An integrated revenue sharing and quantity discounts contract for coordinating a supply chain dealing with short life-cycle
products
G.Partha Sarathi,S.P.Sarmah ?,M.Jenamani
Department of Industrial Engineering and Management,Indian Institute of Technology Kharagpur,India
a r t i c l e i n f o Article history:
Received 1July 2011
Received in revised form 26November 2013Accepted 3February 2014
Available online 15February 2014Keywords:
Supply chain coordination Contracts
Revenue sharing Quantity discounts
Short life-cycle product
a b s t r a c t
This paper develops a combined contract model for coordinating a two stage supply chain where the demand at the retailer’s end is price sensitive and stock dependent.It has been shown that proposed coordination mechanism achieves perfect coordination and win–win situation for both the members of the supply chain.Further,an extensive sensitivity anal-ysis is performed to examine the impact of various parameters on supply chain perfor-mance.It has been found that stock dependency factor has positive impact on order quantity and subsequently on supply chain performance.The paper has also made a com-parative statics analysis to see the impact of certain parameters on the pricing and replen-ishment policies of the retailer.
ó2014Elsevier Inc.All rights reserved.
1.Introduction
Short life cycle products are characterized by a short selling season after which their value reduce drastically.Fashion apparel,electronic goods,personal computers,toys etc.are some of the examples of short life cycle product.The short selling season with uncertain nature of demand poses many challenges to the members of supply chain (SC)dealing with such type of products.For example,the Korean manufacturers of cellular phones introduce more than 50new models each year,and the average life-cycle of cellular phones in Korea is less than 10months [1].The short life-cycle products on one hand,invite the risks of overstocking and understocking while on the other hand,it provides greater opportunities for higher margins [2].These attractive characteristics of this category of products have prompted the researchers to work in the area of managing supply chains of short life-cycle products.In addition to it,since the selling season of such product is small,it is necessary to ensure the availability of stock in the shelf of the retailer and display of it plays an important role in stimulating the demand.Stock dependent demand has been studied in the inventory literatures for quite some times now (Urban,[3])but only few studies have explored the effect of stock dependency on the performance of the SC and it has motivated us to take up the study on price sensitive stock dependent demand of short life cycle product in the context of SC.
The uncertain nature of demand,faulty planning and poor purchasing practices are some of the reasons that increase the risk of under stoking and over stoking of short life cycle products among the members of the SC.To minimize these risks,various contracts such as buy back (BB),quantity discount (QD),revenue sharing (RS)and quantity ?exibility (QF)contract etc.have been cited as coordination mechanism in the SC literature.For example,Agrawal and Seshadri [4]developed risk https://www.doczj.com/doc/ae10546199.html,/10.1016/j.apm.2014.02.003
0307-904X/ó2014Elsevier Inc.All rights reserved.
?Corresponding author.Tel.:+913222283735.
E-mail addresses:gudipatipartha@https://www.doczj.com/doc/ae10546199.html, (G.Partha Sarathi),spsarmah@iem.iitkgp.ernet.in (S.P.Sarmah).
G.Partha Sarathi et al./Applied Mathematical Modelling38(2014)4120–41364121 free contract for a risk averse retailer under news vendor framework to coordinate the SC.In SC contract literature,BB con-tracts have been widely studied,but in reality,BB contract has certain limitations.First,the physical handling of returns may be impractical in certain situations[5].Second,the retailer may not have enough cash to go for one time procurement of the optimum stock[6].Under such situations it is necessary to explore other contract mechanisms to coordinate the SC.RS is one such mechanism that has gained a lot of attention from both academia and industry[7,1].Under RS contract,the retailer pays the supplier a wholesale price for each unit purchased and a certain percentage of revenue the retailer generates [8].This type of contract is attractive to the retailer and can replace BB contract particularly when the items are expensive [6].There are certain advantages with RS contract over BB contract.First,the physical handling of returns is not required. Second,the retailer obtains goods at a lesser wholesale price and need not have to invest huge capital in procuring the items. It is fond that RS contract is widely employed in video rental industries and e-commerce businesses.
In this paper,using newsvendor framework,a combined RS and QD contract model is developed for single manufacturer and single retailer SC dealing with a short life-cycle product considering demand to be price-sensitive and stock dependent. This paper extends the work of Yao et al.[9].They have studied RS contract using price-setting newsvendor framework.Our work differs from Yao et al.[9]in two respects.First,it captures stock dependency of demand in addition to price-sensitivity. Second,it integrates RS and QD contract.
In Section2,we brie?y review the literature relevant to our work.In Section3,we present the mathematical model devel-oped for coordinating the SC.In Section4,we illustrate the model through a numerical example.We also provide the results of sensitivity analysis on this section.Section5concludes the paper with a brief summary and directions for further research work.
2.Literature review
In this section,we have provided a brief review of literature related to our work.Excellent review on SC contract models have been provided by Tsay et al.[10]and Cachon[11]and readers can read those articles to get idea about different types of coordination models.Earlier,many authors have studied SC coordination considering demand as deterministic or constant. But in real practice,it is not true and demand is found to be stochastic in nature.Stochastic model under news vendor frame-work can be classi?ed as?xed price and price setting.In?xed price case,the market price is considered to be?xed and deter-mines only the optimal order quantity(see[12,13]);whereas in case of price setting,optimal price and order quantity are determined simultaneously(see[9,14–17]).Wei and Choi[18]under mean variance framework,SC coordination is explored through wholesale price and pro?t sharing scheme.Chiu et al.[19]have studied target sales rebate as a coordination.mech-anism for a manufacturer and a risk averse retailer under mean variance framework.Tsay[20]analyzed how sensitivity to risk affects the behaviors and outcomes on both sides of a manufacturer-retailer supply relationship,and how these dynam-ics are altered by a manufacturer return policy.
Modeling of demand is an important component to correctly depict the reality.It is well known that price in?uences de-mand and vice versa.To capture this reality,some researchers have introduced a new concept from the principles of ther-modynamics to model demand and price relationship.When demand is price sensitive,they consider price to be analogous to temperature.In such models,demand is considered as the potential or the driving force that creates the difference in mon-etary value or price and for details of such model,one can refer Jaber et al.[21,22].In recent times,some authors have devel-oped models considering sales effort of the retailer,lead time and stock level at the retailer’s end as a means for enhancing the demand[23–25].
Further,it has been well recognized in the literature that demand of many items at the retailer level is proportional to the amount of inventory displayed[26].The real-life examples include short life-cycle products such as,sugar,spices,clothes, gift cards etc.[27].Speci?cally,the supermarkets,where the products are displayed on the aisles,require inventory control models that take stock level into consideration.To model this situation,mainly two approaches have been adopted in the literature.Demand is expressed as a function of initial stock level or is considered as time dependent stock level[3].Though in the inventory literature,the importance of stock level has been recognized for quiet a long time,yet it has not received adequate attention from SC management researchers.A coordination model considering a two stage SC with an initial stock level dependent demand was developed by Wang and Gerchak[28].A coordination model considering stock dependent de-mand is also developed recently by Zhou et al.[29].
To coordinate a SC,in recent times,combined contract models,i.e.,contracts with two or more coordination mechanisms have been proposed in the literature[30,31].Under?xed price newsvendor framework,Shi and Su[13]developed a com-bined BB and QD contract model and have shown that contract is self-enforcing.Burnetas et al.[30]have pointed out that QD contract combined with other mechanisms such as BB offers tremendous scope for future research.Recently,Güler and Bilgi?[32]have used a mixed BB and RS contract to coordinate an assembly system.The manufacturer shares its sales rev-enue with the suppliers.Aydin and Porteous[33]considered two types of rebate for coordinating a two stage SC,one directly to the consumer and another to the retailer and have shown that no members of the SC always favor only one kind of rebate. Demirag et al.[34]also studied two different types of promotion,retailer incentive scheme and customer rebate policy in a two stage SC setting.They have shown that under market uncertainty condition,combined policy performs better and im-proves the sales and increase the pro?t of the manufacturer.Considering demand as a function of sales effort of the retailer, Taylor[24]has shown that SC coordination can be achieved through a properly designed target rebate and return contract.
4122G.Partha Sarathi et al./Applied Mathematical Modelling38(2014)4120–4136
Recently,Chiu et al.[19]have considered additive and multiplicative price dependent demand and shown that combination of wholesale price,channel rebate,and returns together can coordinate a SC.In channel rebate,manufacturer makes pay-ment to the retailer on the basis of his sales to the end customer.The rebate contract has the effect of motivating retailer to lower prices for increasing sales.Here the retailer has to invest a large amount to procure the items from the supplier. The retailer is entitled to get the sales rebate,once he crosses a threshold limit of sales.The inherent de?ciency of return policy cannot be overcome by combining with the rebate policy.Moreover,if the item is of short cycle and the value is high, the cash starved retailer may be unwilling to go for optimum stock.For such kind of product,RS contract has the biggest advantage of making higher availability of product in the market and in combination with QD;it can improve the perfor-mance of such SC.Also,some authors have used two part tariff contract for coordinating a SC.In two part tariff mechanism, there are two components.One?xed component and another per unit(usage)price.The retailer has to pay the?xed amount ?rst and then to pay the additional charges for the purchase of each unit.Retailer not having enough cash will face dif?culty for single period procurement of short life cycle product.Therefore implementation of two part tariff contract under such situation may not be a feasible proposition.
3.Development of the mathematical model
We have considered a two stage SC consists of single manufacturer and single retailer dealing with short life-cycle prod-ucts.The retailer faces a stock dependent and price-sensitive demand which is uncertain in nature.In this model,we develop a coordinating mechanism combining RS and QD contracts.Though,both BB and RS contracts are designed for managing SC risks,there are some inherent differences between them.In case of BB contract,if there is overstocking,the manufacturer bears the risk and buys back the leftover items.Under RS contract,the manufacturer takes the risk of supplying the goods at a lower wholesale price and gets a fraction of the sales revenue.Owing to low wholesale price,the retailer buys more and the risk of understocking is reduced.However,the retailer bears both overage cost,i.e.,the cost incurred by the retailer when order quantity is more than the realized demand;and underage cost,i.e.,the cost incurred by the retailer when the order quantity is less than the realized demand.
Here,we have considered newsvendor model for coordination through contracts.Considering demand D and order quan-tity Q,the expected number of sales,the expected number of leftover items and the expected shortages at the end of the selling season are given by E?mineQ;DT ,E?eQàDTt and E?eDàQTt ,respectively.
Following notation are used in the development of the model:
Decision variables
p Unit sale price at the retailer end
w Unit wholesale price
Q Order quantity of the retailer
Other notation
D Demand rate of the item
w0Unit wholesale price under PO contract
h Unit cost of overage
s Unit cost of underage
m Unit cost of manufacturing
e Random component o
f demand
l Mean demand
r Fraction of revenue kept by the retailer
feáTProbability density function(pdf)of the random demand
FeáTCumulative distribution function(cdf)of the random demand
P x
Expected pro?t of member x under situation y y
(Here x can be M;R or T where M stands for manufacturer,R for retailer and T for the total system;and y can be c or dc where c stands for centralized system,and dc for decentralized system.)
For the development of the model,following assumptions are made:
Both the members of the SC are risk neutral.
The information of market is a common knowledge.
The manufacturer has unlimited capacity and offers product to the retailer before the start of the selling season.
The market demand of end item is uncertain,price sensitive and dependent on initial stock.
The cost parameters follow some straightforward assumptions to ensure internal consistency:
(i)p>w0>m>0.
(ii)06r61.
(iii)h P0;s P0.
3.1.Demand model
The price-sensitivity,stock dependency and uncertainty of demand are captured in the presented demand model.We model demand as a function of initial stock,i.e.,the order quantity and demand is considered as a linear additive function as follows:
D ?a àbp tcQ te :
3.2.Flow of payments in the supply chain
In order to de?ne the pro?t functions of the individual member and that of the entire SC,it is important to understand the ?ow of payments in the supply chain.As shown in Fig.1,under PO contract,the manufacturer produces the product spend-ing a manufacturing cost m per unit and sells the product to the retailer at a wholesale price w per unit.In response to the offer made by the manufacturer,the retailer determines order quantity Q .The retailer sells the products to the end customer at a price p per unit.No returns are allowed in case of leftover items,and no sharing of revenue is permitted when operating under PO contract.The manufacturer spends an amount mQ for producing the products and receives an amount w 0Q from the retailer.Since D is the demand,the expected number of sales at the retailer end can be derived by E ?min eQ ;D T .It means that the retailer receives an amount pE ?min eQ ;D T from the customers due to the proceeds of the sale.The expected number of leftover items and the expected number of shortages are given by E ?eQ àD Tt and E ?eD àQ Tt ,respectively.The retailer pays a penalty of hE ?eQ àD Tt and sE ?eD àQ Tt in case of overage and shortage,respectively.The ?ow of payments with PO contract is shown in Fig.1.
The retailer receives an amount pE ?min eQ ;D T from the proceeds of the sale.When RS contract is in place,the retailer re-tains rpE ?min eQ ;D T and remits the balance e1àr TpE ?min eQ ;D T to the manufacturer as per the contract.The retailer pays a penalty of hE ?eQ àD Tt and sE ?eD àQ Tt in case of overage and shortage,respectively.The ?ow of payments under RS con-tract is shown in Fig.2.
In this study,the combination of QD and RS contracts are used.The combination of QD with RS can be represented in the same way as in Fig.2where the ew ;Q Tpair will be determined as per the model outcome.3.3.Methodology for solution
To analyze and design the proposed combined contract,the following steps are followed:Step 1:Analysis of decentralized non-coordinated SC operating under PO contract
In the ?rst step,the optimal pricing and order strategies of the retailer is determined and based on these optimum values,the pro?ts of SC members are computed.Step 2:Design of an equivalent RS contract
In this step,manufacturer offers RS contract in place of PO contract and accordingly,the manufacturer reduces the whole-sale price to encourage the retailer to opt for RS contract.Step 3:Incorporation of QD contract into the existing RS contract
In the existing RS contract,the wholesale price is discounted to entice the retailer to choose for higher order quantity i.e.,equal to centralized order quantity.
Step 4:Testing the effectiveness of the combined contract
model
G.Partha Sarathi et al./Applied Mathematical Modelling 38(2014)4120–41364123
In this step,to measure the effectiveness of the proposed combined contract model,numerical example is carried out and pro?t of the SC under combined contract and those under PO only contract are compared.The impact of various parameters on SC coordination is investigated by performing extensive sensitivity analysis.3.4.Non-coordinated decentralized setting
Here the problem is analyzed as a Stackelberg game where the manufacturer acts as the leader and the retailer acts as the follower.As a ?rst mover,manufacturer announces his pricing policy and based on it retailer decides his ordering and pricing decisions simultaneously.Manufacturer optimizes his pricing policy considering the rational behavior of the retailer and maximizes his pro?t while the objective of the retailer is to maximize his own pro?t corresponding to the pricing policy of the manufacturer.The equilibrium point that speci?es the pricing and ordering policies of the retailer is found out by the solution of a two-stage non-cooperative game.Retailer’s pro?t can be expressed as
P R ?Expected sales revenue àExpected purchase cost àExpected overage cost àExpected underage cost :
The mathematical expression is as follows
P R ?pE ?min eQ ;D T àw 0Q àhE ?eQ àD Tt àsE ?eD àQ Tt :
e1T
Similarly,pro?t of the manufacturer is given by
P M ?ew 0àm TQ :
e2T
The pro?t function of the SC is given by
P T ?pE ?min eQ ;D T àmQ àhE ?eQ àD Tt àsE ?eD àQ Tt :
e3T
Introducing a transformational variable z ?Q àea àbp tcQ T,the order quantity expression becomes
Q ?
a àbp tz
1àc
:
e4T
Substituting D ?a àbp tcQ te and Q ?ea àbp tcQ Ttz ,we can have the following identities:
E ?eQ àD Tt ?z àl tH ez T;
E ?eD àQ Tt ?H ez Tand subsequently,
E ?min eQ ;D T ?a àbp tc
a àbp tz
1àc
tl àH ez T:Since for two whole numbers x and y ,the minimum of x and y is given by min ex ;y T?x àex ày Ttwhere x tdenotes the maximum of 0and x .
Here H ez Tdenotes the loss function and is given by H ez T?R B
z ex àz Tf ex Tdx .
Therefore,the retailer’s pro?t function given in Eq.(1)can be written as follows.
P R ?p ?a àbp tcQ tel àH ez TT àw 0Q àh ?z àl tH ez T às ?H ez T
?p ?a àbp tel àH ez TT tQ epc àw 0Tàh ?z àl tH ez T às ?H ez T
?p ?a àbp tel àH ez TT t
a àbp tz
epc àw 0Tàh ?z àl tH ez T às ?H ez T :e5T
Also,the pro?t function of the retailer given in Eq.(5)can be rearranged as follows
P R ?p ?a àbp tel àH ez TT ta àbp tz
1àc
pc àw 0eTàh z àel àH ez TT? às l àel àH ez TT? :
Since E ?min ez ;e T ?l àH ez T(see Appendix A ),the above equation can be written
as
4124G.Partha Sarathi et al./Applied Mathematical Modelling 38(2014)4120–4136
P R ?p ea àbp tE ?min ee ;z T Tt
a àbp tz
1àc
epc àw 0Tàh ez àE ?min ee ;z T Tàs el àE ?min ee ;z T T:After rearranging the terms,the pro?t function of the retailer can be expressed as
P R ?p ea àbp Tt
a àbp tz
epc àw 0TàHz às l tep th ts TE ?min ee ;z T :e6T
Now,the objective function of the retailer is
Maximise P R :
e7T
Taking the ?rst and second order derivatives of P R with respect to z and p ,one gets
@P R @z ?pc àw 01àc
àh tep th ts T?1àF ez T ;e8T
@2P R
@z 2
?àep th ts Tf ez T;e9T
@P R @p
?a à2bp tbw 0tzc te1àc Tel àH ez TT
1àc ;
e10T
@2P R @p
2?à2b
1àc ;e11T
The pro?t equation of retailer,P R is concave in nature with respect to p for a speci?c value of z .Under such situation,
retailer’s objective function given in 7can be converted to an optimization problem over a single variable z by ?rst solving for an optimal value of p as a function of z and after that,one can substitute the results back into P R .Exploring the resulting optimal trajectory to maximize P R ep ?;z ;e T,one can obtain z ?.From Eq.(10),one gets
p ??p ez T?
1
2b
?a tbw 0tzc te1àc Tel àH ez TT :Substituting p ??p ez Tin Eq.(7),the optimization problem becomes a maximization problem over a single variable z .
Alternately,one can ?nd the optimal solution of the retailer’s problem by determining p ?and z ?by solving the following two response functions iteratively,which are derived from the Eqs.(8)and (10),respectively.
z ?F
à1
p ts e1àc Tàw 0
e1àc Tep ts th T
!
;
e12T
p ?
1
2b
?a tbw 0tzc te1àc Tel àH ez TT :e13T
The optimum order quantity of the retailer under decentralized scenario can be determined by substituting the values of p ?
and z ?in Eq.(4).The optimal order quantity of the retailer in a non-coordinated scenario is given as
Q ?
?a àbp ?
tz ?
:
e14T
3.5.Decentralized setting with revenue sharing contract
In this section,we have considered a decentralized supply chain where the manufacturer intends to reduce the wholesale price to entice the retailer to order more and in turn shares the sales revenue generated by the retailer to just compensate the reduction in wholesale price of the manufacturer so that the pro?ts of the SC members under this contract are same as that under PO contract.It is assumed that the manufacturer reduces the wholesale price from w 0to w rs to entice the retailer to share his revenue.Under decentralized setting,the expected pro?t of the retailer under RS contract is given by
P R ?P R eQ ?Q ;r ;w ?w rs T?rpE ?min eQ ;D T àw rs Q àhE ?eQ àD Tt àsE ?eD àQ Tt :
e15T
Correspondingly,the pro?t of the manufacturer under RS contract in decentralized setting is given by
P M ?P M eQ ?Q ;r ;w ?w rs T?e1àr TpE ?min eQ ;D T àw rs Q àmQ :e16T
The total system pro?t under RS contract in decentralized setting is the sum of the pro?t of the retailer and the pro?t of the manufacturer.
G.Partha Sarathi et al./Applied Mathematical Modelling 38(2014)4120–41364125
Proposition 1.To facilitate revenue sharing under decentralized system,the manufacturer reduces the wholesale price from w 0to w rs ,where,w rs is given by
w rs ?w 0à
e1àr TpE ?min eQ dc ;D T
Q dc
:
e17T
Proof.Under decentralized system,the manufacturer will reduce the wholesale price and share revenue generated by the retailer only when his pro?t is not suffered due to RS contract.Mathematically,this constraint is expressed as
P M eQ ?Q dc ;r ;w ?w rs TàP M eQ ?Q dc ;r ?1;w ?w 0TP 0;
e1àr Tp dc E ?min eQ dc ;D T tw rs Q dc àmQ dc ?w 0Q dc àmQ dc P 0;e1àr Tp dc E ?min eQ dc ;D T P ew 0àw rs TQ dc ;w rs P w 0à
e1àr Tp dc E ?min eQ dc ;D T
Q dc
:
In order to incorporate revenue sharing in the decentralized setting,the wholesale price should be decreased from w 0to
w rs .It is apparent from the above inequality that w rs min ?w 0àe1àr Tp dc E ?min eQ dc ;D T
Q dc
is the minimum wholesale price that the man-ufacturer can offer to the retailer to accommodate RS contract.This RS contract with w ?w rs min and revenue sharing fraction r is equivalent to PO contract as it provides the same pro?ts as that of PO contract.In the next section,centralized setting is considered.h 3.6.Centralized setting
Centralized setting is equivalent to a situation where the supplier and retailers are considered to be parts of a vertically integrated ?rm with a single controlling body taking the decision to maximize the overall SC pro?t.Therefore,the total supply chain pro?t is the sum of the pro?ts of both the members as shown below
Max P T ?P R tP M ;e18T
P T ?pE ?min eQ ;D T àmQ àhE ?eQ àD Tt àsE ?eD àQ Tt
?p ea àbp tcQ tE ?min ez ;e T TàmQ àhE ?ez àe Tt àsE ?ee àz Tt ?p ea àbp tE ?min ez ;e T Tàepc àm TQ àhE ?ez àe Tt àsE ?ee àz Tt :
Finally,after simpli?cation,the total pro?t becomes
P T ?p ea àbp tl àH ez TTàepc àm T
a àbp tz
àh ez àl tH ez TTàs eH ez TT:e19T
Again,the centralized system pro?t is given by
P T ?p ea àbp tl àH ez TTàepc àm T
a àbp tz
àh ez àl tH ez TTàs eH ez TT?p ea àbp tE ?min ez ;e T Tàepc àm T
a àbp tz
1àc àh ez àE ?min ez ;e T Tàs el àE ?min ez ;e T T;P T ?p ea àbp Tàepc àm Ta àbp tz
àhz às l tep th ts TE ?min ez ;e T :
e20T
Taking the ?rst and second order derivatives of P T with respect to z and p ,one gets
@P T @z
?
pc àm 1àc
àh tep th ts T?1àF ez T ;e21T
@2P T
@z 2
?àep th ts Tf ez T;e22T@P T @p
?a à2bp tbm tzc te1àc Tel àH ez TT
1àc ;
e23T
4126G.Partha Sarathi et al./Applied Mathematical Modelling 38(2014)4120–4136
@2P T @p 2
?à2b
1àc ;e24T
Again,here P R is concave in p for a speci?c value of z .Therefore,objective function presented in 18can be changed to an
optimization problem over a single variable z by initially solving it for an optimal value of p as a function of z and then one can substitute the results back into P R .Finding the resulting optimal trajectory to maximize P R ep ?;z ;e T,one can get z ?.From the Eq.(17),one gets
p ??p ez T?
1
?a tbw 0tzc te1àc Tel àH ez TT :Substituting p ??p ez Tinto 18,the optimization problem becomes a maximization problem over a single variable z .
Alternately,as discussed earlier,the optimum solution under centralized can be obtained by solving the following two response functions iteratively,which are derived from Eqs.(21)and (23),respectively.
z ?F
à1
p ts e1àc Tàm
!
;
e25T
p ?1
2b
a tbm tzc te1àc Tel àH ez TT!:e26T
Substituting these optimum values denoted by p ?c and z ?
c in the Eq.(4),one can obtain the optimum centralize
d order quan-tity.Therefore,th
e optimum order quantity under centralized setting is given by
Q ?c
?a àbp ?
c tz ?c
1àc
:
e27T
Comparing order quantities under centralized and decentralized settings using the Eqs.(14)and (27),one can easily estab-lish that when Q c >Q dc as w 0>m .Therefore,the manufacturer can use quantity discount mechanism to entice the retailer to opt for higher order quantity,i.e.,Q c .
Finally,the optimum retail price and order quantity for centralized SC operating under price-sensitive and stock dependent demand environment are given by the following equations (see Appendix C ).
p c ?12b
a tbm tz c àe1àc TE ?ez c àe Tt ?
?;e28TQ c ?
a àbm tz c 2e1àc T
t12E ez c àe Tt?
?;e29T
where z c ?F à1p
ts e1àc Tàm
h i
.In the next section we perform comparative statics analysis to investigate the impact of various factors on the pricing and replenishment decisions under centralized setting using the Eqs.(28)and (29)and derive the following https://www.doczj.com/doc/ae10546199.html,parative statics analysis
Comparative statics is the determination of the changes in the endogenous variables of a model that will result from a change in the exogenous variables or parameters of that model.It is a powerful tool for establishing theoretical deductions.These deductions can be determined by simply differentiating the ?rst order conditions with respect to parameters.Proposition 2.Retail price increases with stock dependency factor under centralized setting.
Proof.It follows from the Eq.(28)that
@p c ?@1ea tbm tz c àe1àc TE ?ez c àe Tt T !?1@z c à@ee1àc TE ?ez c àe Tt
T !
?
12b @z c @c àe1àc T@E ?ez c àe Tt @c tE ?ez c àe Tt !?12b @z c @c 1àe1àc T@E ?ez c àe Tt @z c tE ?ez c àe Tt
!?
12b @z c @c
e1àe1àc TF ez c TTtE ?ez c àe Tt
!:Since
@z c >0,F ez c T<1and E ?ez c àe Tt >0(see Appendixes D and E ),it follows that
@p c >0.h
Proposition 3.Order quantity increases with stock dependency factor under centralized setting.Proof.It follows from the Eq.(29)that
G.Partha Sarathi et al./Applied Mathematical Modelling 38(2014)4120–41364127
@Q c @c ?
aàbm
2e1àcT2
t
1
2e1àcT2
e1àcT
@z c
@c
àz ceà1T
"#
t
1
2
@E?ez càeTt
@c
?
aàbm
2e1àcT2
t
1
2e1àcT2
e1àcT
@z c
@c
tz c
"#
t
1
2
@E?ez càeTt
@z c
@z c
@c
:
Hence,@Q c
@c >0,since@z c
@c
>0(see Appendix D).h
Proposition4.Retail price decreases with price-sensitivity factor under centralized setting. Proof.It follows from the Eq.(28)that
@p
c?@1eatbmtz
c àe1àcTE?ez càeTt T
?@atz c
t
m
à
e1àcT
E?ez càeTt
?àeatz cT
2b2
t
e1àcTE?ez càeTt
2b2
?
àaàz cte1àcTE?ez càeTt
2b2
:
Since z c>E?ez càeTt ,it follows that@p c<0(see Appendixes D and E).h
Proposition5.Order quantity decreases with price-sensitivity factor under centralized setting.
Proof.It follows from the Eq.(29)that@Q c
@b ?àm
2e1àcT
<0.h
Proposition6.Retail price increases with primary demand(market base)under centralized setting. Proof.It follows from the Eq.(28)that@p c?1>0.h
Proposition7.Order quantity increases with primary demand(market base)under centralized setting.
Proof.It follows from the Eq.(29)that@Q c
@a ?1
2e1àcT
>0.h
Proposition8.Order quantity increases with retail price under centralized setting. Proof.It follows from the Eqs.(28)and(29)that
@Q c @p
c ?
1
2e1àcT
@z c
@p
c
t
1
2
@
@p
c
eE?ez càeTt T;
@Q c @p
c ?
1
2e1àcT
@z c
@p
c
t
1
2
@eE?ez càeTt T
@z c
@z c
@p
c
;
@Q c @p
c ?
@z c
@p
c
1
2e1àcT
t
1
2
@eE?ez càeTt T
@z c
:
Since@z c
@p c >0and@eE?ez càeTt T
@z c
>0,it follows that@Q c
@p c
>0.h
Proposition9.Order quantity decreases with manufacturing cost under centralized setting. Proof.It follows from the Eq.(29)that
@Q c @m ?
1
2e1àcT
àbt
@z c
@m
t
1
2
@E?ez càeTt
@m
?
1
2e1àcT
àbt
@z c
@m
t
1
2
@E?ez càeTt
@z c
@z c
@m
:
Since@z c
@m <0,it follows that@Q c
@m
<0(see Appendix D).h
4128G.Partha Sarathi et al./Applied Mathematical Modelling38(2014)4120–4136
3.7.Coordination under decentralized setting–incorporation of quantity discounts into revenue sharing contract
It is worth mentioning here that the retailer will show his interest in raising his order quantity provided he receives dis-count for ordering larger quantity.In other words,the retailer will opt for centralized order quantity provided his pro?t is more than or equal to that under decentralized order quantity.In the same line,the manufacturer will be interested for cen-tralized order quantity provided his pro?t is higher than or equal to that under decentralized-no revenue sharing scenario. Looking at the problem from the perspective of both the members,one can determine the upper and lower bound of whole-sale price for coordination(for complete derivation see Appendix B).
w06
1
Q c
er p c E?mineQ c;DT àp dc E?mineQ dc;DT
? tw rs Q dcàh E?eQ càDTt àE?eQ dcàDTt
??
às EeDàQ cTt
??
àEeDàQ dcTt
??
??
T
and
W0P
1
c
?meQ càQ dcTtw rs Q dcte1àrT?p dc E?mineQ dc;DT àp c E?mineQ c;DT : One can determine the wholesale price rangeew min;w maxTwhere,
w min?
1
Q c
emeQ càQ dcTtw rs Q dcte1àrT?p dc E?mineQ dc;DT àp c E?mineQ c;DT Te30T
and
w max?
1
Q c
r?p
c
E?mineQ c;DT àp
dc
E?mineQ dc;DT tw rs Q dcàh?E?eQ càDTt àE?eQ dcàDTt às?E?eDàQ cTt àE?eDàQ dcTt àá
:
e31T
Any wholesale price chosen from the feasible range will not only coordinate the SC but also allocate the pro?ts arbitrarily to both the members of the SC.
The retailer will be interested to choose the centralized order quantity provided the retailer gets some form of incentives from the manufacturer.Therefore,the manufacturer offers incentive in the form of quantity discounts to encourage the re-tailer to opt for the centralized order quantity.Let w00be the new reduced wholesale price offered by the manufacturer to the retailer so that the retailer is interested for the centralized order quantity.The expected pro?t of the retailer under central-ized order quantity can be written as
P R
c
?P ReQ?Q c;r;w?w00T?rp c E?mineQ?c;DT àw00Q?càhE?eQ?càDTt àsE?eDàQ?cTt :e32TThe expected pro?t of the manufacturer can be written as
P M
c
?P MeQ?Q c;r;w?w00T?e1àrTp c E?mineQ?c;DT tw00Q?càmQ?ce33TThe total expected pro?t under centralized order quantity is obtained by the sum of the pro?t of the retailer and the man-ufacturer.The coordination bene?t can be computed by assuming that the manufacturer charges the maximum wholesale price under coordinated setting.In that case,w?w max,and the manufacturer will receive all the bene?ts leaving the pro?t of the retailer unaffected from the initial situation.The increase in total pro?t due to coordination is equal to the increase in the pro?t of the manufacturer only.Under this situation,the expected pro?t of the retailer is given as P R
c
?P ReQ?Q c;r;w?w maxT?rp c E?mineQ c;DT àw max Q càhE?eQ càDTt àsE?eDàQ cTt :e34TAnd,the expected pro?t of the manufacturer is given by
P M
c
?P MeQ?Q c;r;w?w maxT?e1àrTp c E?mineQ c;DT tw max Q càmQ c:e35TThe total system expected pro?t is the sum of the pro?t of the retailer and the manufacturer and the increase in pro?t of the manufacturer is equal to the increase in total system pro?t due to coordination.This extra pro?t can be shared through certain pro?t sharing mechanism which is agreeable to both the parties.The model under consideration achieves perfect coordination and leads to a win–win situation for both the entities of the SC.
4.Numerical example
Here the coordination model discussed above is illustrated through a numerical example.The SC comprises of one man-ufacturer and one retailer and the demand at the retailer’s end is assumed to be stock dependent and it is modeled as: D?200à25pt0:1Qte where,e is uniformly distributed in the range?0;10 .The cost parameters are given below: w0?3:25;m?1;s?0:25;h?0:25;r?0:65:
To determine the solution for the joint pricing problem and perform sensitivity analysis,all the equations are coded in MATLAB7.
G.Partha Sarathi et al./Applied Mathematical Modelling38(2014)4120–41364129
4130G.Partha Sarathi et al./Applied Mathematical Modelling38(2014)4120–4136
Table1
Optimum values.
Optimum values Decentralized setting Centralized setting
p? 5.70 4.60
z? 4.798.34
Q?69.21103.59
Table2
Pro?ts under different contract model settings.
Contract model Pro?t in rupees
P R P M P T
Price-only contract162.40155.72318.12
Benchmark RS contract162.40155.72318.12
Combined RS and QD contract*162.40194.06356.46
Combined RS and QD contract**181.57174.89356.46
*In this case,all the coordination bene?t goes to the manufacturer.
**In this case,both retailer and manufacturer are equally powerful and the coordination bene?t is shared
equally.
https://www.doczj.com/doc/ae10546199.html,parison of contracts under different settings
Here four settings of the SC are considered and they are:decentralized SC with PO contract,decentralized SC with RS con-tract that yields the same pro?ts as that of PO contract,decentralized SC with combined RS and QD contract and?nally, decentralized coordinated SC with combined RS and QD contract.The optimum values of retail price and order quantity under both decentralized and centralized setting are determined and results are shown in Table1,and the corresponding pro?ts are given in Table2.
Using Eqs.(12)–(14),the optimal retail price and order quantity under PO contract under decentralized non-coordinated setting are determined and are found to be Rs.5.7and69.21units,respectively(see Table1)and the pro?ts of the retailer, the manufacturer and the total SC are Rs.162.40,155.72and318.12,respectively(see Table2).The manufacturer wishes to offer RS contract with a reduced wholesale price to encourage the retailer and in turn shares a certain fraction of revenue generated by the retailer.If r?0:65is the agreed fraction of revenue retained by the retailer then the rest0.35fraction of revenue is given to the manufacturer.With this RS agreement,using the Eq.(17),the manufacturer decreases the whole-sale price from Rs.3.25to Rs.1.288and in turn shares the revenue with r?0:65.The pro?ts of SC members under this scheme are same as that of PO contract.
Similarly,the optimum retail price and order quantity under centralized setting are determined using Eqs.(25)–(27).Ta-ble1presents the optimum values under decentralized and centralized settings and the total pro?ts under both the situa-tions are found to be Rs.318.12and Rs.356.46,respectively.
To improve the SC pro?ts,the manufacturer wishes to adopt QD contract by which the centralized order quantity can be offered at a discounted wholesale https://www.doczj.com/doc/ae10546199.html,ing the Eqs.(30)and(31),the wholesale price range0.9458–1.3159is determined.When the wholesale price is reduced to0.9458then all the coordination bene?t goes to the retailer and when it is discounted to1.3159entire bene?t of coordination goes to the manufacturer.To quantify the bene?t of coor-dination,we assume that the manufacturer discounts to Rs.1.3159taking all the bene?t of coordination and leaving the retailer’s pro?t unaffected.The coordination bene?t can be shared equally by the manufacturer and the retailer.The pro?ts of both the retailer and the manufacturer after equal division of the coordination bene?t are shown in the last row of the Table2.
The results shown in Table2reveal that after coordination,the SC pro?t is improved by12%.When equal bargaining power is assumed,the retailer’s pro?t is improved by11.8%and the manufacturer pro?t is improved by12.31%by the combined RS and QD contract.It is observed that both the members of SC are better off by following combined contract.
4.2.Sensitivity analysis
Sensitivity analysis is carried out to study the impact of stock dependency,price-sensitivity and demand uncertainty on SC performance.While coordination bene?t is de?ned as the improvement in SC pro?t,SC performance is de?ned as the per-centage improvement in SC pro?t due to coordination.We have varied one parameter while keeping all other parameters constant to perform sensitivity analysis.
Table3
Impact of stock dependency factor.
c w rs w min w max p?
dc Q?dc p?c Q?c Coordination
bene?t
Supply chain performance
in percentage
0 1.28780.9469 1.3162 5.6962.0 4.5992.734.2312.01 0.1 1.28810.9458 1.3159 5.7069.2 4.60103.638.3312.05 0.2 1.28840.9445 1.3155 5.7178.3 4.62117.443.5512.10 0.3 1.28880.9428 1.3151 5.7290.1 4.63135.450.4012.17 0.4 1.28930.9406 1.3145 5.74106.1 4.66159.959.7812.26 0.5 1.29010.9377 1.3138 5.76128.9 4.69195.173.3812.39 0.6 1.29110.9335 1.3127 5.79164.3 4.73250.094.8112.58 0.7 1.29270.9270 1.3110 5.85226.2 4.81347.2133.2912.86 0.8 1.29530.9159 1.3078 5.96361.7 4.96563.1220.6513.36 0.9 1.29750.8932 1.2989 6.33872.6 5.441392.0564.77
14.28
Table4
Impact of price-sensitivity factor.
b w rs w min w max p?
dc Q?dc p?c Q?c Coordination
bene?t
Supply chain
performance in
percentage
150.36860.34630.56138.4588.97.35110.123.68 3.69
160.51190.45860.68868.0286.9 6.92109.525.17 4.25
170.63850.55400.79907.6484.9 6.54108.826.65 4.86
180.75100.63520.89547.3082.9 6.20108.128.13 5.53
190.85180.70440.97987.0080.9 5.90107.529.60 6.26
200.94260.7633 1.0542 6.7379.0 5.63106.831.077.04
21 1.02470.8135 1.1198 6.4977.0 5.39106.232.537.90
22 1.09950.8559 1.1779 6.2675.1 5.17105.533.998.82
23 1.16780.8916 1.2295 6.0673.1 4.96104.935.449.81
24 1.23040.9213 1.2752 5.8771.2 4.78104.236.8910.89
25 1.28810.9458 1.3159 5.7069.2 4.60103.638.3312.05
G.Partha Sarathi et al./Applied Mathematical Modelling38(2014)4120–41364131
(i)Effect of variation of stock dependency factor
Considering the demand function D ?200àbp tcQ te where e is uniformly distributed in the range ?0;10 ,we vary stock dependency factor c while keeping all other parameters constant.The results are shown in Table 3,Figs.3and 4.It can be seen that as the stock dependency factor increased,retail price,order quantity,coordination bene?t and SC perfor-mance are increased.Examining the Eqs.(14)and (27)for order quantity in both decentralized and centralized setting,we observe that the term e1àc Tappears in the denominator,therefore,as c approaches 1,there is obvious increase in coor-dination bene?t and SC performance,but,at c ?1,the solution becomes infeasible.Sudden rise in the slope of both the curves are the evidences of this phenomenon.This phenomenon is quite practical as there can be no situation that guarantees that as soon as the product is displayed it will be sold.
Table 5
Impact of demand uncertainty.B
w rs
w min
w max
p ?dc
Q ?dc
p ?c
Q ?c
Coordination
bene?t Supply chain performance in percentage 10 1.28810.9458 1.3159 5.7069.2 4.60103.638.3312.0520 1.29410.9408 1.3181 5.7872.6 4.71110.141.5212.5230 1.29950.9366 1.3199 5.8676.1 4.81116.644.7112.9240 1.30410.9331 1.3215 5.9479.7 4.92123.347.8913.2850 1.30800.9302 1.3227 6.0283.4 5.03130.151.0613.5760 1.31130.9278 1.3237 6.1087.3 5.13137.054.2213.8270 1.31400.9258 1.3244 6.1991.2 5.24143.957.3514.0280 1.31600.9242 1.3247 6.2795.3 5.35150.960.4614.1890 1.31740.9228 1.3248 6.3699.6 5.45158.063.5314.29100 1.31820.9217 1.3247 6.45103.9 5.56165.2
66.57
14.37
4132G.Partha Sarathi et al./Applied Mathematical Modelling 38(2014)4120–4136
G.Partha Sarathi et al./Applied Mathematical Modelling38(2014)4120–41364133
(ii)Effect of variation of price-sensitivity factor
We now consider the demand function D?200àbpt0:1Qte where e is uniformly distributed in the range?0;10 and vary the price-sensitivity factor b,keeping all other parameters constant and the results are shown in Table4,Figs.5and6.It is clear from the Table4that as the price-sensitivity is increased,the retail price and order quantity are decreased as one can see that in the Eqs.(13)and(26),b appears in the denominator.But it may be noted that the coordination bene?t and SC performance are improved.It can be interpreted as follows.When the product is more price-sensitive,it negatively affects the demand(please see the above demand function).Since the order quantity is low,the corresponding pro?ts of the channel members will also decrease.Therefore,for such a product,coordination through the proposed contract can help improving their pro?ts.
(iii)Effect of variation of demand uncertainty
Finally,the impact of the demand uncertainty is investigated considering the demand function D?200à25pt0:1Qte where e is uniformly distributed in the range?0;B .Here,we vary B while keeping all other parameters constant and the re-sults are shown in Table5,Figs.7and8.A closer look at the Table5reveals that as the demand uncertainty increased the retail price,order quantity,coordination bene?t and SC performance are increased when the SC operates under the com-bined RS and QD contract.To interpret this result,one may observe that with discounting,the wholesale price decreases enticing the retailer to order more.As a result,the service level of the retailer goes up and the retailer is better prepared to handle the demand uncertainty,consequently SC performance also improves.Therefore,coordination through contract can be bene?cial in an uncertain demand environment.
5.Conclusions
In this paper,we have proposed a combined RS and QD contract model to coordinate a decentralized SC operating under price-sensitive and stock dependent demand.Such models are preferred over BB contract where the items are expensive and physical return of unsold items is not practical.We demonstrate that the combined contract model improves the SC perfor-
mance and ensures win–win situation to both manufacturer and the retailer.The following conclusions are drawn from the sensitivity analysis.
First,as the stock dependency coef?cient increases,both the retail price and the order quantity tend to increase.Second,as the price-sensitivity increases,both the retail price and order quantity decrease.Third,as the demand uncertainty increases,both price and order quantity increase.Finally,with respect to stock dependency,price-sensitivity and demand uncertainty,it is found that coordination bene?t and SC performance under RS and QD contract increases with the increase of the values.Further,the above observations emphasize the potential of the proposed contract for most SCs dealing with short lifecycle products such as agro products.For example,one can consider the case of a factory supplying ?our to a retail chain.The ?our like other perishable agro-products has a de?nite expiry date.If the ?our packets are displayed in large quantities in the aisles,it creates an impression of freshly arrived stock,draws attention of the customers,and hence likely to be sold faster.So it is wiser for the factory to provide discount on the existing wholesale price and entice the retailer to buy more.As a result,the sales is likely to go up,increasing retailer’s pro?t which later on can be shared with the factory.While the service level of the retailer increases,the retailer has to share the risk of salvaging the item and the loss due to expiration if any.On the contrary,when there is no contract,retailer’s service level will be low and factory bears the risk of product expiration.Finally,this work can be extended in many ways.First,the risk neutrality assumption can be relaxed considering other alternative objectives such as expected utility maximization,satis?cing or aspiration level objectives,or mean–variance objectives.Second,it is assumed that the demand is in?uenced by the initial stock level.But,in most instances the demand depends on the current inventory levels and not only on the initial inventory levels.Third,it is assumed that the demand and cost structures are known to all members of the SC,which seldom is true in reality.The current situation can be modeled as a game with incomplete information,considering private information of each player.Fourth,the limitation of two echelon SCs can be extended to include multi level case.Fifth,the single-period setting can be relaxed to multi period situation to represent more than one opportunity for the buyer to procure the https://www.doczj.com/doc/ae10546199.html,stly,other demand enhancing factors such as retailer’s reputation,location,advertisement,sales promotion,service and effort can be included in the future work.Appendix A
Since D ?a àbp tcQ te and Q ?ea àbp tcQ Ttz ,we can have the following identities:
E ?min eQ ;D T ?E ?a àbp tcQ tmin ez ;e T ?a àbp tcQ tE ?min ez ;e T ?a àbp tc
a àbp tz
1àc tE ?min ez ;e T ;E ?eQ àD Tt ?E ?ez àe Tt ?E ?z àmin ez ;e T ?E ?z àE ?min ez ;e T ?z àE ?min ez ;e T ;E ?eD àQ Tt ?E ?ee àz Tt ?E ?e àmin ee ;z T ?E ?e àE ?min ee ;z T ?l àE ?min ee ;z T :
But,E ?min ez ;e T ?R z A xf ex Tdx tR B z zf ex Tdx ?R B A xf ex Tdx àR B
z ex àz Tf ex Tdx .
Therefore,the term E ?min ez ;e T can be written as l àH ez Twhere H ez Tis de?ned as H ez T?R B
z ex àz Tf ex Tdx .Therefore,
E ?eQ àD Tt ?E ?ez àe Tt ?z àE ?min ez ;e T ?z àel àH ez TT?z àl tH ez T;E ?eD àQ Tt ?E ?ee àz Tt ?l àE ?min ee ;z T ?l àel àH ez TT?H ez T;
E ?min eQ ;D T ?E ?a àbp tcQ tmin ez ;e T ?a àbp tcQ tE ?min ez ;e T ?a àbp tc
a àbp tz
1àc
tE ?min ez ;e T ?a àbp tc
a àbp tz
1àc
tl àH ez T:Appendix B
From the perspectives of the manufacturer and the retailer,we have the following constraints,known as participation constraints .
P M eQ ?Q c ;r ;w ?w 0TàP M eQ ?Q dc ;r ?1;w ?w 0TP 0;P R eQ ?Q c ;r ;w ?w 0TàP R eQ ?Q dc ;r ?1;w ?w 0TP 0;
The above participating constraints can also be written as follows.
P M eQ ?Q c ;r ;w ?w 0TàP M eQ ?Q r ;r ;w ?w rs TP 0P R eQ ?Q c ;r ;w ?w 0TàP R eQ ?Q r ;r ;w ?w rs TP 0:
4134G.Partha Sarathi et al./Applied Mathematical Modelling 38(2014)4120–4136
From the manufacturer’s constraint,we get
e1àr Tp c E ?min eQ c ;D T tw 0Q c àmQ c à?e1àr Tp dc E ?min eQ dc ;D T tw rs Q dc àmQ dc P 0;W 0Q c P m eQ c àQ dc Ttw rs Q dc te1àr T?p dc E ?min eQ dc ;D T àp c E ?min eQ c ;D T ;w 0P
1
Q c
?m eQ c àQ dc Ttw rs Q dc te1àr T?p dc E ?min eQ dc ;D T àp c E ?min eQ c ;D T :Similarly,from the retailer’s constraint,we obtain
rp c E ?min eQ c ;D T àw 0Q c àhE ?eQ c àD Tt àsE ?eD àQ c Tt à
áàrp dc E ?min eQ dc ;D T àw rs Q dc àhE ?eQ dc àD Tt
à
àsE ?eD àQ dc Tt á
P 0;
w 0Q c 6r ?p c E ?min eQ c ;D T àp dc E ?min eQ dc ;D T tw rs Q dc àh ?E ?eQ c àD Tt àE ?eQ dc àD Tt às ?E ?eD àQ c Tt àE ?eD àQ dc Tt ;w 06
1Q c
r ?p c E ?min eQ c ;D T àp dc E ?min eQ dc ;D T tw rs Q dc àh ?E ?eQ c àD Tt àE ?eQ dc àD Tt às ?E ?eD àQ c Tt àE ?eD àQ dc Tt à
á:From the above two participating constraints,one can ?nd the wholesale price range w min àw max which can coordinate the SC and can also facilitate the win–win situation,where,
w min ?
1
c
?m eQ c àQ dc Ttw rs Q dc te1àr T?p dc E ?min eQ dc ;D T àp c E ?min eQ c ;D T and
w max ?
1c
/?p c E ?min eQ c ;D T àp dc E ?min eQ dc ;D T tw rs Q dc àh ?E ?eQ c àD Tt àE ?eQ dc àD Tt às ?E ?eD àQ c Tt àE ?eD àQ dc Tt :à
áAppendix C
Retail price under centralized setting is given by
p c ?
1?a tbm tl àH ez c Ttc ez c àl tH ez c TT ?1?a tbm tcz c te1àc Tel àH ez c TT ?1?a tbm tcz c te1àc TE ?min ez c ;e T ?1?a tbm tcz c te1àc TE ?ez c àez c àe TtT ?1
2b ?a tbm tz c àe1àc TE ?ez c àe Tt :Substituting
p c ?
1
?a tbm tz c àe1àc TE ?ez c àe Tt in Q c ?a àbp c tz c
1àc
,we get Q c ?
a àbm tz c 2e1àc T
t1
2E ?ez c àe Tt :Appendix D
We know that F ez c T?p c ts e1àc Tàm c
.Let
p c ts e1àc Tàm
e1àc Tep c th ts T
?q so that z c ?F à1eq T
@z c ?@z c q @q ?@F à1
eq Tq @q ?1f eF eTT@q ?1f eF eTT@p c ts e1àc Tàm c
?1f eF eTTh tm tsc
e1àc Tep c th ts T !
>0;@z c @m ?@z c @q @q @m ?@F à1eq T@q @q @m ?1f eF à1eq TT@q @m ?
1f eF à1eq TT@@m p c ts e1àc Tàm e1àc Tep c th ts T ?1f eF à1eq TTà1
e1àc Tep c th ts T
<0;@z c ?@z c q @q ?@F à1eq Tq @q ?1f eF eTT@q ?1f eF eTT@p c ts e1àc Tàm c
?1
f eF eTT
e0T?0;
G.Partha Sarathi et al./Applied Mathematical Modelling 38(2014)4120–41364135
@z c @b ?
@z c
@q
@q
@b
?
@Fà1eqT
@q
@q
@b
?
1
feFà1eqTT
@q
@b
?
1
feFà1eqTT
@
@b
p
c
tse1àcTàm
e1àcTep cthtsT
?
1
feFà1eqTT
e0T?0;
@z c @c ?
@z c
@q
@q
@c
?
@Fà1eqT
@q
@q
@c
?
1
feFeTT
@q
@c
?
1
feFeTT
@
@c
p
c
tse1àcTàm
e1àcTep cthtsT
?
1
feFeTT
p
c
àm
e1àcTep cthtsT
!
>0:
Appendix E
E?ezàeTt ?Z z
A
ezàxTfexTdx?ezàxTFexTt
Z
FexTdx
!z
A
??ezàxTFexT z
A
t
Z
FexTdx
!z
A
?0à0t
Z z
A
FexTdx?
Z z
A
FexTdx:
For uniform distribution,E?ezàeTt ?R z
A
xàA
àá
dx?exàAT2
h i z
A
?ezàAT2.
Therefore,
@ @z E?ezàeTt ?
@
@z
ezàAT2
2eBàAT
"#
?
zàA
BàA
?FezT<1:
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