= (119908 minus 119888) (119886

1+ 1198862minus 2120573119901 minus 120583

1119910 + 1205832119910) (25)

subject to

maxΠ119877= (119901 minus 119908) (119886

1minus 120573119901 minus 120583

1119910)

+ (119901 + 120572119910 minus 119908) (1198862minus 120573119901 + 120583

2119910) minus

1205781199102

2

(26)

We can see that based on the analysis stated above in thismodel the manufacturer determines the wholesale price 119908and the retailer determines the retail price 119901 and the optionalservice level 119910 respectively

6 Results and Discussions forthe Retailer Case

We now derive chain membersrsquo optimal strategies by maxi-mizing their profits in the above general optimization model

Firstly the following parameters hold

Lemma 10 The inequalities 4120573(120578 minus 21205721205832) gt (minus120572120573 minus 120583

1+ 1205832)2

and 21205721205832lt 120578 hold for any practical problem

Proof As the retailerrsquos operation problem its profit musthave a maximum value in any practical case Consider thefollowing derivatives

120597Π119877

120597119901

= 1198861+ 1198862minus 4120573119901 + 2120573119908 minus 120572120573119910 minus 120583

1119910 + 1205832119910

120597Π119877

120597119910

= 1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 + 2120572120583

2119910

minus 120578119910

(

1205972Π119877

1205971199012

1205972Π119877

120597119901120597119910

1205972Π119877

120597119910120597119901

1205972Π119877

1205971199102

)

= (

minus4120573 minus120572120573 minus 1205831+ 1205832

minus120572120573 minus 1205831+ 1205832

21205721205832minus 120578

)

(27)

From the existence condition of extreme function formultivariable function there must be

Δ3= 2120572120583

2minus 120578 lt 0

Δ4= 4120573 (120578 minus 2120572120583

2) minus (minus120572120573 minus 120583

1+ 1205832)2

gt 0

(28)

Therefore the lemma holds

6 Mathematical Problems in Engineering

Proposition 11 For any given 119908 the retailer best responsefunctions satisfy

1199011015840=

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

1199101015840=

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(29)

Proof Let 120597Π119877120597119910 = 120572119886

2minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 +

21205721205832119910 minus 120578119910 = 0 we can get

119910 =

1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901

120578 minus 21205721205832

(30)

Bring it into 120597Π119877120597119901 = 119886

1+1198862minus4120573119901+2120573119908minus120572120573119910minus120583

1119910+

1205832119910 Making it zero we can get the retailerrsquos best response

decisions119901 =

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

(31)

It is easy to get that

119910 =

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(32)

Among them119860 = 1198861+1198862 120590 = 120583

1minus1205832minus120572120573 120585 = minus120583

1+1205832

andΔ4= 4120573120577minus120582

2 that is 120577 = 120578minus21205721205832 120582 = minus120572120573minus120583

1+1205832

Paying attention to the impact of wholesale price on theretailerrsquos best response function we derive the first derivativesof 1199011015840 with respect to wholesale price and the first derivativesof 1199101015840 with respect to wholesale price that is 120597119901120597119908 = (2120573120577 +120582(1205831minus 1205832))Δ4and 120597119910120597119908 = 2120573120577(120583

1minus 1205832minus 120572120573)120577Δ

4lt 0 We

can easily see that the higher the wholesale price the higherthe optional service level However the effect of the wholesaleprice on the retail price is not very evident

Theorem 12 The profit function Π119872(119908) is concave with

respect to the wholesale price 119908

Proof Bringing the retailer best response functions into themanufacturer profit function Π

119872(119908) we can get

Π119872= (119908 minus 119888)

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

(33)

Consider the following derivatives

120597Π119872

120597119908

=

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

+ (119908 minus 119888)

2120573 (120582120577 (minus1205831+ 1205832) minus 2120573120577

2) + (120583

1minus 1205832) (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582)

120577Δ4

(34)

1205972Π119872

1205971199082=

minus812057321205772minus 8120572120577120573

2(minus1205831+ 1205832)

120577Δ4

lt 0 (35)

Because 1205972Π1198721205971199082= (minus8120573

21205772minus8120572120577120573

2(minus1205831+1205832))120577Δ

4lt

0 Π119872(119908) is concave with respect to the wholesale price 119908

that is to say there does exist wholesale price119908lowast maximizingΠ119872(119908)

Proposition 13 The manufacturer equilibrium decisions sat-isfy

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(36)

Proof Let (34) be equal to zero we can get

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(37)

Bringing 119908lowast into (31) and (32) we can get the retailerequilibrium decisions

119901lowast=

81205732120577 (120577 + 120572120585) (120577119860 + 120582120572119886

2) + (2120573120577 minus 120582120585) (119860Δ

4120577 + (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888))

8Δ4(12057321205772+ 120572120577120573

2120585)

119910lowast=

812057321205772(120577 + 120572120585) (120582119860 + 4120572120573119886

2) + 2120573119860Δ

41205772120590 + 2120573120577120590 (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 4120573

212057721205851205902(1205721198862+ 120585119888)

8120577Δ4(12057321205772+ 120572120577120573

2120585)

(38)

Mathematical Problems in Engineering 7

95

100

105

110

115

120

125

130

80 10060 80 1006080 1006080 1006080 10060122

124

126

128

130

132

134

136

138

0

05

1

15

2

25

3

35

200

400

600

800

1000

1200

1400

100

200

300

400

500

600

700

800

Figure 1 Effects of unit production cost

7 Numerical Simulation

In this section we will analyze the changes of differentparameters of supply chain members by numerical simu-lation The objective of numerical analyses is to examinehow the equilibrium profit functions change when the valuesof parameters change And we compare the equilibriumdecisions and the equilibrium profits under two models Thebaseline parameters are set as follows 119888 = 80 119886

1= 100

1198862= 50 120572 = 02 120583

1= 01 120583

2= 04 120578 = 3 and

120573 = 05 In order to compare directly we will use a diagramshowing the curves of two models The red curve representsthe equilibrium decisions and the equilibrium profits wherethe optional service level is determined by the manufacturerand the blue curve represents the equilibrium decisions andthe equilibrium profits where the optional service level isdetermined by the retailer

The ordinate axes in Figures 1ndash8 from left to rightrespectively represent the equilibrium wholesale price theequilibrium retail price the equilibrium optional servicelevel the manufacturerrsquos profit and the retailerrsquos profit

Figure 1 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119888It is easy to see that as 119888 increases the retail price andthe wholesale price are linearly increasing but the optimaloptional service level and the profits of all chain membersare linearly decreasing In Figure 1 119888 is from 50 to 100 Wecan see that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer

Figure 2 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120578 Itis easy to see that as 120578 increases in the model where theoptional service level is determined by manufacturer theretail price and the wholesale price and the optional servicelevel are decreasing but in the model where the optionalservice level is determined by the retailer the retail price andthe wholesale price are increasing and the optional servicelevel is decreasing In Figure 2 120578 is from 2 to 6 We cansee that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer For 120583

1 there are similar results

It is represented in Figure 3 The only difference is thatthe retailerrsquos profit in the model where the optional servicelevel is determined by the retailer firstly decreases and thenincreases

Figure 4 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119886

1

It is easy to see that as 1198861increases all of the equilibrium

decisions and the profits of all chain members are increasingIn Figure 4 119886

1is from 50 to 200 We can see that the

optimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by manufacturer are greater than theoptimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by the retailer But the retailerrsquos profitin the model where the optional service level is determinedby the manufacturer is less than the retailerrsquos profit in themodel where the optional service level is determined by the

8 Mathematical Problems in Engineering

50 1000 50 1000 50 1000 50 1000 50 1000106

107

108

109

110

111

112

113

114

115

116

128

129

130

131

132

133

134

0

05

1

15

2

25

3

35

4

580

585

590

595

600

605

610

615

620

625

300

320

340

360

380

400

420

440

460

480

500

Figure 2 Effects of optional service cost coefficient

02 040 02 040 02 040 02 040 02 040109

110

111

112

113

114

115

116

1295

130

1305

131

1315

132

1325

133

minus1

minus05

0

05

1

15

2

25

602

604

606

608

610

612

614

616

618

620

622

300

320

340

360

380

400

420

Figure 3 Effects of service elastic coefficient of the first category of customers

retailer Moreover the optimal optional service level in themodel where the optional service level is determined by themanufacturer is firstly less and then greater than the optimaloptional service level in the model where the optional servicelevel is determined by the retailer For 119886

2 there are similar

results It is represented in Figure 5 The only difference isthat the optimal optional service level in the model wherethe optional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer

Figure 6 indicates how the equilibrium decisions and theprofits of all chainmembers changewith respect to120573 It is easyto see that as 120573 increases all of the equilibrium decisions andthe profits of all chain members are decreasing In Figure 6120573 is from 01 to 05 We can see that the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model where the

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

From (20) we can see that when 1198861is increasing 119908lowast is

increasing so 119908lowast minus 119888 is increasing and is greater than zeroHence the equilibrium optional service level 119910lowast is decreasingin 1198861 And the equilibriumwholesale price119908lowast is increasing in

1198862From (14) we can see that when 120578 is increasing the

equilibrium optional service level 119910lowast is decreasing in 120578

Proposition 9 (i) If 6120578 gt 120572(minus1205831+ 1205832+ 120572120573) the equilibrium

retail price 119901lowast is increasing in 1198861and 119886

2 otherwise the

equilibrium retail price 119901lowast is decreasing in 1198861and 119886

2 (ii) If

2120573120578 gt (minus1205831+ 1205832)(minus1205831+ 1205832+ 120572120573) the equilibrium retail price

119901lowast is increasing in 119888 otherwise the equilibrium retail price 119901lowast

is decreasing in 119888

Proof From (15) it is easily got that

120597119901lowast

1205971198861

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

120597119901lowast

1205971198862

=

6120578 minus 120572 (minus1205831+ 1205832+ 120572120573)

2 (8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2

)

(21)

It is easy to know that if 6120578 gt 120572(minus1205831+ 1205832+ 120572120573)

the equilibrium retail price 119901lowast is increasing in 1198861and 119886

2

otherwise the equilibrium retail price 119901lowast is decreasing in 1198861

and 1198862

From (15) it is easily got that

120597119901lowast

120597119888

=

2120573120578 minus (minus1205831+ 1205832) (minus1205831+ 1205832+ 120572120573)

8120573120578 minus (minus1205831+ 1205832+ 120572120573)

2 (22)

It is easy to know that if 2120573120578 gt (minus1205831+1205832)(minus1205831+1205832+120572120573)

the equilibrium retail price 119901lowast is increasing in 119888 otherwisethe equilibrium retail price 119901lowast is decreasing in 119888

5 The Optional Service Level Decidedby the Retailer

In this part the optional service level 119910 is determined by theretailer the retailer also determines the retail price 119901 and thewholesale price 119908 is determined by the manufacturer

Hence the manufacturer profit function is

Π119872= (119908 minus 119888) (119889

1(119901 119910) + 119889

2(119901 119910)) (23)

Then the profit function of retailer can be formulated bythe following

Π119877= (119901 minus 119908) 119889

1(119901 119910) + (119901 + 120572119910 minus 119908) 119889

2(119901 119910)

minus

1205781199102

2

(24)

In the above equation the first item is sales revenue andthe second is the cost of the optional service as Tsay andAgrawal [10] does

Based on (1) (23) and (24) the general optimizationmodel can be described as follows

maxΠ119872= (119908 minus 119888) (119886

1+ 1198862minus 2120573119901 minus 120583

1119910 + 1205832119910) (25)

subject to

maxΠ119877= (119901 minus 119908) (119886

1minus 120573119901 minus 120583

1119910)

+ (119901 + 120572119910 minus 119908) (1198862minus 120573119901 + 120583

2119910) minus

1205781199102

2

(26)

We can see that based on the analysis stated above in thismodel the manufacturer determines the wholesale price 119908and the retailer determines the retail price 119901 and the optionalservice level 119910 respectively

6 Results and Discussions forthe Retailer Case

We now derive chain membersrsquo optimal strategies by maxi-mizing their profits in the above general optimization model

Firstly the following parameters hold

Lemma 10 The inequalities 4120573(120578 minus 21205721205832) gt (minus120572120573 minus 120583

1+ 1205832)2

and 21205721205832lt 120578 hold for any practical problem

Proof As the retailerrsquos operation problem its profit musthave a maximum value in any practical case Consider thefollowing derivatives

120597Π119877

120597119901

= 1198861+ 1198862minus 4120573119901 + 2120573119908 minus 120572120573119910 minus 120583

1119910 + 1205832119910

120597Π119877

120597119910

= 1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 + 2120572120583

2119910

minus 120578119910

(

1205972Π119877

1205971199012

1205972Π119877

120597119901120597119910

1205972Π119877

120597119910120597119901

1205972Π119877

1205971199102

)

= (

minus4120573 minus120572120573 minus 1205831+ 1205832

minus120572120573 minus 1205831+ 1205832

21205721205832minus 120578

)

(27)

From the existence condition of extreme function formultivariable function there must be

Δ3= 2120572120583

2minus 120578 lt 0

Δ4= 4120573 (120578 minus 2120572120583

2) minus (minus120572120573 minus 120583

1+ 1205832)2

gt 0

(28)

Therefore the lemma holds

6 Mathematical Problems in Engineering

Proposition 11 For any given 119908 the retailer best responsefunctions satisfy

1199011015840=

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

1199101015840=

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(29)

Proof Let 120597Π119877120597119910 = 120572119886

2minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 +

21205721205832119910 minus 120578119910 = 0 we can get

119910 =

1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901

120578 minus 21205721205832

(30)

Bring it into 120597Π119877120597119901 = 119886

1+1198862minus4120573119901+2120573119908minus120572120573119910minus120583

1119910+

1205832119910 Making it zero we can get the retailerrsquos best response

decisions119901 =

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

(31)

It is easy to get that

119910 =

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(32)

Among them119860 = 1198861+1198862 120590 = 120583

1minus1205832minus120572120573 120585 = minus120583

1+1205832

andΔ4= 4120573120577minus120582

2 that is 120577 = 120578minus21205721205832 120582 = minus120572120573minus120583

1+1205832

Paying attention to the impact of wholesale price on theretailerrsquos best response function we derive the first derivativesof 1199011015840 with respect to wholesale price and the first derivativesof 1199101015840 with respect to wholesale price that is 120597119901120597119908 = (2120573120577 +120582(1205831minus 1205832))Δ4and 120597119910120597119908 = 2120573120577(120583

1minus 1205832minus 120572120573)120577Δ

4lt 0 We

can easily see that the higher the wholesale price the higherthe optional service level However the effect of the wholesaleprice on the retail price is not very evident

Theorem 12 The profit function Π119872(119908) is concave with

respect to the wholesale price 119908

Proof Bringing the retailer best response functions into themanufacturer profit function Π

119872(119908) we can get

Π119872= (119908 minus 119888)

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

(33)

Consider the following derivatives

120597Π119872

120597119908

=

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

+ (119908 minus 119888)

2120573 (120582120577 (minus1205831+ 1205832) minus 2120573120577

2) + (120583

1minus 1205832) (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582)

120577Δ4

(34)

1205972Π119872

1205971199082=

minus812057321205772minus 8120572120577120573

2(minus1205831+ 1205832)

120577Δ4

lt 0 (35)

Because 1205972Π1198721205971199082= (minus8120573

21205772minus8120572120577120573

2(minus1205831+1205832))120577Δ

4lt

0 Π119872(119908) is concave with respect to the wholesale price 119908

that is to say there does exist wholesale price119908lowast maximizingΠ119872(119908)

Proposition 13 The manufacturer equilibrium decisions sat-isfy

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(36)

Proof Let (34) be equal to zero we can get

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(37)

Bringing 119908lowast into (31) and (32) we can get the retailerequilibrium decisions

119901lowast=

81205732120577 (120577 + 120572120585) (120577119860 + 120582120572119886

2) + (2120573120577 minus 120582120585) (119860Δ

4120577 + (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888))

8Δ4(12057321205772+ 120572120577120573

2120585)

119910lowast=

812057321205772(120577 + 120572120585) (120582119860 + 4120572120573119886

2) + 2120573119860Δ

41205772120590 + 2120573120577120590 (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 4120573

212057721205851205902(1205721198862+ 120585119888)

8120577Δ4(12057321205772+ 120572120577120573

2120585)

(38)

Mathematical Problems in Engineering 7

95

100

105

110

115

120

125

130

80 10060 80 1006080 1006080 1006080 10060122

124

126

128

130

132

134

136

138

0

05

1

15

2

25

3

35

200

400

600

800

1000

1200

1400

100

200

300

400

500

600

700

800

Figure 1 Effects of unit production cost

7 Numerical Simulation

In this section we will analyze the changes of differentparameters of supply chain members by numerical simu-lation The objective of numerical analyses is to examinehow the equilibrium profit functions change when the valuesof parameters change And we compare the equilibriumdecisions and the equilibrium profits under two models Thebaseline parameters are set as follows 119888 = 80 119886

1= 100

1198862= 50 120572 = 02 120583

1= 01 120583

2= 04 120578 = 3 and

120573 = 05 In order to compare directly we will use a diagramshowing the curves of two models The red curve representsthe equilibrium decisions and the equilibrium profits wherethe optional service level is determined by the manufacturerand the blue curve represents the equilibrium decisions andthe equilibrium profits where the optional service level isdetermined by the retailer

The ordinate axes in Figures 1ndash8 from left to rightrespectively represent the equilibrium wholesale price theequilibrium retail price the equilibrium optional servicelevel the manufacturerrsquos profit and the retailerrsquos profit

Figure 1 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119888It is easy to see that as 119888 increases the retail price andthe wholesale price are linearly increasing but the optimaloptional service level and the profits of all chain membersare linearly decreasing In Figure 1 119888 is from 50 to 100 Wecan see that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer

Figure 2 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120578 Itis easy to see that as 120578 increases in the model where theoptional service level is determined by manufacturer theretail price and the wholesale price and the optional servicelevel are decreasing but in the model where the optionalservice level is determined by the retailer the retail price andthe wholesale price are increasing and the optional servicelevel is decreasing In Figure 2 120578 is from 2 to 6 We cansee that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer For 120583

1 there are similar results

It is represented in Figure 3 The only difference is thatthe retailerrsquos profit in the model where the optional servicelevel is determined by the retailer firstly decreases and thenincreases

Figure 4 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119886

1

It is easy to see that as 1198861increases all of the equilibrium

decisions and the profits of all chain members are increasingIn Figure 4 119886

1is from 50 to 200 We can see that the

optimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by manufacturer are greater than theoptimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by the retailer But the retailerrsquos profitin the model where the optional service level is determinedby the manufacturer is less than the retailerrsquos profit in themodel where the optional service level is determined by the

8 Mathematical Problems in Engineering

50 1000 50 1000 50 1000 50 1000 50 1000106

107

108

109

110

111

112

113

114

115

116

128

129

130

131

132

133

134

0

05

1

15

2

25

3

35

4

580

585

590

595

600

605

610

615

620

625

300

320

340

360

380

400

420

440

460

480

500

Figure 2 Effects of optional service cost coefficient

02 040 02 040 02 040 02 040 02 040109

110

111

112

113

114

115

116

1295

130

1305

131

1315

132

1325

133

minus1

minus05

0

05

1

15

2

25

602

604

606

608

610

612

614

616

618

620

622

300

320

340

360

380

400

420

Figure 3 Effects of service elastic coefficient of the first category of customers

retailer Moreover the optimal optional service level in themodel where the optional service level is determined by themanufacturer is firstly less and then greater than the optimaloptional service level in the model where the optional servicelevel is determined by the retailer For 119886

2 there are similar

results It is represented in Figure 5 The only difference isthat the optimal optional service level in the model wherethe optional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer

Figure 6 indicates how the equilibrium decisions and theprofits of all chainmembers changewith respect to120573 It is easyto see that as 120573 increases all of the equilibrium decisions andthe profits of all chain members are decreasing In Figure 6120573 is from 01 to 05 We can see that the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model where the

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Proposition 11 For any given 119908 the retailer best responsefunctions satisfy

1199011015840=

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

1199101015840=

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(29)

Proof Let 120597Π119877120597119910 = 120572119886

2minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901 +

21205721205832119910 minus 120578119910 = 0 we can get

119910 =

1205721198862minus 120572120573119901 + 120583

1119908 minus 120583

2119908 minus 120583

1119901 + 1205832119901

120578 minus 21205721205832

(30)

Bring it into 120597Π119877120597119901 = 119886

1+1198862minus4120573119901+2120573119908minus120572120573119910minus120583

1119910+

1205832119910 Making it zero we can get the retailerrsquos best response

decisions119901 =

120577 (119860 + 2120573119908) + 120582 (1205721198862+ 1205831119908 minus 120583

2119908)

Δ4

(31)

It is easy to get that

119910 =

120582120577 (119860 + 2120573119908) + 4120573120577 (1205721198862+ 1205831119908 minus 120583

2119908)

120577Δ4

(32)

Among them119860 = 1198861+1198862 120590 = 120583

1minus1205832minus120572120573 120585 = minus120583

1+1205832

andΔ4= 4120573120577minus120582

2 that is 120577 = 120578minus21205721205832 120582 = minus120572120573minus120583

1+1205832

Paying attention to the impact of wholesale price on theretailerrsquos best response function we derive the first derivativesof 1199011015840 with respect to wholesale price and the first derivativesof 1199101015840 with respect to wholesale price that is 120597119901120597119908 = (2120573120577 +120582(1205831minus 1205832))Δ4and 120597119910120597119908 = 2120573120577(120583

1minus 1205832minus 120572120573)120577Δ

4lt 0 We

can easily see that the higher the wholesale price the higherthe optional service level However the effect of the wholesaleprice on the retail price is not very evident

Theorem 12 The profit function Π119872(119908) is concave with

respect to the wholesale price 119908

Proof Bringing the retailer best response functions into themanufacturer profit function Π

119872(119908) we can get

Π119872= (119908 minus 119888)

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

(33)

Consider the following derivatives

120597Π119872

120597119908

=

119860120577Δ4+ (120582120577 (minus120583

1+ 1205832) minus 2120573120577

2) (119860 + 2120573119908) + (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582) (120572119886

2+ 1205831119908 minus 120583

2119908)

120577Δ4

+ (119908 minus 119888)

2120573 (120582120577 (minus1205831+ 1205832) minus 2120573120577

2) + (120583

1minus 1205832) (4120573120577 (minus120583

1+ 1205832) minus 2120573120577120582)

120577Δ4

(34)

1205972Π119872

1205971199082=

minus812057321205772minus 8120572120577120573

2(minus1205831+ 1205832)

120577Δ4

lt 0 (35)

Because 1205972Π1198721205971199082= (minus8120573

21205772minus8120572120577120573

2(minus1205831+1205832))120577Δ

4lt

0 Π119872(119908) is concave with respect to the wholesale price 119908

that is to say there does exist wholesale price119908lowast maximizingΠ119872(119908)

Proposition 13 The manufacturer equilibrium decisions sat-isfy

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(36)

Proof Let (34) be equal to zero we can get

119908lowast

=

119860120577Δ4+ (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888)

812057321205772+ 8120572120577120573

2120585

(37)

Bringing 119908lowast into (31) and (32) we can get the retailerequilibrium decisions

119901lowast=

81205732120577 (120577 + 120572120585) (120577119860 + 120582120572119886

2) + (2120573120577 minus 120582120585) (119860Δ

4120577 + (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 2120573120577120590 (120572119886

2+ 120585119888))

8Δ4(12057321205772+ 120572120577120573

2120585)

119910lowast=

812057321205772(120577 + 120572120585) (120582119860 + 4120572120573119886

2) + 2120573119860Δ

41205772120590 + 2120573120577120590 (120582120577120585 minus 2120573120577

2) (119860 minus 2120573119888) + 4120573

212057721205851205902(1205721198862+ 120585119888)

8120577Δ4(12057321205772+ 120572120577120573

2120585)

(38)

Mathematical Problems in Engineering 7

95

100

105

110

115

120

125

130

80 10060 80 1006080 1006080 1006080 10060122

124

126

128

130

132

134

136

138

0

05

1

15

2

25

3

35

200

400

600

800

1000

1200

1400

100

200

300

400

500

600

700

800

Figure 1 Effects of unit production cost

7 Numerical Simulation

In this section we will analyze the changes of differentparameters of supply chain members by numerical simu-lation The objective of numerical analyses is to examinehow the equilibrium profit functions change when the valuesof parameters change And we compare the equilibriumdecisions and the equilibrium profits under two models Thebaseline parameters are set as follows 119888 = 80 119886

1= 100

1198862= 50 120572 = 02 120583

1= 01 120583

2= 04 120578 = 3 and

120573 = 05 In order to compare directly we will use a diagramshowing the curves of two models The red curve representsthe equilibrium decisions and the equilibrium profits wherethe optional service level is determined by the manufacturerand the blue curve represents the equilibrium decisions andthe equilibrium profits where the optional service level isdetermined by the retailer

The ordinate axes in Figures 1ndash8 from left to rightrespectively represent the equilibrium wholesale price theequilibrium retail price the equilibrium optional servicelevel the manufacturerrsquos profit and the retailerrsquos profit

Figure 1 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119888It is easy to see that as 119888 increases the retail price andthe wholesale price are linearly increasing but the optimaloptional service level and the profits of all chain membersare linearly decreasing In Figure 1 119888 is from 50 to 100 Wecan see that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer

Figure 2 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120578 Itis easy to see that as 120578 increases in the model where theoptional service level is determined by manufacturer theretail price and the wholesale price and the optional servicelevel are decreasing but in the model where the optionalservice level is determined by the retailer the retail price andthe wholesale price are increasing and the optional servicelevel is decreasing In Figure 2 120578 is from 2 to 6 We cansee that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer For 120583

1 there are similar results

It is represented in Figure 3 The only difference is thatthe retailerrsquos profit in the model where the optional servicelevel is determined by the retailer firstly decreases and thenincreases

Figure 4 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119886

1

It is easy to see that as 1198861increases all of the equilibrium

decisions and the profits of all chain members are increasingIn Figure 4 119886

1is from 50 to 200 We can see that the

optimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by manufacturer are greater than theoptimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by the retailer But the retailerrsquos profitin the model where the optional service level is determinedby the manufacturer is less than the retailerrsquos profit in themodel where the optional service level is determined by the

8 Mathematical Problems in Engineering

50 1000 50 1000 50 1000 50 1000 50 1000106

107

108

109

110

111

112

113

114

115

116

128

129

130

131

132

133

134

0

05

1

15

2

25

3

35

4

580

585

590

595

600

605

610

615

620

625

300

320

340

360

380

400

420

440

460

480

500

Figure 2 Effects of optional service cost coefficient

02 040 02 040 02 040 02 040 02 040109

110

111

112

113

114

115

116

1295

130

1305

131

1315

132

1325

133

minus1

minus05

0

05

1

15

2

25

602

604

606

608

610

612

614

616

618

620

622

300

320

340

360

380

400

420

Figure 3 Effects of service elastic coefficient of the first category of customers

retailer Moreover the optimal optional service level in themodel where the optional service level is determined by themanufacturer is firstly less and then greater than the optimaloptional service level in the model where the optional servicelevel is determined by the retailer For 119886

2 there are similar

results It is represented in Figure 5 The only difference isthat the optimal optional service level in the model wherethe optional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer

Figure 6 indicates how the equilibrium decisions and theprofits of all chainmembers changewith respect to120573 It is easyto see that as 120573 increases all of the equilibrium decisions andthe profits of all chain members are decreasing In Figure 6120573 is from 01 to 05 We can see that the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model where the

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

95

100

105

110

115

120

125

130

80 10060 80 1006080 1006080 1006080 10060122

124

126

128

130

132

134

136

138

0

05

1

15

2

25

3

35

200

400

600

800

1000

1200

1400

100

200

300

400

500

600

700

800

Figure 1 Effects of unit production cost

7 Numerical Simulation

In this section we will analyze the changes of differentparameters of supply chain members by numerical simu-lation The objective of numerical analyses is to examinehow the equilibrium profit functions change when the valuesof parameters change And we compare the equilibriumdecisions and the equilibrium profits under two models Thebaseline parameters are set as follows 119888 = 80 119886

1= 100

1198862= 50 120572 = 02 120583

1= 01 120583

2= 04 120578 = 3 and

120573 = 05 In order to compare directly we will use a diagramshowing the curves of two models The red curve representsthe equilibrium decisions and the equilibrium profits wherethe optional service level is determined by the manufacturerand the blue curve represents the equilibrium decisions andthe equilibrium profits where the optional service level isdetermined by the retailer

The ordinate axes in Figures 1ndash8 from left to rightrespectively represent the equilibrium wholesale price theequilibrium retail price the equilibrium optional servicelevel the manufacturerrsquos profit and the retailerrsquos profit

Figure 1 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119888It is easy to see that as 119888 increases the retail price andthe wholesale price are linearly increasing but the optimaloptional service level and the profits of all chain membersare linearly decreasing In Figure 1 119888 is from 50 to 100 Wecan see that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer

Figure 2 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120578 Itis easy to see that as 120578 increases in the model where theoptional service level is determined by manufacturer theretail price and the wholesale price and the optional servicelevel are decreasing but in the model where the optionalservice level is determined by the retailer the retail price andthe wholesale price are increasing and the optional servicelevel is decreasing In Figure 2 120578 is from 2 to 6 We cansee that the equilibrium decisions and the manufacturerprofit where the optional service level is determined by themanufacturer are greater than the equilibrium decisions andthe manufacturer profit where the optional service level isdetermined by the retailer But the retailer profit where theoptional service level is determined by the manufacturer isless than the retailer profit where the optional service level isdetermined by the retailer For 120583

1 there are similar results

It is represented in Figure 3 The only difference is thatthe retailerrsquos profit in the model where the optional servicelevel is determined by the retailer firstly decreases and thenincreases

Figure 4 indicates how the equilibrium decisions andthe profits of all chain members change with respect to 119886

1

It is easy to see that as 1198861increases all of the equilibrium

decisions and the profits of all chain members are increasingIn Figure 4 119886

1is from 50 to 200 We can see that the

optimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by manufacturer are greater than theoptimal retail price and the optimal wholesale price and themanufacturer profit in the model where the optional servicelevel is determined by the retailer But the retailerrsquos profitin the model where the optional service level is determinedby the manufacturer is less than the retailerrsquos profit in themodel where the optional service level is determined by the

8 Mathematical Problems in Engineering

50 1000 50 1000 50 1000 50 1000 50 1000106

107

108

109

110

111

112

113

114

115

116

128

129

130

131

132

133

134

0

05

1

15

2

25

3

35

4

580

585

590

595

600

605

610

615

620

625

300

320

340

360

380

400

420

440

460

480

500

Figure 2 Effects of optional service cost coefficient

02 040 02 040 02 040 02 040 02 040109

110

111

112

113

114

115

116

1295

130

1305

131

1315

132

1325

133

minus1

minus05

0

05

1

15

2

25

602

604

606

608

610

612

614

616

618

620

622

300

320

340

360

380

400

420

Figure 3 Effects of service elastic coefficient of the first category of customers

retailer Moreover the optimal optional service level in themodel where the optional service level is determined by themanufacturer is firstly less and then greater than the optimaloptional service level in the model where the optional servicelevel is determined by the retailer For 119886

2 there are similar

results It is represented in Figure 5 The only difference isthat the optimal optional service level in the model wherethe optional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer

Figure 6 indicates how the equilibrium decisions and theprofits of all chainmembers changewith respect to120573 It is easyto see that as 120573 increases all of the equilibrium decisions andthe profits of all chain members are decreasing In Figure 6120573 is from 01 to 05 We can see that the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model where the

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

50 1000 50 1000 50 1000 50 1000 50 1000106

107

108

109

110

111

112

113

114

115

116

128

129

130

131

132

133

134

0

05

1

15

2

25

3

35

4

580

585

590

595

600

605

610

615

620

625

300

320

340

360

380

400

420

440

460

480

500

Figure 2 Effects of optional service cost coefficient

02 040 02 040 02 040 02 040 02 040109

110

111

112

113

114

115

116

1295

130

1305

131

1315

132

1325

133

minus1

minus05

0

05

1

15

2

25

602

604

606

608

610

612

614

616

618

620

622

300

320

340

360

380

400

420

Figure 3 Effects of service elastic coefficient of the first category of customers

retailer Moreover the optimal optional service level in themodel where the optional service level is determined by themanufacturer is firstly less and then greater than the optimaloptional service level in the model where the optional servicelevel is determined by the retailer For 119886

2 there are similar

results It is represented in Figure 5 The only difference isthat the optimal optional service level in the model wherethe optional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer

Figure 6 indicates how the equilibrium decisions and theprofits of all chainmembers changewith respect to120573 It is easyto see that as 120573 increases all of the equilibrium decisions andthe profits of all chain members are decreasing In Figure 6120573 is from 01 to 05 We can see that the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail priceand the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model where the

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

100 2000 100 2000 100 2000 100 2000 100 200080

90

100

110

120

130

140

150

160

170

80

100

120

140

160

180

200

220

0

1

2

3

4

5

6

7

8

0

500

1000

1500

2000

2500

3000

3500

4000

0

500

1000

1500

2000

2500

Figure 4 Effects of market base of the first category of customers

50 1000 50 1000 50 1000 50 1000 50 100080

90

100

110

120

130

140

150

90

100

110

120

130

140

150

160

170

180

minus2

minus1

0

1

2

3

4

5

6

7

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0

200

400

600

800

1000

1200

1400

Figure 5 Effects of market base of the second category of customers

optional service level is determined by the manufacturer isless than the retailerrsquos profit in the model where the optionalservice level is determined by the retailer Moreover theoptimal optional service level in themodel where the optionalservice level is determined by the manufacturer is firstlygreater and then less than the optimal optional service levelin the model where the optional service level is determinedby the retailer

Figure 7 indicates how the equilibrium decisions and theprofits of all chain members change with respect to 120583

2 It is

easy to see that as1205832increases all of the equilibriumdecisions

and the profits of all chain members in the model where theoptional service level is determined by the manufacturer areincreasing and the optimal optional service level and theretailerrsquos profit in the model where the optional service levelis determined by the retailer are increasing In Figure 7 120583

2

is from 03 to 08 We can see that the optimal retail priceand the optimal wholesale price and themanufacturerrsquos profitin the model where the optional service level is determinedby the manufacturer are greater than the optimal retail price

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

050 050 050 050 050100

150

200

250

300

350

400

450

100

200

300

400

500

600

700

0

5

10

15

20

25

0

2000

4000

6000

8000

10000

12000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 6 Effects of price sensitive coefficient

05 10 05 10 05 10 05 10 05 1090

95

100

105

110

115

120

122

124

126

128

130

132

134

136

0

1

2

3

4

5

6

7

8

300

350

400

450

500

550

600

650

300

400

500

600

700

800

900

1000

Figure 7 Effects of service elastic coefficient of the second category of customers

and the optimal wholesale price and the manufacturer profitin the model where the optional service level is determinedby the retailer But the retailerrsquos profit in the model wherethe optional service level is determined by the manufactureris less than the retailerrsquos profit in the model where theoptional service level is determined by the retailer Moreoverthe optimal optional service level in the model where theoptional service level is determined by the manufactureris firstly greater and then less than the optimal optionalservice level in the model where the optional service level isdetermined by the retailer For 120572 there are similar results Itis represented in Figure 8 The difference is that the optimaloptional service level in the model where the optional servicelevel is determined by the retailer and the retailerrsquos profit in

the model where the optional service level is determined bythe manufacturer are decreasing and the optimal optionalservice level in the model where the optional service levelis determined by the manufacturer is firstly greater than theoptimal optional service level in themodel where the optionalservice level is determined by the retailer

8 Conclusions and the FutureResearch Directions

Considering the existence of price and service competitionthe supply chain equilibrium decision problem has becomean important problem in the field of management science

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

050 050 050 050 05098

100

102

104

106

108

110

112

114

116

118

124

125

126

127

128

129

130

131

132

133

134

15

2

25

3

35

4

480

500

520

540

560

580

600

620

640

250

300

350

400

450

500

550

600

650

700

Figure 8 Effects of service elastic coefficient

Optional service is one aspect of the service and alsobecomes more and more common in real life In this paperwe assume that the basic service level is known and weuse a general demand function to describe the impact ofproduct price and optional service level on market demandin such a competitive environment Moreover a Stackelbergmodel structure is proposed between the retailer and themanufacturer in a two-echelon supply chain We obtain theoptimal wholesale price the optimal retail price and theoptimal optional service level when the optional service levelis respectively determined by the manufacturer and theretailer Finally in order to compare the two models in theoptimal equilibrium decision the profit of the members andthe whole supply chain we carried out numerical simulationwhich has certain guiding significance to the enterprisemanagement

However our research leaves several unanswered ques-tions for future research In this paper we assume thatthe demand function is determined while in real life themarket demand is often random Assuming that the demandfunction is random is a good extension of this paper Wealso add some constraints to some known parameters Theother limitation of the model is that we cannot comparethe equilibrium decisions of the two models in theorybecause of the complexity of the equilibriumdecision Hencehow to simplify the model is another important researchdirection

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (NSFC) Research Fund no 71302112

References

[1] S Kameshwaran N Viswanadham and V Desai ldquoBundlingand pricing of product with after-sale servicesrdquo InternationalJournal of Operational Research vol 6 no 1 pp 92ndash109 2009

[2] J-C Lu Y-C Tsao and C Charoensiriwath ldquoCompetitionunder manufacturer service and retail pricerdquo Economic Mod-elling vol 28 no 3 pp 1256ndash1264 2011

[3] T Xiao and D Yang ldquoPrice and service competition of supplychains with risk-averse retailers under demand uncertaintyrdquoInternational Journal of Production Economics vol 114 no 1 pp187ndash200 2008

[4] M A Cohen and S Whang ldquoCompeting in product andservice a product life-cycle modelrdquo Management Science vol43 no 4 pp 535ndash545 1997

[5] G Li F F Huang T C E Cheng Q Zheng and P Ji ldquoMake-or-buy service capacity decision in a supply chain providingafter-sales servicerdquo European Journal of Operational Researchvol 239 no 2 pp 377ndash388 2014

[6] H Kurata and S-H Nam ldquoAfter-sales service competition ina supply chain optimization of customer satisfaction level orprofit or bothrdquo International Journal of Production Economicsvol 127 no 1 pp 136ndash146 2010

[7] H Kurata and S-H Nam ldquoAfter-sales service competition in asupply chain does uncertainty affect the conflict between profitmaximization and customer satisfactionrdquo International Journalof Production Economics vol 144 no 1 pp 268ndash280 2013

[8] M Xu G Yu and H Zhang ldquoGame analysis in a supplychain with service provisionrdquo Journal of Management Sciencesin China vol 2 no 4 pp 18ndash27 2006

[9] G Iyer ldquoCoordinating channels under price and nonpricecompetitionrdquoMarketing Science vol 17 no 4 pp 338ndash355 1998

[10] A A Tsay and N Agrawal ldquoChannel dynamics under price andservice competitionrdquo Manufacturing and Service OperationsManagement vol 2 no 4 pp 372ndash391 2000

[11] Y Xia and SMGilbert ldquoStrategic interactions between channelstructure and demand enhancing servicesrdquo European Journal ofOperational Research vol 181 no 1 pp 252ndash265 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of