research article coordinating contracts for two-stage...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 259164 12 pageshttpdxdoiorg1011552013259164

Research ArticleCoordinating Contracts for Two-Stage Fashion Supply Chainwith Risk-Averse Retailer and Price-Dependent Demand

Minli Xu1 Qiao Wang1 and Linhan Ouyang2

1 School of Business Central South University Changsha 410083 China2 School of Management Nanjing University of Science and Technology Nanjing 210094 China

Correspondence should be addressed to Minli Xu xu minli163com

Received 7 December 2012 Accepted 11 January 2013

Academic Editor Tsan-Ming Choi

Copyright copy 2013 Minli Xu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

When the demand is sensitive to retail price revenue sharing contract and two-part tariff contract have been shown to be able tocoordinate supply chains with risk neutral agents We extend the previous studies to consider a risk-averse retailer in a two-echelonfashion supply chain Based on the classic mean-variance approach in finance the issue of channel coordination in a fashion supplychain with risk-averse retailer and price-dependent demand is investigated We propose both single contracts and joint contractsto achieve supply chain coordination We find that the coordinating revenue sharing contract and two-part tariff contract in thesupply chain with risk neutral agents are still useful to coordinate the supply chain taking into account the degree of risk aversionof fashion retailer whereas a more complex sales rebate and penalty (SRP) contract fails to do so When using combined contractsto coordinate the supply chain we demonstrate that only revenue sharing with two-part tariff contract can coordinate the fashionsupply chain The optimal conditions for contract parameters to achieve channel coordination are determined Numerical analysisis presented to supplement the results and more insights are gained

1 Introduction

Fashion supply chain is characterized by short product lifecycle high volatile customer demand and clientsrsquo varyingtastes [1] Within such supply chains it is difficult to pre-dict the demand accurately Because of the highly demanduncertainty the fashion retailer must suffer risks from thetrading off between overstocks and stock-outs [2] Besidesthe complex features of fashion supply chain make supplychain coordination increasingly significant for supply chainagents in fashion industry

Coordination among supply chain agents via settingincentive alignment contracts is a hot topic in supply chainmanagement Under the coordinating contracts the incen-tives of supply chain agents are aligned with the objectiveof the whole supply chain so that the decentralized supplychain behaves as well as the vertically integrated supply chainWithout supply chain coordination problems involving dou-ble marginalization will prevail [3] reducing the supplychainrsquos efficiency tremendously Over the past two decades

many forms of contracts with reasonable contract parametershave been studied to achieve supply chain coordination withrisk-neutral agents by fighting against the issue of doublemarginalization These traditional contracts include returnspolicy [4 5] revenue-sharing contract [6] quantity flexibilitycontract [7 8] two-part tariff contract [9] and sales rebatecontract [10ndash13] For more detailed information of papers onthese and some other supply chain contracts please refer to[14]

Revenue-sharing contract indicates that the newsvendorretailer pays the upstream manufacturer a unit wholesaleprice for each unit ordered plus a proportion of his revenuefrom selling the product Both theoretical and empiricalstudies have been carried out on the effect of revenue-sharingcontract in the video cassette rental industry [15 16] Underthe classic newsvendor models such contracts have beenshown to be capable of coordinating the newsvendor [6 1718]

Under a two-part tariff contract the retailer gives themanufacturer a fixed transfer payment apart from the unit

2 Mathematical Problems in Engineering

wholesale price for each unit purchased And it has also beenshown that a two-part tariff contract coordinates the supplychain when the optimal value of unit wholesale price equalsthe manufacturerrsquos unit production cost [9]

Sales rebate and penalty (SRP) is based on the retailerrsquossales performance With a SRP contract the manufacturerwill specify a certain sales target prior to the selling seasonDifferent from sales rebate which executes rebate only foreach unit sold above the target level the retailer will begranted a unit rebate or else the retailer must pay themanufacturer a penalty In supply chain management bothSRP contract and sales rebate contract have been demon-strated unable to coordinate the channel when the demand issensitive to retail price or the retailerrsquos sales effort [10 19ndash21]

Early studies considered retail price exogenous leavingthe retailer with the decision of order quantity alone inorder to maximize expected profit As retail price plays animportant role inmarketing channel a new steam of researchon supply chain coordination and contracting integratespricing into the order quantity decision of the retailer underdifferent demand models Reviews of this work [12 22]explicitly stated that revenue-sharing contract and two-parttariff contract are able to coordinate the newsvendor withprice-dependent demand while many other traditional con-tracts arenrsquot And there is an increasing interest in examiningcombined contracts consisting of two or more traditionalcontracts to achieve channel coordination [19ndash21 23]

However the common results derived from the previousstudies may not be precise in operations management sincein the real world different decisionmakersmay have differentdegrees of risk aversion in light of this we extend theresults of proceeding studies to explore the issue of supplychain coordination with risk-averse fashion retailer andprice-dependent demand Specifically we investigate a singleperiod one-manufacturer one-retailer fashion supply chainwith a variety of contracts The manufacturer acting as theleader in the Stackelberg game offers the retailer a contractwith a set of contract parameters The fashion retailer actingas the follower sets self-interest order quantity and retailprice as a response We propose both single contracts andcombined contracts with the optimal values of contractparameters to achieve channel coordination within fashionsupply chain

The main objectives of our study cover the followingfirstly to explore whether the coordinating revenue-sharingcontract and two-part tariff contract in supply chains withrisk neural retailer can still coordinate the fashion supplychain with risk-averse retailer who has to choose retail pricein addition to stocking quantity secondly to compare theperformance of a more complicated sales rebate and penaltycontract in supply chain coordination with the performancesof revenue-sharing contract and two-part tariff contractfinally when joint contracts are got by taking advantages ofthe three single contracts to probe whether the resultingcombined contracts are useful to coordinate the supply chain

In recent years an increasing number of researchershave noticed the importance and the impact of risk aver-sion in supply chain contracting and coordination andsought in succession for the criteria to depict supply chain

agentsrsquo risk aversion attitude or preference In the literaturethe measures for describing risk aversion involve mean-variance (MV) [24] Neumann-Morgenstern utility function(VNUM) mean-downside-risk (MDR) [25] Value-at-risk(VaR) [26 27] and Conditional Value-at-risk (CVaR) [2829] Since MV is simple implementable and is easily under-stood by managers and practitioners compared with othermeasures we adopt mean-variance formulation to capturethe fashion retailerrsquos risk aversion in this paper

This paper is closely linked to the literature on supplychain coordinating and contracting with price-dependentdemand [30 31] in terms of a random and price-dependentdemand It is also correlated to studies of supply chaincoordination with agents having risk preferences in whichwe consider a risk-averse retailer [32ndash36] But our studyis the first to investigate the issue of channel coordinationfor the supply chain with risk-averse retailer and price-dependent demand We firstly investigate the problem ofcoordinating a two-stage fashion supply chain under singlecontracts including revenue-sharing contract two-part tariffcontract and sales rebate and penalty contract After provingthat revenue-sharing contract and two-part tariff contractcould still achieve channel coordination in this contextwhile a more complex sales rebate and penalty cannot wefurther explore the role of combined contracts (sales rebateand penalty with revenue-sharing contract sales rebate andpenalty with two-part tariff contract and revenue sharingwith two-part tariff contract) in supply chain coordinationBy identifying the coordination conditions and mechanismsof various contracts our work contributes to supplement thecurrent literature on supply chain coordination and contract-ing We also provide meaningful guidance to managers inreal operations management on how to choose the type ofcontract and determine the optimal contract parameters inorder to coordinate fashion supply chain inmore complicatednewsvendor frameworks

The paper is organized as followsModel formulation andnotation definition are presented in Section 2 The bench-mark case of integrated fashion supply chain is studied inSection 3 Supply chain coordination under single contractsand combined contracts is investigated in Sections 4 and5 Numerical study to supplement the analytical results andgain more insights is given in Section 6 Section 7 providesmanagerial insights and concluding comments

2 Model Formulation and Notation Definition

Consider a two-echelon fashion supply chain with a risk-neutral manufacturer and a risk-averse retailer The retailersells a fashion product whose demand is sensitive to retailprice The upstream manufacturer produces the product andsells it through a vertically separated retailer The sequenceof events in the supply chain is as follows The manufactureras the leader of a Stackelberg game offers the retailer acontract After knowing the details of the contract thefashion retailer commits his order quantity and retail priceThen themanufacturer organizes the production and deliversthe finished products to the retailer prior to the selling seasonAfterwards the selling season starts and the demand is

Mathematical Problems in Engineering 3

realized At the end of the selling season based on the agreedcontract both the manufacturer and the retailer performthe respective contract terms and achieve transfer paymentsbetween each other

Let 119901 be the retail price 119888 the production cost incurred bythe manufacturer 119908(119908 ge 119888) the wholesale price 119907(119907 lt 119888) thesalvage value of unsold inventory and 119902 the productionorderquantity Use 119905 gt 0 as the sales target level and 119906 gt 0 as therebate (and penalty) for sales rebate and penalty contract Use120582 isin (0 1) as the fraction of revenue earned by the retailerin revenue-sharing contract and 119866 gt 0 as the fixed transferpayment from the retailer to the manufacturer in two-parttariff contract

In the literature there are two fashions in which thedemand 119909 depends on the selling price 119901 (1) the additiveform 119909 = 119863(119901) + 120585 (2) the multiplicative form 119909 = 119863(119901)120585where 119863(119901) ge 0 is a function of 119901 representing the expecteddemand and 120585 is a nonnegative variable representing therandom proportion of the demand 120585 is independent ofselling price 119901 with a probability density function 119891(sdot) anda cumulative distribution function 119865(sdot) It is assumed that119891(sdot) gt 0 has a continuous derivative 1198911015840(sdot) 119865(sdot) is continuousstrictly increasing and differentiable Let119865minus1(sdot) be the reversefunction of119865(sdot) and119865(sdot) = 1minus119865(sdot)119863(119901) is strictly decreasingin 119901 and 1198631015840(119901) lt 0 In this paper we only considerthe additive demand model For the multiplicative one webelieve similar results would be derived

In order to ensure the existence and uniqueness ofmodel results we give the following definitions of 119863(119901) and120585

Definition 1 By definition 119890 = minus1199011198631015840(119901)119863(119901) is the priceelasticity of 119863(119901) 119863(119901) has an increasing price elasticity(IPE) in 119901 if

119889119890

119889119901ge 0 (1)

Price elasticity 119890 measures the percentage change indemand with respect to one percentage change in sellingprice The IPE property is intuitive In the literature manydemand forms own IPE property such as the simplest lineardemand isoelastic demand and exponential demand

For the ease of position in this paper we suppose a lineardemand of 119863(119901) Let 119863(119901) = 119886 minus 119896119901 where 119886 gt 0 is the basedemand and 119896 gt 0 is the price elasticity of demand Thus wehave 119901 isin [119888 119886119896]

Definition 2 Define 119903(120585) = 119891(120585)(1 minus 119865(120585)) as the failure rateof the 120585distribution then 120585has an increasing failure rate (IFR)if for 120585 ge 0

1199031015840

(120585) ge 0 (2)

It is noted that in the literature various random distri-butions exhibit IFR property involving uniform and normaldistributions

To capture the decision making of risk-averse fashionretailer we adopt the same risk aversion decision model asin [34]

min 119881119903(119902 119901)

st 119864119903(119902 119901) ge 119896

119903

(P-1)

where119864119903(119902 119901) and119881

119903(119902 119901) denote themean and the variance

of the retailerrsquos profit respectively and 119896119903gt 0 denotes the

retailerrsquos expected profit threshold 119896119903can be considered to

be the indicator of the retailerrsquos risk aversion degree sincelarger values of 119896

119903indicate that the retailer does not want

to earn a low expected profit leading to a more risk-averseretailer Define 119864

119903= 119864119903(119902119903

lowast 119901119903

lowast) as the retailerrsquos attainable

maximum expected profit Then from (P-1) 119896119903le 119864119903must

establish otherwise there is no feasible solution for (P-1)

3 The Integrated Fashion Supply Chain

First we offer a benchmark by analyzing the case whenthe fashion supply chain is vertically integrated so that themanufacturer owns its own retailer Note that the type ofcontract does not affect the performance of the integratedfashion supply chain The optimal solutions to this modelare production level 119902 and retail price 119901 which provide uswith guidelines to the optimal policy for the whole systemDefine 119876 = 119902 minus119863(119901) 120578(119876) = 2119876int119876

0119865(120585)119889120585 minus 2 int

119876

0120585119865(120585)119889120585 minus

(int119876

0119865(120585)119889120585)

2 The integrated fashion supply chainrsquos profitexpected profit and the variance of profit are given as follows

prodsc (119876 119901) = (119901 minus 119888)119863 (119901) + (119901 minus 119888)119876 minus (119901 minus 119907) (119876 minus 120585)+

(3)

119864sc (119876 119901) = (119901 minus 119888)119863 (119901) + (119901 minus 119888)119876 minus (119901 minus 119907)int119876

0

119865 (120585) 119889120585

(4)

119881sc (119876 119901) = (119901 minus 119907)2

120578 (119876) (5)

Let (119876sclowast 119901sclowast) and 119902sc

lowast= 119863(119901sc

lowast) +119876sc

lowast be the optimaljoint decision and optimal production level for the integratedsupply chain

Proposition 3 Under the additive price-dependent demandthe integrated supply chainrsquos optimal joint decision (119876sclowast 119901sclowast)and optimal production quantity 119902sclowast exist and are uniquesatisfying

(119901sclowastminus 119888) minus (119901sc

lowastminus 119907) 119865 (119876sc

lowast) = 0 (6)

119886 minus 119896 (2119901sclowastminus 119888) + 119876sc

lowastminus int

119876sclowast

0

119865 (120585) 119889120585 = 0 (7)

119902sclowast= 119886 minus 119896119901sc

lowast+ 119865minus1(119901sclowastminus 119888

119901sclowast minus 119907) (8)

Proof For any given 119901 isin [119888 119886119896] from (4) by taking the firstand second differentials of 119864sc(119876 119901)with respect to119876 we get


120597119864sc(119876 119901)120597119876 = (119901 minus 119888) minus (119901 minus 119907)119865(119876) 1205972119864sc(119876 119901)120597119876

2

=

minus(119901 minus 119907)119891(119876) lt 0 Thus for any given 119901 isin [119888 119886119896] 119864sc(119876 119901)is a concave function of 119876 and 119876sc

lowast is finite and uniquesatisfying

(119901 minus 119888) minus (119901 minus 119907) 119865 (119876sclowast) = 0 (9)

From (9) we know that119876sclowast is a function of 119901 According

to the implicit function theorem we have

119889119876sclowast

119889119901= minus1205972119864sc (119876sc

lowast 119901) 120597119876120597119901

1205972119864sc (119876sclowast 119901) 1205971198762

=1 minus 119865 (119876sc

lowast)

(119901 minus 119907) 119891 (119876sclowast)

(10)

Therefore from (10) 119876sclowast is strictly increasing in 119901

By taking119876sclowast into 119864sc(119876 119901) we get 119864sc(119876sc

lowast 119901) Taking

the first and second derivatives of119864sc(119876sclowast 119901)with respect to

119901 we derive

119889119864sc (119876sclowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119888) + 119876sc

lowastminus int

119876sclowast

0

119865 (120585) 119889120585

(11)

1198892119864sc (119876sc

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876sc

lowast))119889119876sclowast

119889119901 (12)

Substituting (10) into (12) we get

1198892119864sc (119876sc

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876sclowast))2

(119901 minus 119907) 119891 (119876sclowast) (13)

Define 119867(119876) = 119891(119876)(1 minus 119865(119876))2 and taking thederivative of119867(119876) with respect to 119876 we have

119889119867 (119876)

119889119876=(1 minus 119865 (119876)) 119891

1015840(119876) + 2(119891 (119876))

2

(1 minus 119865 (119876))3

(14)

From Definition 2 we know that 119867(119876sclowast) is strictly

increasing in 119901 Therefore 1198892119864sc(119876sclowast 119901)119889119901

2 is strictlydecreasing in 119901 Let 1199010 satisfy 1198892119864sc(119876sc

lowast 119901)119889119901

2= 0 If 1199010

does not exist then we can know 1198892119864sc(119876sclowast 119901)119889119901

2lt 0

since lim119901rarrinfin1198892119864sc(119876sc

lowast 119901)119889119901

2=minus2119896lt0 and 119864sc(119876sc

lowast 119901)

is a concave function of 119901 If 1199010 exists for 119901 lt 11990101198892119864sc(119876sc

lowast 119901)119889119901

2gt 0 and for 119901gt1199010 1205972119864sc(119876sc

lowast 119901)120597119901

2lt

0 That is 119864sc(119876sclowast 119901) is convex in 119901 for 119901lt1199010 and concave

in 119901 for 119901 gt 1199010 Because 119889119864sc(119876sclowast 119901)119889119901|

119901=119888= (119886 minus

119896119888) + int119876sclowast

(119888)

0119865(120585)119889120585 gt 0 119864sc(119876sc

lowast 119901 119890) is unimodal in

119901 isin [119888 119886119896]Hence there exists a unique retail price 119901sc

lowastisin [119888 119886119896]

that maximizes 119864sc(119876sclowast 119901) and is given by (7)

Since 119902sclowast= 119863(119901sc

lowast) + 119876sc

lowast it is natural to conclude thatthe fashion supply chainrsquos optimal production quantity 119902sc

lowast isunique and satisfies (8)

Remark 4 Proposition 3 reveals the optimal solutions ofthe integrated fashion supply chain with the additive price-dependent demand Correspondingly the entire supplychainrsquos maximum expected profit is 119864sc = 119864sc(119876sc

lowast 119901sclowast)

4 The Decentralized Fashion Supply Chainunder Single Contracts

Now we consider the case when the fashion supply chain isdecentralized In the decentralized supply chain the manu-facturer and the fashion retailer are independent and enter aStackelberg game as described before Specifically the retaileris assumed to be risk-averse with an expected profit threshold119896119903gt 0 and 119896

119903lt 119864sc (otherwise there would be no incentive

for the manufacturer to offer a contact) As an extensionof prior works in the following sections we will considerthe optimal joint pricing-inventory decisions of a risk-averseretailer in the decentralized fashion supply chain under singlecontracts such as revenue sharing contact sales rebate andpenalty contract and two-part tariff contract For the purposeof simplification define 119876 = 119902 minus 119863(119901) 119879 = 119905 minus 119863(119901) and119879lowast= 119905 minus 119863(119901

lowast) Let (119876

119903

lowast 119901119903

lowast) and (119876

119903119898119907

lowast 119901119903119898119907

lowast) be the

optimal joint decisions for the risk-neutral retailer and risk-averse retailer respectively

41 SRP Contract With a SRP contract 120579SRP(119908 119905 119906) themanufacturer offers a sales target 119905 gt 0 to the retailer priorto the selling season At the end of the selling season foreach unit sold above 119905 the manufacturer will give the retailera unit rebate 119906 gt 0 otherwise the retailer must pay themanufacturer a penalty 119906

In this setting the fashion retailerrsquos profit expected profitand the variance of profit are

prod119903(119876 119901)

= (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876 minus (119901 minus 119907) (119876 minus 120585)+

+ 119906 (min (120585 119876) minus 119879)

(15)

119864119903(119876 119901)

= (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876 minus (119901 minus 119907)int

119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(16)

119881119903(119876 119901)

= 119864(prod119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

119864(int

119876

0

119865 (120585) 119889120585 minus (119876 minus 120585)+

)

2

+ 1199062119864((min (120585 119876) minus 119879) minus (119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585))

2

+ 2119906 (119901 minus 119907) 119864(int

119876

0

119865 (120585) 119889120585 minus (119876 minus 120585)+

)

times ((min (120585 119876) minus 119879) minus (119876 minus 119879 minus int119876

0

119865 (120585) 119889120585))


= (119901 minus 119907)2

(int

119876

0

(119876 minus 120585)2

119891 (120585) 119889120585 minus (int

119876

0

119865 (120585) 119889120585)

2

)

+ 1199062(int

119876

0

(120585 minus 119879)2

119891 (120585) 119889120585 + int

infin

119876

(119876 minus 119879)2119891 (120585) 119889120585

minus(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

2

)

+ 2119906 (119901 minus 119907) (int

119876

0

119865 (120585) 119889120585 (119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

minusint

119876

0

(119876 minus 120585) (120585 minus 119879) 119891 (120585) 119889120585)

= (119901 minus 119907 + 119906)2

(2119876int

119876

0

119865 (120585) 119889120585 minus 2int

119876

0

120585119865 (120585) 119889120585

minus(int

119876

0

119865 (120585) 119889120585)

2

)

= (119901 minus 119907 + 119906)2

120578 (119876)

(17)

Proposition 5 For a given SRP contract 120579119878119877119875(119908 119905 119906) offered

by the manufacturer the risk-neutral retailerrsquos optimal jointdecision (119876

119903

lowast 119901119903

lowast) is given by

(119901119903

lowastminus 119908 + 119906) minus (119901

119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (18)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (19)

Proof For any given 119901 isin (119908 minus 119906 119901] from (16) by taking thefirst and second differentials of 119864

119903(119876 119901)with respect to119876 we

can derive that 120597119864119903(119876 119901)120597119876 = (119901minus119908+119906)minus (119901minus119907+119906)119865(119876)

and 1205972119864119903(119876 119901)120597119876

2= minus(119901 minus 119907 + 119906)119891(119876) lt 0 Thus 119864

119903(119876 119901)

is a concave function of 119876 119876119903

lowastcan be given by

(119901 minus 119908 + 119906) minus (119901 minus 119907 + 119906) 119865 (119876119903

lowast) = 0 (20)

From (20) we can get to know that119876119903

lowast is a function of 119901By making use of the implicit function theorem we have

119889119876119903

lowast

119889119901= minus1205972119864119903(119876119903

lowast 119901) 120597119876120597119901

1205972119864119903(119876119903

lowast 119901) 1205971198762

=1 minus 119865 (119876

119903

lowast)

(119901 minus 119907 + 119906) 119891 (119876119903

lowast)

(21)

Thus we know that 119876119903

lowast is strictly increasing in 119901Substituting 119876

119903

lowast into 119864119903(119876 119901) we get

119864119903(119876119903

lowast 119901)

= (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876119903

lowastminus (119901 minus 119907)int

119876119903

lowast

0

119865 (120585) 119889120585

+ 119906(119876119903

lowastminus 119879 minus int

119876119903

lowast

0

119865 (120585) 119889120585)

(22)

From (22) 119864119903(119876119903

lowast 119901) can be regarded as a function of

variable 119901 alone Taking the first and second derivatives of119864119903(119876119903

lowast 119901) with respect to 119901 we get

119889119864119903(119876119903

lowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585

(23)

1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876

119903

lowast))119889119876119903

lowast

119889119901 (24)

By taking (21) into (24) we have

1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876119903

lowast))2

(119901 minus 119907 + 119906) 119891 (119876119903

lowast) (25)

Similar to Proposition 3 we know that 119864119903(119876119903

lowast 119901) is

unimodal in 119901 If 119908 minus 119906 gt 119888 119889119864119903(119876119903

lowast 119901)119889119901|

119901=119908minus119906= (119886 minus

119896(119908 minus 119906)) + int119876119903

lowast

(119908minus119906)

0119865(120585)119889120585 gt 0 Thus there exists a unique

119901119903

lowast which satisfies (19)

Remark 6 By Comparing (19) with (7) and (18) with (6)we find that (119876sc

lowast 119901sclowast) is the risk-neutral fashion retailerrsquos

optimal joint decision if and only if 119906 = 0 and 119908 = 119888However a SRP contract with 119906 = 0 and 119908 = 119888 gives themanufacturer zero profit So SRP contract cannot coordinatethe supply chain with risk-neutral fashion retailer and price-dependent demand

42 Revenue-Sharing Contract A revenue-sharing contract120579RS(119908 120582) stipulates that the fashion retailer pays the upstreammanufacturer a unit wholesale price 119908 for each unit orderedplus a proportion of his revenue from selling the product Let120582 isin (0 1) be the fraction of supply chain revenue earnedby the retailer and thus (1 minus 120582) is the fraction shared by themanufacturer Under the revenue-sharing contract 120579RS(119908 120582)the retailerrsquos expected profit and the variance of profit aregiven as follows

119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585

(26)

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876) (27)

Proposition 7 For a given revenue-sharing contract 120579119877119878(119908 120582)

offered by the manufacturer the risk-neutral fashion retailerrsquosoptimal joint decision (119876

119903

lowast 119901119903

lowast) is given by

(120582119901119903

lowastminus 119908) minus 120582 (119901

119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (28)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908) + 120582119876

119903

lowastminus 120582int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (29)

Proof Similar to Proposition 5

Remark 8 Comparing (28) with (6) and (29) with (7) wefind that (119876sc

lowast 119901sclowast) can be the risk-neutral retailerrsquos optimal


ordering quantity and retail price if and only if 119908 = 120582119888 lt 119888which is equal to the optimal conditions for the contractparameters to coordinate the supply chain when retail priceis given exogenously Therefore consistent with the findingin the literature [12] when the random demand is sensitiveto pricing revenue sharing contact with reasonable contractparameters is sufficient to coordinate the supply chain withrisk-neutral retailer

43 Two-Part Tariff Contract With a two-part tariff contract120579TPT(119908 119866) the fashion retailer gives themanufacturer a fixedtransfer payment 119866 gt 0 apart from the unit wholesale pricefor each unit ordered And the retailerrsquos expected profit andthe variance of profit are

119864119903(119876 119901) = (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876

minus (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

(30)

119881119903(119876 119901) = 119864(prod

119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

120578 (119876)

(31)

Proposition 9 For a given two-part tariff contract 120579119879119875119879(119908 119866)


119903

lowast 119901119903

lowast) is given by

(119901119903

lowastminus 119908) minus (119901

119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (32)

119886 minus 119896 (2119901119903

lowastminus 119908) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (33)


Remark 10 By comparing (32) with (6) and (33) with (7)it is easy to get 119908 = 119888 such that the independent retailerrsquosoptimal decisions (119876

119903

lowast 119901119903

lowast) are equal to the integrated

fashion supply chainrsquos optimal solution (119876sclowast 119901sclowast) Hence

a two-part tariff contract 120579TPT(119908 119866) could perfectly achievechannel coordination for a fashion supply chain with risk-neutral retailer and price-dependent demand

Now by considering the risk aversion decision modelas given in (P-1) we establish the following propositions toattain the optimal joint decision for the risk-averse retailerunder single contracts

Proposition 11 Under single contracts for any given 119901 isin[119908 119886119896] 119881

119903(119876 119901) is strictly increasing in 119876 For any given

119876 ge 0 119881119903(119876 119901) is strictly increasing in 119901

Proof From (17) (27) and (31) taking differentials of119881119903(119876 119901) with respect to 119876 and 119901 and since 119889120578(119876)119889119876 =2(1 minus 119865(119876)) int

119876

0119865(120585)119889120585 gt 0 it can be easily verified that

for any given 119901 isin [119908 119886119896] 119881119903(119876 119901) is strictly increasing in

119876 and for any given 119876 ge 0 119881119903(119876 119901) is strictly increasing

in 119901

Proposition 12 Given the retailerrsquos expected threshold 119896119903le

119864119903 the risk-averse fashion retailerrsquos optimal joint decision(119876119903119898119907

lowast 119901119903119898119907

lowast) satisfies

119864119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903

119876119903119898119907

lowastle 119876119903

lowast

119901119903119898119907

lowastle 119901119903

lowast

(34)

Proof From the proceeding analysis we know that under sin-gle contracts such as SRP contract revenue-sharing contractand two-part tariff contract 119864

119903(119876 119901) is a concave function

of 119876 and is unimodal in 119901 isin [119908 119886119896] Besides 119881119903(119876 119901) is

strictly increasing in 119876 and 119901 Therefore according to (P-1)the optimal pricing-inventory decision for the risk-aversefashion retailer is obtained by solving119864

119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903

Moreover since 119896119903le 119864119903 in each region of (119876 le 119876

119903

lowast 119901 le

119901119903

lowast) (119876 gt 119876

119903

lowast 119901 le 119901

119903

lowast) (119876 le 119876

119903

lowast 119901 gt 119901

119903

lowast) and

(119876 gt 119876119903

lowast 119901 gt 119901

119903

lowast) there exists a corresponding decision

pair (119876 119901) that could make 119864119903(119876 119901) = 119896

119903established Since

119881119903(119876 119901) is strictly increasing in119876 and 119901 the optimal solution(119876119903119898119907

lowast 119901119903119898119907

lowast) for (P-1) can only fall in the region (119876 le

119876119903

lowast 119901 le 119901

119903

lowast) otherwise (119876

119903119898119907

lowast 119901119903119898119907

lowast) cannot be the risk-

averse fashion retailerrsquos optimal joint decision So we have119876119903119898119907

lowastle 119876119903

lowast and 119901119903119898119907

lowastle 119901119903

lowast

Remark 13 From Proposition 12 we can know that themaximum expected profit of the risk-averse fashion retailergenerated under single contracts is always no greater thanthat of a risk-neutral retailer This is the loss of profit broughtout by the retailerrsquos risk aversion attitude or preference Inaddition it can be seen from Proposition 12 that under theadditive price-dependent demand the risk-averse fashionretailer tends to order less and charge a lower price incomparison with a risk-neutral retailer which is consistentwith the known results derived from the studies on jointpricing and inventory decisions of a risk-averse newsvendor[29 33]

A contract provided by the upstreammanufacturer is saidto coordinate the supply chain if it is able to align the incen-tives of the manufacturer and the retailer so that the inde-pendent retailer makes the same decisions as the integratedsupply chain namely (119876

119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc

lowast 119901sclowast) Now

we present the following proposition to explore the necessaryconditions for a contract to achieve channel coordination

Proposition 14 For any given 119896119903lt 119864sc a contract achieves

supply chain coordination if and only if the contract satisfies(1) 119864119903(119876sclowast 119901sclowast) = 119896

119903 (2) 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast ge 0 (3)120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0

Proof If a contract achieves supply chain coordinationthen (119876

119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc

lowast 119901sclowast) stands According to

Proposition 12 we have 119864119903(119876sclowast 119901sclowast) = 119896

119903 On the other

hand since a contract coordinates the supply chain from(P-1) we know that119864

119903(119876sclowast 119901sclowast)ge119896119903We know that119881

119903(119876 119901)

is strictly increa-sing in119876 and 119901 and119864119903(119876 119901) is a continuous

function of 119876 and 119901 If 119864119903(119876sclowast 119901sclowast) gt 119896119903establishes then

there always exists an optimal joint decision 119876 lt 119876sclowast and


119901lt119901sclowast such that 119864

119903(119876 119901)ge119896

119903and 119881

119903(119876 119901)lt119881

119903(119876sclowast 119901sclowast)

which contradicts the fact that (119876sclowast 119901sclowast) is the optimal joint

pricing and inventory decisions for the risk-averse fashionretailer Therefore we have 119864

119903(119876sclowast 119901sclowast) = 119896119903

If (119876119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc

lowast 119901sclowast) then according to

Proposition 12 119876sclowastle119876119903

lowast and 119901sclowastle119901119903

lowast Since 119864119903(119876 119901) is a

concave function of119876 and strictly increasing in119876 isin (0 119876119903

lowast]

from 119876sclowastle 119876119903

lowast we have 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast ge 0Otherwise if 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast lt 0 and because forany given 119901 isin [119908 119886119896] 119881

119903(119876 119901) is strictly increasing in

119876 then there exists 119876 lt 119876sclowast such that 119864

119903(119876sclowast 119901sclowast) lt

119864119903(119876 119901sc

lowast) and 119881

119903(119876 119901sc

lowast) lt 119881

119903(119876sclowast 119901sclowast) It contra-

dicts the fact that (119876sclowast 119901sclowast) is the optimal joint deci-

sion of the risk-averse fashion retailer Similarly 119864119903(119876 119901)

is unimodal in 119901 isin [119908 119886119896] then from 119901sclowastle

119901119903

lowast we have 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0 Otherwise if

120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast lt 0 and because 119881

119903(119876 119901) is strictly

increasing in 119901 for any given 119876 ge 0 then there exists119901 lt 119901sc

lowast such that 119864119903(119876sclowast 119901sclowast) lt 119864

119903(119876sclowast 119901) and

119881119903(119876sclowast 119901) lt 119881

119903(119876sclowast 119901sclowast) It contradicts the fact that

(119876sclowast 119901sclowast) is the optimal joint decision of the risk-averse

fashion retailer As a result we have 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast ge

0 and 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0

Remark 15 From Proposition 14 it can be derived that whenthe supply chain is coordinated the risk-averse fashionretailerrsquos expected profit is equal to 119896

119903 and hence the man-

ufacturerrsquos expected profit is equal to 119864sc minus 119896119903Next we investigate in more detail whether the single

contracts above could achieve supply chain coordination

Proposition 16 For any given 119896119903lt 119864sc SRP contract cannot

achieve supply chain coordination

Proof From Proposition 14 we can get that the supply chainachieves coordination if and only if SRP contract satisfies119864119903(119876sclowast 119901sclowast) = 119896

119903 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast ge 0 and120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast ge 0 we can get (119901sclowastminus119908)minus

(119901sclowastminus 119907)119865(119876sc

lowast) + 119906119865(119876sc

lowast) ge 0 And from (6) we have

119906119865(119876sclowast) ge 119908 minus 119888 From 120597119864

119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0 we have

119886minus119896(2119901sclowastminus119908+119906)+119876sc

lowastminusint119876sclowast

0119865(120585)119889120585 ge 0 And from (7) it

can be obtained that 119906 le 119908minus119888 Since 119906119865(119876sclowast) lt 119906 there does

not exist some value of 119906 such that 119906119865(119876sclowast) ge 119908 minus 119888 and 119906 le

119908 minus 119888 establish simultaneously In other words SRP contract120579SRP(119908 119905 119906) cannot achieve channel coordination

Proposition 17 For any given 119896119903lt 119864sc revenue-sharing

contract and two-part tariff contract can achieve channelcoordination Specifically the optimal conditions satisfied bythe contract parameters of these two contracts to coordinate thesupply chain are as follows

(1) for revenue-sharing contract 119908 = 120582119888 120582 = 119896119903119864sc

(2) for two-part tariff contract 119908 = 119888 119866 = 119864sc minus 119896119903

Proof According to Proposition 14 for the revenue-sharingcontract from 119864


119903 we get the expression

119908 = 120582119901sclowastminus (119896119903+ 120582(119901sc

lowastminus 119907) int

119876sclowast

0119865(120585)119889120585)119902sc

lowast From120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast ge 0 we have 120582119901sclowastminus 119908 minus 120582(119901sc

lowastminus

119907)119865(119876sclowast) ge 0 Substituting (6) into 120582119901sc

lowastminus 119908 minus 120582(119901sc

lowastminus

119907)119865(119876sclowast) ge 0 it can be calculated that 119908 le 120582119888 From

120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0 we get 120582119886 minus 119896(2120582119901sc

lowastminus 119908) +

120582119876sclowastminus 120582int119876sclowast

0119865(120585)119889120585 ge 0 and taking (7) into it we know

that 119908 ge 120582119888 Combining 119908 le 120582119888 and 119908 ge 120582119888 we have119908 = 120582119888 and by taking 119908 = 120582119888 into the expression of 119908 wehave 120582 = 119896

119903119864sc Hence revenue-sharing contract can still

coordinate the supply chain when the fashion retailer is riskaverse

Similarly for two-part tariff contract from 119864119903(119876sclowast

119901sclowast) = 119896119903 we have 119908=119901sc

lowastminus(119896119903+ (119901sc


119876sclowast

0119865(120585)119889120585 +

119866)119902sclowast From 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast ge 0 we get 119901sclowastminus 119908 minus


lowast) ge 0 and from (6) 119908 le 119888 is derived From

120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0 we get 119886minus119896(2119901sc

lowastminus119908) + 119876sc

lowastminus

int119876sclowast

0119865(120585)119889120585 ge 0 and from (7) we have 119908 ge 119888 Thus we

have 119908 = 119888 Comparing 119908 = 119888 and 119908 = 119901sclowastminus[(119896119903+ (119901sc

lowastminus

119907) int119876sclowast

0119865(120585)119889120585 + 119866)119902sc

lowast] we can get 119866=119864sc minus 119896119903 Therefore

two-part tariff also could achieve supply chain coordinationwith risk sensitive retailer

Remark 18 From Propositions 16 and 17 we find that whenthe end demand depends on retail price and the fashionretailer is risk sensitive a more complex SRP contract (withthree parameters) cannot achieve supply chain coordinationwhereas simpler revenue-sharing contract and two-part tariffcontract (with two parameters) can

From Proposition 17 the values of 120582 and 119866 can beregarded as indicators of the fashion retailerrsquos risk aversionlevel Specifically with a larger 120582 the expected profit thresh-old of the retailer 119896

119903is greater and the retailer is more

risk averse Contrarily a larger value of 119866 means a smallerexpected profit threshold for the retailer indicating a less risksensitive retailer As a result if the fraction of sales revenueor the value of fixed transfer payment which the fashionretailer is willing to offer to the manufacturer is small thenthe retailer is relatively more risk averse

5 The Decentralized Fashion Supply Chainunder Combined Contracts

In the above section we investigate the role of three singlecontracts in coordinating fashion supply chains and findthat a more complicated SRP contract fails to coordinatethe supply chain while two other simpler contracts perfectlyachieve channel coordination In this section we furtherexplore contracts that combine the advantages of the abovecontracts Specifically we try to explore whether the resultingcontracts are effective to coordinate the supply chain whenthe coordinating contracts and the failed contract combinewith each other Define similarly119876 = 119902minus119863(119901) 119879 = 119905minus119863(119901)and 119879lowast = 119905 minus 119863(119901lowast)


51 SRP with Revenue-Sharing Contract Under this contractthe fashion retailerrsquos profit expected profit and the varianceof profit are

prod119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876 minus 120582 (119901 minus 119907) (119876 minus 120585)+

+ 119906 (min (120585 119876) minus 119879) (35)

119864119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 + 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(36)

119881119903(119876 119901) = [120582 (119901 minus 119907) + 119906]

2

120578 (119876) (37)

Proposition 19 For a given SRP with revenue-sharing con-tract offered by the manufacturer the risk-neutral fashionretailerrsquos optimal joint decision (119876

119903

lowast 119901119903

lowast) is given by

(120582119901119903

lowastminus 119908 + 119906) minus [120582 (119901

119903

lowastminus 119907) + 119906] 119865 (119876

119903

lowast) = 0 (38)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908 + 119906) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (39)


Remark 20 By comparing (39) with (7) and (38) with (6)we find that when 119906 = 0 119908 = 120582119888 (119876

119903

lowast 119901119903

lowast) = (119876sc


establishes However it contradicts the assumption of 119906 gt 0in SRPwith revenue-sharing contractThus when the fashionretailer is risk-neutral SRP with revenue-sharing contractcannot achieve channel coordination

52 SRP with Two-Part Tariff Contract In this setting thefashion retailerrsquos expected profit and the variance of profit aregiven by

119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585) minus 119866

119881119903(119876 119901) = (119901 minus 119907 + 119906)

2

120578 (119876)

(40)

Proposition 21 For a given SRP with two-part tariff contractoffered by the manufacturer the risk-neutral fashion retailerrsquosoptimal joint decision (119876

119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (41)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (42)

Remark 22 Comparing (41) with (18) and (42) with (19) wediscover that under SRP with two-part tariff contract therisk-neutral fashion retailerrsquos optimal decisions are equal tothose under a single SRP contract Therefore consistent withthe analysis in Section 4 SRP with two-part tariff contractcannot coordinate the supply chain with risk-averse retailerand price-dependent demand

53 Revenue Sharing with Two-Part Tariff Contract Undera revenue sharing with two-part tariff contract the fashionretailerrsquos expected profit and the variance of profit are asfollows

119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876)

(43)

Similar to SRP with two-part tariff contract the optimaljoint ordering-pricing decision for the risk-neutral fashionretailer under revenue sharing with two-part tariff contract isequal to that under a single revenue-sharing contract Hencerevenue sharingwith two-part tariff contract is able to achievechannel coordination in the fashion supply chain with risk-averse retailer and price-dependent demand

As a result it only remains uncertain whether SRPwith revenue-sharing contract could achieve supply chaincoordination with risk-averse retailer Following the similarapproach as presented in Section 4 we now investigatethe role of SRP with revenue-sharing contract in channelcoordination

From (37) by some simple deductions we know thatunder SRP with revenue-sharing contract 119881

119903(119876 119901) is strictly

increasing in 119876 and 119901 Therefore with any given expectedthreshold 119896

119903le 119864119903 the risk-averse retailerrsquos optimal joint

decision (119876119903119898119907

lowast 119901119903119898119907

lowast) satisfies (34)

Proposition 23 For any given 119896119903lt 119864sc SRP with revenue-

sharing contract cannot achieve supply chain coordination

Proof From Proposition 14 we know that channel coor-dination is obtained if and only if SRP with revenue-sharing contract satisfies 119864


119903 120597119864119903(119876

119901sclowast)120597119876|119876=119876sc

lowast ge 0 and 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc

lowast ge 0 we have (120582119901sclowastminus 119908) minus

120582(119901sclowastminus 119907)119865(119876sc

lowast) + 119906119865(119876sc

lowast) ge 0 Combining with (6)

we can derive that 119906119865(119876sclowast) ge 119908 minus 120582119888 Nonetheless from

120597119864119903(119876sclowast 119901)120597119901|


lowastminus 119908 +

119906) + 120582(119876sclowastminus int119876sclowast

0119865(120585)119889120585) ge 0 and from (7) by some

simplifications we have 119906 le 119908 minus 120582119888 Since 119906119865(119876sclowast) lt

119906 there does not exist some value of 119906 that could satisfy119906119865(119876sc

lowast)ge119908 minus 120582119888 and 119906 le 119908 minus 120582119888 simultaneously Thus SRP

with revenue-sharing contract cannot coordinate the fashionsupply chain


Table 1 Optimal values of contract parameters for different values of 119896119903

119896119903

Revenue-sharing contract Two-part tariff contract Revenue sharing with two-part tariff contract119908 120582 119908 119866 120582 119908 119866

1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363

Remark 24 It is interesting to discover that although a singlerevenue-sharing contract itself could coordinate the quantityand pricing decisions in the fashion supply chain with risksensitive retailer the combined SRP with revenue-sharingcontract cannot optimize the whole supply chainrsquos profit Tosome extent this means that when faced with more intricatesupply chain circumstance perhaps a simpler contract ismore effective and efficient to achieve channel coordinationin comparison with a more complicated one

Furthermore when a single revenue-sharing contract anda single two-part tariff contract can coordinate the supplychain with risk-averse retailer and price-dependent demanda composite contract of these two contracts would still beeffective to coordinate the supply chain Instead a single SRPcontract cannot achieve channel coordination thus when itcombines with revenue-sharing contract or two-part tariffcontract the resulting combined contract is still unable tocoordinate the fashion supply chain

6 Numerical Analysis

In this section we present numerical analysis to gain moreinsights on supply chain coordination with contracts Wefocus on the coordinating revenue-sharing contract two-part tariff contract and the combined revenue sharing withtwo-part tariff contract here Numerical analysis can bedecomposed into two parts one is to investigate how todetermine the optimal values of contract parameters andthe other is sensitivity analysis to explore the impacts ofsome important parameters on supply chain coordinationand objectives of supply chain members

61 Determine the Values of Contract Parameters First wegive the values of parameters used in this section Suppose thebase demand 119886 = 600 and the price elasticity of demand 119896 =10The randomvariable 120585 follows a uniformdistributionwitha lower bound 119860 = 0 and an upper bound 119861 = 160 The unitproduction cost 119888 = 15 and the unit salvage value 119907 = 2Withthese parameters the optimal joint decision that maximizesthe expected profit of the integrated fashion supply chainis 119876sc

lowast= 10674 and119901sc

lowast= 4106 and the supply chainrsquos

optimal production level is given by 119902sclowast= 29618 The

respective expected profit and the variance of profit for thefashion supply chain are 119864sc = 632670 and119881sc(119876sc

lowast 119901sclowast) =

193126113 Since the expected profit threshold for the risk-averse retailer must be smaller than the maximum expectedprofit gained by the fashion supply chain we assume 119896

119903lt

632670 in the following analysisWe consider six values of 119896

119903= 1000 2000 3000 4000

5000 and 6000 to explore the optimal values of contractparameters for the coordinating contracts above It should benoted that for the combined revenue sharing with two-parttariff contract 120582 gt 119896

119903119864sc must establish to ensure that119866 gt 0

The results are summarized in Table 1From Table 1 we can see the effectiveness of revenue

sharing two-part tariff and their combined contract incoordinating the fashion supply chain Consistent withProposition 17 120582 in revenue-sharing contract and 119866 in two-part tariff contract can be used to judge the downstreamfashion retailerrsquos risk aversion level For revenue-sharingcontract with a larger 119896

119903 the retailer is more risk averse

thus leading to a higher fraction of sales revenue kept by theretailer himself By anticipating the retailerrsquos response themanufacturer would react by setting a higher wholesale price119908 For two-part tariff contract a wholesale price 119908 equalingto the unit production 119888 gives the manufacturer zero profitbut the fixed transfer payment 119866 ensures a positive profitfor the manufacturer And a higher value of 119866 which theretailer is willing to pay indicates a less risk-averse retailerIn combined contract for the retailerrsquos same risk aversiondegrees the proposition of sales revenue kept by the retailerhimself must be larger than that in the single revenue-sharingcontract owing to the fact that the fashion retailer mustpay the manufacturer an additional fixed payment in thecombined contract

62 Sensitivity Analysis Now we study the effects of someimportant parameters on supply chain coordination andobjectives of supply chain members Firstly we focus onrevenue-sharing contract with parameters such as basedemand 119886 price elasticity of demand 119896 and demand uncer-tainty The results are given in Table 2

From Table 2 we find that with the increase of basedemand 119886 the optimal production quantity 119902sc

lowast and pricing119901sclowast for the integrated fashion supply chain also increase

leading to larger expected profit 119864sc and the variance ofprofit 119881sc This is consistent with Proposition 3 that 119864sc is aconcave function of 119902 and is unimodal in 119901 and119881sc is strictlyincreasing in 119902 and 119901 Taking into account 119896

119903lt 119864sc we


Table 2 Sensitivity analysis for revenue-sharing contract

Parameter 119876sclowast

119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117

consider the appropriate values of 119896119903and find that the optimal

values of 119908 and 119881119903do not exhibit some rule of changes since

120582 changes randomly However when we fix the value of 119896119903

in the region 119896119903lt 119864sc for all cases of 119886 we could intuitively

reach the conclusion that the values of 120582 119908 and 119881119903tend to

decreaseHowever it can be found from Table 2 that with larger

values of price elasticity 119896 the entire supply chainrsquos optimalproduction quantity and retail price become smaller so dothe supply chainrsquos expected profit 119864sc and the variance ofprofit 119881sc On the contrary by fixing values of 119896

119903subject to

119896119903lt 119864sc we discover that the values of119908 120582 and119881119903 all become

largerIn addition we also try to illustrate the effect of different

degrees of demand uncertainty We define demand uncer-tainty as CV = 120579120575 where 120575 represents the mean and 120579denotes the standard variance of the random demand Wecould derive from Table 2 that when the level of demanduncertainty increases the optimal joint quantity and pricingdecisions for the fashion supply chain incline to firstlyincrease and then decrease Thus the supply chainrsquos expectedprofit 119864sc and the variance of profit 119881sc also have the samerule of changes With respect to the values of 119908 and 120582 theychange toward the opposite directionThe combined changesof 120582 and 119881sc cause the changes of 119881119903

Similarly following the same method we could alsoget the results of sensitivity analysis for the other twocoordinating contractsmdashtwo-part tariff contract and revenuesharing with two-part tariff contractThey are summarized inTables 3 and 4 respectively

From Table 3 it can be easily discovered that no matterhow the values of parameter 119886 119896 and CV change the optimalvalues of119908 are always equal to 15 But the optimal values of119866are dependent upon the change in values of those parameters

Table 3 Sensitivity analysis for two-part tariff contract

Parameter 119896119903

119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919

Specifically the optimal119866 equals119864scminus119896119903When parameters 119896and CV change the integrated fashion supply chainrsquos optimalexpected profit 119864sc also change as shown in Table 2 If thevalues of 119896

119903are fixed 119866 changes positively with the changes

of 119864sc That is the values of 119866 decrease in the case of 119896119903and

firstly increase and then decrease in the case of CVSimilar analysis could be realized for the joint revenue

sharing with two-part tariff contract What is worth notinghere is that in Table 4 we could find that the values of 120582are larger than the corresponding values in Table 2 for thesingle revenue-sharing contract whereas the values of 119866 are


Table 4 Sensitivity analysis for revenue-sharing with two-part tariffcontract


120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884

smaller than the corresponding values in the single two-parttariff contract This is because in the combined contract thefashion retailer cannot keep all the sales revenue but has topay an additional fixed transfer payment to themanufacturerAccordingly the values of the variance of profit119881

119903in revenue

sharing with two-part tariff contract are always larger thanthe respective values in revenue-sharing contract alone whilethey obtain their largest values in two-part tariff contractsince they are equal to the according values of 119881sc

7 Management Insightsand Concluding Remark

In this paper the issue of supply chain coordination withrisk-averse retailer and price-dependent demand is studiedWe extend in this paper the previous works to considera risk-averse retailer in a two-stage fashion supply chainAdopting the additive price-sensitive demand model weconstruct the benchmark solution to the integrated fashionsupply chain Using the classic MV formulation in portfoliomanagement in finance to characterize the risk sensitivefashion retailerrsquos decision models we propose both singlecontracts and combined contracts to achieve channel coordi-nationWe find that under single contracts the coordinatingrevenue-sharing contract and two-part tariff contract insupply chains with risk-neutral agents could still coordinatethe supply chain with risk sensitive retailer However a morecomplicated sales rebate and penalty contract fails to doso Then we try to combine traditional single contracts toexplore whether the resulting joint contracts are useful tocoordinate the supply chain It is found that only the jointrevenue sharingwith two-part tariff contract is able to achievesupply chain coordination By presenting numerical analysisto illustrate analytical results we discuss the determination

of optimal values of contract parameters in coordinatingcontracts as well as sensitivity analysis to explore the effectsof base demand price elasticity and the degree of demanduncertainty on supply chain coordination and objectives ofsupply chain members

We highlight the managerial insights of our results inthe following Mean-variance formulation for risk analysisin supply chains is intuitive to decisions makers making iteasy and applicable for managers and practitioners in fashionsupply chains to use proposed models to determine the typeof contract and the optimal values of contract parametersto achieve channel coordination This paper captures thefundamental feature of retailing fashion channel that thefashion retailer could influence the end demand by settingappropriate retail price Our findings indicate that in suchcomplicated retailing situation it may be more effective forreal managers to adopt relatively simple contracts so as toachieve coordination Moreover the theoretical results offerreferences to decision makers on the conditions a contractmust satisfy to coordinate fashion supply chains which issignificant for improving efficiency in such supply chainswitheffective retailing channels and fashion retailerrsquos risk aversionpreference or attitude

In this paper we just focus on channel coordination insingle period single manufacture and single retailer fashionsupply chain Future research on supply chain coordinationover multiple periods or with multiple competing risk-averseretailers would be a meaningful direction and could producemore insights

Acknowledgments

The authors sincerely thank the Editor-in-Chief and thethree anonymous reviewers for their valuable comments andsuggestions on the revision of the paper This paper wassupported in part by (1) the Fund for Humanity and SocialScience of the Ministry of Education China under Grant09YJC630230 (2) the Natural Science Foundation of HunanProvince China under Grant 10JJ3023

References

[1] T M Choi Fashion Supply Chain Management Industry andBusiness Analysis IGI Global 2011

[2] T M Choi and C H Chiu ldquoMean-downside-risk and mean-variance newsvendor models implications for sustainable fash-ion retailingrdquo International Journal of Production Economicsvol 135 no 2 pp 552ndash560 2010

[3] J J Spengler ldquoVertical integration and antitrust policyrdquo TheJournal of Political Economy vol 58 no 4 pp 347ndash352 1950

[4] B A Pasternack ldquoOptimal pricing and return policies forperishable commoditiesrdquo Marketing Science vol 4 no 2 pp166ndash176 1985

[5] T M Choi D Li H Yan and C H Chiu ldquoChannel coor-dination in supply chains with agents having mean-varianceobjectivesrdquo Omega vol 36 no 4 pp 565ndash576 2008

[6] G P Cachon and M A Lariviere ldquoSupply chain coordinationwith revenue-sharing contracts strengths and limitationsrdquoManagement Science vol 51 no 1 pp 30ndash44 2005


[7] A A Tsay S Nahmias and N Agrawal ldquoModeling supplychain contracts a reviewrdquo in International Series in OperationsResearch and Management Science pp 299ndash336 1999

[8] Z Lian and A Deshmukh ldquoAnalysis of supply contracts withquantity flexibilityrdquo European Journal of Operational Researchvol 196 no 2 pp 526ndash533 2009

[9] C J Corbett D Zhou and C S Tang ldquoDesigning supply con-tracts contract type and information asymmetryrdquoManagementScience vol 50 no 4 pp 550ndash559 2004

[10] T A Taylor ldquoSupply chain coordination under channel rebateswith sales effort effectsrdquoManagement Science vol 48 no 8 pp992ndash1007 2002

[11] W K Wong J Qi and S Y S Leung ldquoCoordinating supplychains with sales rebate contracts and vendor-managed inven-toryrdquo International Journal of Production Economics vol 120 no1 pp 151ndash161 2009

[12] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research and Management Sciencevol 11 no 1 pp 229ndash340 2003

[13] CHChiu TMChoiH T Yeung andYX Zhao ldquoSales rebatecontracts in fashion supply chainsrdquo Mathematical Problems inEngineering vol 2012 Article ID 908408 19 pages 2012

[14] J D Dana and K E Spier ldquoRevenue sharing and verticalcontrol in the video rental industryrdquo The Journal of IndustrialEconomics vol 49 no 3 pp 223ndash245 2001

[15] J H Mortimer ldquoVertical contracts in the video rental industryrdquoReview of Economic Studies vol 75 no 1 pp 165ndash199 2008

[16] E Cao C Wan and M Lai ldquoCoordination of a supply chainwith one manufacturer and multiple competing retailers undersimultaneous demand and cost disruptionsrdquo International Jour-nal of Production Economics vol 141 no 1 pp 425ndash433 2013

[17] D Hu and Q Li ldquoSupply chain coordination under revenue-sharing contract with sales effort effectrdquo in Proceedings ofthe International Conference on Computer and Management(CAMAN rsquo11) pp 1ndash4 May 2011

[18] K Chen and T Xiao ldquoOrdering policy and coordination of asupply chain with two-period demand uncertaintyrdquo EuropeanJournal of Operational Research vol 215 no 2 pp 347ndash357 2011

[19] C H Chiu T M Choi and C S Tang ldquoPrice rebateand returns supply contracts for coordinating supply chainswith price-dependent demandsrdquo Production and OperationsManagement vol 20 no 1 pp 81ndash91 2011

[20] Y He X Zhao L Zhao and J He ldquoCoordinating a supply chainwith effort and price dependent stochastic demandrdquo AppliedMathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[21] H Krishnan R Kapuscinski and D A Butz ldquoCoordinatingcontracts for decentralized supply chains with retailer promo-tional effortrdquo Management Science vol 50 no 1 pp 48ndash632004

[22] H Emmons and S M Gilbert ldquoNote The role of returnspolicies in pricing and inventory decisions for catalogue goodsrdquoManagement Science vol 44 no 2 pp 276ndash283 1998

[23] Y Qin H Tang and C Guo ldquoChannel coordination andvolume discounts with price-sensitive demandrdquo InternationalJournal of Production Economics vol 105 no 1 pp 43ndash53 2007

[24] H M Markowitz Portfolio Selection Efficient Diversification ofInvestments 1959

[25] P C Fishburn ldquoMean-risk analysis with risk associated withbelow-target returnsrdquo The American Economic Review vol 67no 1 pp 116ndash126 1977

[26] S Choi and A Ruszczynski ldquoA risk-averse newsvendor withlaw invariant coherent measures of riskrdquo Operations ResearchLetters vol 36 no 1 pp 77ndash82 2008

[27] C H Chiu and T M Choi ldquoOptimal pricing and stockingdecisions for newsvendor problem with value-at-risk consider-ationrdquo IEEE Transactions on Systems Man and Cybernetics Avol 40 no 5 pp 1116ndash1119 2010

[28] J Y Gotoh and Y Takano ldquoNewsvendor solutions via condi-tional value-at-risk minimizationrdquo European Journal of Opera-tional Research vol 179 no 1 pp 80ndash96 2007

[29] Y H ChenM H Xu and Z G Zhang ldquoA risk-averse newsven-dor model under the CVaR criterionrdquo Operations Research vol57 no 4 pp 1040ndash1044 2009

[30] K S Sundar S Narayanan and D Nagaraju ldquoCoordination intwo level supply chain with price dependent demandrdquo AppliedMathematical Sciences vol 6 no 74 pp 3687ndash3703 2012

[31] J-M Chen H-L Cheng and M-C Chien ldquoOn channelcoordination through revenue-sharing contracts with priceand shelf-space dependent demandrdquo Applied MathematicalModelling vol 35 no 10 pp 4886ndash4901 2011

[32] F J Arcelus S Kumar and G Srinivasan ldquoEvaluating man-ufacturerrsquos buyback policies in a single-period two-echelonframework under price-dependent stochastic demandrdquoOmegavol 36 no 5 pp 808ndash824 2008

[33] V Agrawal and S Seshadri ldquoImpact of uncertainty and riskaversion on price and order quantity in the newsvendor prob-lemrdquo Manufacturing and Service Operations Management vol2 no 4 pp 410ndash423 2000

[34] C-H Chiu T-M Choi and X Li ldquoSupply chain coordinationwith risk sensitive retailer under target sales rebaterdquo Automat-ica vol 47 no 8 pp 1617ndash1625 2011

[35] X Gan S P Sethi and H Yan ldquoCoordination of supplychains with risk-averse agentsrdquo Supply Chain Coordinationunder Uncertainty vol 3 no 1 pp 135ndash149 2004

[36] YWei and TM Choi ldquoMean-variance analysis of supply chainsunder wholesale pricing and profit sharing schemesrdquo EuropeanJournal of Operational Research vol 204 no 2 pp 255ndash2622010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


wholesale price for each unit purchased And it has also beenshown that a two-part tariff contract coordinates the supplychain when the optimal value of unit wholesale price equalsthe manufacturerrsquos unit production cost [9]

Sales rebate and penalty (SRP) is based on the retailerrsquossales performance With a SRP contract the manufacturerwill specify a certain sales target prior to the selling seasonDifferent from sales rebate which executes rebate only foreach unit sold above the target level the retailer will begranted a unit rebate or else the retailer must pay themanufacturer a penalty In supply chain management bothSRP contract and sales rebate contract have been demon-strated unable to coordinate the channel when the demand issensitive to retail price or the retailerrsquos sales effort [10 19ndash21]

Early studies considered retail price exogenous leavingthe retailer with the decision of order quantity alone inorder to maximize expected profit As retail price plays animportant role inmarketing channel a new steam of researchon supply chain coordination and contracting integratespricing into the order quantity decision of the retailer underdifferent demand models Reviews of this work [12 22]explicitly stated that revenue-sharing contract and two-parttariff contract are able to coordinate the newsvendor withprice-dependent demand while many other traditional con-tracts arenrsquot And there is an increasing interest in examiningcombined contracts consisting of two or more traditionalcontracts to achieve channel coordination [19ndash21 23]

However the common results derived from the previousstudies may not be precise in operations management sincein the real world different decisionmakersmay have differentdegrees of risk aversion in light of this we extend theresults of proceeding studies to explore the issue of supplychain coordination with risk-averse fashion retailer andprice-dependent demand Specifically we investigate a singleperiod one-manufacturer one-retailer fashion supply chainwith a variety of contracts The manufacturer acting as theleader in the Stackelberg game offers the retailer a contractwith a set of contract parameters The fashion retailer actingas the follower sets self-interest order quantity and retailprice as a response We propose both single contracts andcombined contracts with the optimal values of contractparameters to achieve channel coordination within fashionsupply chain

The main objectives of our study cover the followingfirstly to explore whether the coordinating revenue-sharingcontract and two-part tariff contract in supply chains withrisk neural retailer can still coordinate the fashion supplychain with risk-averse retailer who has to choose retail pricein addition to stocking quantity secondly to compare theperformance of a more complicated sales rebate and penaltycontract in supply chain coordination with the performancesof revenue-sharing contract and two-part tariff contractfinally when joint contracts are got by taking advantages ofthe three single contracts to probe whether the resultingcombined contracts are useful to coordinate the supply chain

In recent years an increasing number of researchershave noticed the importance and the impact of risk aver-sion in supply chain contracting and coordination andsought in succession for the criteria to depict supply chain

agentsrsquo risk aversion attitude or preference In the literaturethe measures for describing risk aversion involve mean-variance (MV) [24] Neumann-Morgenstern utility function(VNUM) mean-downside-risk (MDR) [25] Value-at-risk(VaR) [26 27] and Conditional Value-at-risk (CVaR) [2829] Since MV is simple implementable and is easily under-stood by managers and practitioners compared with othermeasures we adopt mean-variance formulation to capturethe fashion retailerrsquos risk aversion in this paper

This paper is closely linked to the literature on supplychain coordinating and contracting with price-dependentdemand [30 31] in terms of a random and price-dependentdemand It is also correlated to studies of supply chaincoordination with agents having risk preferences in whichwe consider a risk-averse retailer [32ndash36] But our studyis the first to investigate the issue of channel coordinationfor the supply chain with risk-averse retailer and price-dependent demand We firstly investigate the problem ofcoordinating a two-stage fashion supply chain under singlecontracts including revenue-sharing contract two-part tariffcontract and sales rebate and penalty contract After provingthat revenue-sharing contract and two-part tariff contractcould still achieve channel coordination in this contextwhile a more complex sales rebate and penalty cannot wefurther explore the role of combined contracts (sales rebateand penalty with revenue-sharing contract sales rebate andpenalty with two-part tariff contract and revenue sharingwith two-part tariff contract) in supply chain coordinationBy identifying the coordination conditions and mechanismsof various contracts our work contributes to supplement thecurrent literature on supply chain coordination and contract-ing We also provide meaningful guidance to managers inreal operations management on how to choose the type ofcontract and determine the optimal contract parameters inorder to coordinate fashion supply chain inmore complicatednewsvendor frameworks

The paper is organized as followsModel formulation andnotation definition are presented in Section 2 The bench-mark case of integrated fashion supply chain is studied inSection 3 Supply chain coordination under single contractsand combined contracts is investigated in Sections 4 and5 Numerical study to supplement the analytical results andgain more insights is given in Section 6 Section 7 providesmanagerial insights and concluding comments

2 Model Formulation and Notation Definition

Consider a two-echelon fashion supply chain with a risk-neutral manufacturer and a risk-averse retailer The retailersells a fashion product whose demand is sensitive to retailprice The upstream manufacturer produces the product andsells it through a vertically separated retailer The sequenceof events in the supply chain is as follows The manufactureras the leader of a Stackelberg game offers the retailer acontract After knowing the details of the contract thefashion retailer commits his order quantity and retail priceThen themanufacturer organizes the production and deliversthe finished products to the retailer prior to the selling seasonAfterwards the selling season starts and the demand is







119889119890

119889119901ge 0 (1)




1199031015840

(120585) ge 0 (2)



min 119881119903(119902 119901)

st 119864119903(119902 119901) ge 119896

119903

(P-1)

where119864119903(119902 119901) and119881







119903= 119864119903(119902119903

lowast 119901119903






0119865(120585)119889120585 minus 2 int

119876

0120585119865(120585)119889120585 minus

(int119876

0119865(120585)119889120585)



(3)


0

119865 (120585) 119889120585

(4)

119881sc (119876 119901) = (119901 minus 119907)2

120578 (119876) (5)


lowast= 119863(119901sc

lowast) +119876sc




lowastminus 119907) 119865 (119876sc

lowast) = 0 (6)


lowastminus int

119876sclowast

0

119865 (120585) 119889120585 = 0 (7)







2

=







119889119901= minus1205972119864sc (119876sc

lowast 119901) 120597119876120597119901

1205972119864sc (119876sclowast 119901) 1205971198762

=1 minus 119865 (119876sc

lowast)

(119901 minus 119907) 119891 (119876sclowast)

(10)





119901 we derive

119889119864sc (119876sclowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119888) + 119876sc

lowastminus int

119876sclowast

0

119865 (120585) 119889120585

(11)

1198892119864sc (119876sc

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876sc


119889119901 (12)


1198892119864sc (119876sc

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876sclowast))2

(119901 minus 119907) 119891 (119876sclowast) (13)


119889119867 (119876)

119889119876=(1 minus 119865 (119876)) 119891

1015840(119876) + 2(119891 (119876))

2

(1 minus 119865 (119876))3

(14)




lowast 119901)119889119901

2= 0 If 1199010


2lt 0


lowast 119901)119889119901

2=minus2119896lt0 and 119864sc(119876sc

lowast 119901)


lowast 119901)119889119901

2gt 0 and for 119901gt1199010 1205972119864sc(119876sc

lowast 119901)120597119901

2lt



119901=119888= (119886 minus

119896119888) + int119876sclowast

(119888)

0119865(120585)119889120585 gt 0 119864sc(119876sc



lowastisin [119888 119886119896]



lowast) + 119876sc









lowast) Let (119876

119903

lowast 119901119903

lowast) and (119876

119903119898119907

lowast 119901119903119898119907

lowast) be the




prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879)

(15)

119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(16)

119881119903(119876 119901)

= 119864(prod119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

119864(int

119876

0

119865 (120585) 119889120585 minus (119876 minus 120585)+

)

2


119876

0

119865 (120585) 119889120585))

2

+ 2119906 (119901 minus 119907) 119864(int

119876

0

119865 (120585) 119889120585 minus (119876 minus 120585)+

)


0

119865 (120585) 119889120585))


= (119901 minus 119907)2

(int

119876

0

(119876 minus 120585)2

119891 (120585) 119889120585 minus (int

119876

0

119865 (120585) 119889120585)

2

)

+ 1199062(int

119876

0

(120585 minus 119879)2

119891 (120585) 119889120585 + int

infin

119876

(119876 minus 119879)2119891 (120585) 119889120585


119876

0

119865 (120585) 119889120585)

2

)

+ 2119906 (119901 minus 119907) (int

119876

0

119865 (120585) 119889120585 (119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

minusint

119876

0

(119876 minus 120585) (120585 minus 119879) 119891 (120585) 119889120585)

= (119901 minus 119907 + 119906)2

(2119876int

119876

0

119865 (120585) 119889120585 minus 2int

119876

0

120585119865 (120585) 119889120585

minus(int

119876

0

119865 (120585) 119889120585)

2

)

= (119901 minus 119907 + 119906)2

120578 (119876)

(17)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (18)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (19)




and 1205972119864119903(119876 119901)120597119876


119903(119876 119901)




lowast) = 0 (20)



119889119876119903

lowast

119889119901= minus1205972119864119903(119876119903

lowast 119901) 120597119876120597119901

1205972119864119903(119876119903

lowast 119901) 1205971198762

=1 minus 119865 (119876

119903

lowast)

(119901 minus 119907 + 119906) 119891 (119876119903

lowast)

(21)



119903


119864119903(119876119903

lowast 119901)

= (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876119903


119876119903

lowast

0

119865 (120585) 119889120585

+ 119906(119876119903


119876119903

lowast

0

119865 (120585) 119889120585)

(22)

From (22) 119864119903(119876119903




119889119864119903(119876119903

lowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585

(23)

1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876

119903

lowast))119889119876119903

lowast

119889119901 (24)


1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876119903

lowast))2

(119901 minus 119907 + 119906) 119891 (119876119903

lowast) (25)


lowast 119901) is


lowast 119901)119889119901|

119901=119908minus119906= (119886 minus

119896(119908 minus 119906)) + int119876119903

lowast

(119908minus119906)


119901119903






119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585

(26)

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876) (27)



119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (28)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (29)







119864119903(119876 119901) = (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876


119876

0

119865 (120585) 119889120585 minus 119866

(30)

119881119903(119876 119901) = 119864(prod

119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

120578 (119876)

(31)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (32)

119886 minus 119896 (2119901119903

lowastminus 119908) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (33)



119903

lowast 119901119903









119876




in 119901



lowast 119901119903119898119907

lowast) satisfies

119864119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903

119876119903119898119907

lowastle 119876119903

lowast

119901119903119898119907

lowastle 119901119903

lowast

(34)





119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903


119903

lowast 119901 le

119901119903

lowast) (119876 gt 119876

119903

lowast 119901 le 119901

119903

lowast) (119876 le 119876

119903

lowast 119901 gt 119901

119903

lowast) and

(119876 gt 119876119903

lowast 119901 gt 119901

119903





lowast 119901119903119898119907


119876119903

lowast 119901 le 119901

119903


119903119898119907

lowast 119901119903119898119907



lowastle 119876119903

lowast and 119901119903119898119907

lowastle 119901119903

lowast



119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc





119903 (2) 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0


119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc



119903 On the other



119903(119876 119901)






119903(119876 119901)ge119896

119903and 119881

119903(119876 119901)lt119881





If (119876119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc




lowast Since 119864119903(119876 119901) is a


lowast]


lowast we have 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119903(119876 119901sc

lowast)120597119876|119876=119876sc





119864119903(119876 119901sc

lowast) and 119881

119903(119876 119901sc

lowast) lt 119881





119901119903



120597119864119903(119876sclowast 119901)120597119901|


119903(119876 119901) is strictly



119903(119876sclowast 119901) and

119881119903(119876sclowast 119901) lt 119881




lowast)120597119876|119876=119876sc

lowast ge

0 and 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0








119903 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



119903(119876sclowast 119901)120597119901|




0119865(120585)119889120585 ge 0 And from (7) it













119876sclowast

0119865(120585)119889120585)119902sc

lowast From120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


lowastminus



lowastminus


120597119864119903(119876sclowast 119901)120597119901|










lowastminus(119896119903+ (119901sc


119876sclowast

0119865(120585)119889120585 +

119866)119902sclowast From 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc




120597119864119903(119876sclowast 119901)120597119901|



lowastminus

int119876sclowast



lowastminus


0119865(120585)119889120585 + 119866)119902sc











prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879) (35)

119864119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 + 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(36)

119881119903(119876 119901) = [120582 (119901 minus 119907) + 119906]

2

120578 (119876) (37)


119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) + 119906] 119865 (119876

119903

lowast) = 0 (38)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908 + 119906) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (39)



119903

lowast 119901119903

lowast) = (119876sc




119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585) minus 119866

119881119903(119876 119901) = (119901 minus 119907 + 119906)

2

120578 (119876)

(40)


119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (41)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (42)



119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876)

(43)




119903(119876 119901) is strictly



decision (119876119903119898119907

lowast 119901119903119898119907






119903 120597119864119903(119876

119901sclowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



120597119864119903(119876sclowast 119901)120597119901|











119896119903


1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363












119903lt


119903= 1000 2000 3000 4000











119903lt 119864sc we




119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119889119890

119889119901ge 0 (1)




1199031015840

(120585) ge 0 (2)



min 119881119903(119902 119901)

st 119864119903(119902 119901) ge 119896

119903

(P-1)

where119864119903(119902 119901) and119881







119903= 119864119903(119902119903

lowast 119901119903






0119865(120585)119889120585 minus 2 int

119876

0120585119865(120585)119889120585 minus

(int119876

0119865(120585)119889120585)



(3)


0

119865 (120585) 119889120585

(4)

119881sc (119876 119901) = (119901 minus 119907)2

120578 (119876) (5)


lowast= 119863(119901sc

lowast) +119876sc




lowastminus 119907) 119865 (119876sc

lowast) = 0 (6)


lowastminus int

119876sclowast

0

119865 (120585) 119889120585 = 0 (7)







2

=







119889119901= minus1205972119864sc (119876sc

lowast 119901) 120597119876120597119901

1205972119864sc (119876sclowast 119901) 1205971198762

=1 minus 119865 (119876sc

lowast)

(119901 minus 119907) 119891 (119876sclowast)

(10)





119901 we derive

119889119864sc (119876sclowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119888) + 119876sc

lowastminus int

119876sclowast

0

119865 (120585) 119889120585

(11)

1198892119864sc (119876sc

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876sc


119889119901 (12)


1198892119864sc (119876sc

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876sclowast))2

(119901 minus 119907) 119891 (119876sclowast) (13)


119889119867 (119876)

119889119876=(1 minus 119865 (119876)) 119891

1015840(119876) + 2(119891 (119876))

2

(1 minus 119865 (119876))3

(14)




lowast 119901)119889119901

2= 0 If 1199010


2lt 0


lowast 119901)119889119901

2=minus2119896lt0 and 119864sc(119876sc

lowast 119901)


lowast 119901)119889119901

2gt 0 and for 119901gt1199010 1205972119864sc(119876sc

lowast 119901)120597119901

2lt



119901=119888= (119886 minus

119896119888) + int119876sclowast

(119888)

0119865(120585)119889120585 gt 0 119864sc(119876sc



lowastisin [119888 119886119896]



lowast) + 119876sc









lowast) Let (119876

119903

lowast 119901119903

lowast) and (119876

119903119898119907

lowast 119901119903119898119907

lowast) be the




prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879)

(15)

119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(16)

119881119903(119876 119901)

= 119864(prod119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

119864(int

119876

0

119865 (120585) 119889120585 minus (119876 minus 120585)+

)

2


119876

0

119865 (120585) 119889120585))

2

+ 2119906 (119901 minus 119907) 119864(int

119876

0

119865 (120585) 119889120585 minus (119876 minus 120585)+

)


0

119865 (120585) 119889120585))


= (119901 minus 119907)2

(int

119876

0

(119876 minus 120585)2

119891 (120585) 119889120585 minus (int

119876

0

119865 (120585) 119889120585)

2

)

+ 1199062(int

119876

0

(120585 minus 119879)2

119891 (120585) 119889120585 + int

infin

119876

(119876 minus 119879)2119891 (120585) 119889120585


119876

0

119865 (120585) 119889120585)

2

)

+ 2119906 (119901 minus 119907) (int

119876

0

119865 (120585) 119889120585 (119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

minusint

119876

0

(119876 minus 120585) (120585 minus 119879) 119891 (120585) 119889120585)

= (119901 minus 119907 + 119906)2

(2119876int

119876

0

119865 (120585) 119889120585 minus 2int

119876

0

120585119865 (120585) 119889120585

minus(int

119876

0

119865 (120585) 119889120585)

2

)

= (119901 minus 119907 + 119906)2

120578 (119876)

(17)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (18)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (19)




and 1205972119864119903(119876 119901)120597119876


119903(119876 119901)




lowast) = 0 (20)



119889119876119903

lowast

119889119901= minus1205972119864119903(119876119903

lowast 119901) 120597119876120597119901

1205972119864119903(119876119903

lowast 119901) 1205971198762

=1 minus 119865 (119876

119903

lowast)

(119901 minus 119907 + 119906) 119891 (119876119903

lowast)

(21)



119903


119864119903(119876119903

lowast 119901)

= (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876119903


119876119903

lowast

0

119865 (120585) 119889120585

+ 119906(119876119903


119876119903

lowast

0

119865 (120585) 119889120585)

(22)

From (22) 119864119903(119876119903




119889119864119903(119876119903

lowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585

(23)

1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876

119903

lowast))119889119876119903

lowast

119889119901 (24)


1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876119903

lowast))2

(119901 minus 119907 + 119906) 119891 (119876119903

lowast) (25)


lowast 119901) is


lowast 119901)119889119901|

119901=119908minus119906= (119886 minus

119896(119908 minus 119906)) + int119876119903

lowast

(119908minus119906)


119901119903






119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585

(26)

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876) (27)



119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (28)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (29)







119864119903(119876 119901) = (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876


119876

0

119865 (120585) 119889120585 minus 119866

(30)

119881119903(119876 119901) = 119864(prod

119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

120578 (119876)

(31)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (32)

119886 minus 119896 (2119901119903

lowastminus 119908) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (33)



119903

lowast 119901119903









119876




in 119901



lowast 119901119903119898119907

lowast) satisfies

119864119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903

119876119903119898119907

lowastle 119876119903

lowast

119901119903119898119907

lowastle 119901119903

lowast

(34)





119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903


119903

lowast 119901 le

119901119903

lowast) (119876 gt 119876

119903

lowast 119901 le 119901

119903

lowast) (119876 le 119876

119903

lowast 119901 gt 119901

119903

lowast) and

(119876 gt 119876119903

lowast 119901 gt 119901

119903





lowast 119901119903119898119907


119876119903

lowast 119901 le 119901

119903


119903119898119907

lowast 119901119903119898119907



lowastle 119876119903

lowast and 119901119903119898119907

lowastle 119901119903

lowast



119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc





119903 (2) 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0


119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc



119903 On the other



119903(119876 119901)






119903(119876 119901)ge119896

119903and 119881

119903(119876 119901)lt119881





If (119876119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc




lowast Since 119864119903(119876 119901) is a


lowast]


lowast we have 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119903(119876 119901sc

lowast)120597119876|119876=119876sc





119864119903(119876 119901sc

lowast) and 119881

119903(119876 119901sc

lowast) lt 119881





119901119903



120597119864119903(119876sclowast 119901)120597119901|


119903(119876 119901) is strictly



119903(119876sclowast 119901) and

119881119903(119876sclowast 119901) lt 119881




lowast)120597119876|119876=119876sc

lowast ge

0 and 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0








119903 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



119903(119876sclowast 119901)120597119901|




0119865(120585)119889120585 ge 0 And from (7) it













119876sclowast

0119865(120585)119889120585)119902sc

lowast From120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


lowastminus



lowastminus


120597119864119903(119876sclowast 119901)120597119901|










lowastminus(119896119903+ (119901sc


119876sclowast

0119865(120585)119889120585 +

119866)119902sclowast From 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc




120597119864119903(119876sclowast 119901)120597119901|



lowastminus

int119876sclowast



lowastminus


0119865(120585)119889120585 + 119866)119902sc











prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879) (35)

119864119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 + 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(36)

119881119903(119876 119901) = [120582 (119901 minus 119907) + 119906]

2

120578 (119876) (37)


119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) + 119906] 119865 (119876

119903

lowast) = 0 (38)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908 + 119906) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (39)



119903

lowast 119901119903

lowast) = (119876sc




119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585) minus 119866

119881119903(119876 119901) = (119901 minus 119907 + 119906)

2

120578 (119876)

(40)


119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (41)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (42)



119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876)

(43)




119903(119876 119901) is strictly



decision (119876119903119898119907

lowast 119901119903119898119907






119903 120597119864119903(119876

119901sclowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



120597119864119903(119876sclowast 119901)120597119901|











119896119903


1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363












119903lt


119903= 1000 2000 3000 4000











119903lt 119864sc we




119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2

=







119889119901= minus1205972119864sc (119876sc

lowast 119901) 120597119876120597119901

1205972119864sc (119876sclowast 119901) 1205971198762

=1 minus 119865 (119876sc

lowast)

(119901 minus 119907) 119891 (119876sclowast)

(10)





119901 we derive

119889119864sc (119876sclowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119888) + 119876sc

lowastminus int

119876sclowast

0

119865 (120585) 119889120585

(11)

1198892119864sc (119876sc

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876sc


119889119901 (12)


1198892119864sc (119876sc

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876sclowast))2

(119901 minus 119907) 119891 (119876sclowast) (13)


119889119867 (119876)

119889119876=(1 minus 119865 (119876)) 119891

1015840(119876) + 2(119891 (119876))

2

(1 minus 119865 (119876))3

(14)




lowast 119901)119889119901

2= 0 If 1199010


2lt 0


lowast 119901)119889119901

2=minus2119896lt0 and 119864sc(119876sc

lowast 119901)


lowast 119901)119889119901

2gt 0 and for 119901gt1199010 1205972119864sc(119876sc

lowast 119901)120597119901

2lt



119901=119888= (119886 minus

119896119888) + int119876sclowast

(119888)

0119865(120585)119889120585 gt 0 119864sc(119876sc



lowastisin [119888 119886119896]



lowast) + 119876sc









lowast) Let (119876

119903

lowast 119901119903

lowast) and (119876

119903119898119907

lowast 119901119903119898119907

lowast) be the




prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879)

(15)

119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(16)

119881119903(119876 119901)

= 119864(prod119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

119864(int

119876

0

119865 (120585) 119889120585 minus (119876 minus 120585)+

)

2


119876

0

119865 (120585) 119889120585))

2

+ 2119906 (119901 minus 119907) 119864(int

119876

0

119865 (120585) 119889120585 minus (119876 minus 120585)+

)


0

119865 (120585) 119889120585))


= (119901 minus 119907)2

(int

119876

0

(119876 minus 120585)2

119891 (120585) 119889120585 minus (int

119876

0

119865 (120585) 119889120585)

2

)

+ 1199062(int

119876

0

(120585 minus 119879)2

119891 (120585) 119889120585 + int

infin

119876

(119876 minus 119879)2119891 (120585) 119889120585


119876

0

119865 (120585) 119889120585)

2

)

+ 2119906 (119901 minus 119907) (int

119876

0

119865 (120585) 119889120585 (119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

minusint

119876

0

(119876 minus 120585) (120585 minus 119879) 119891 (120585) 119889120585)

= (119901 minus 119907 + 119906)2

(2119876int

119876

0

119865 (120585) 119889120585 minus 2int

119876

0

120585119865 (120585) 119889120585

minus(int

119876

0

119865 (120585) 119889120585)

2

)

= (119901 minus 119907 + 119906)2

120578 (119876)

(17)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (18)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (19)




and 1205972119864119903(119876 119901)120597119876


119903(119876 119901)




lowast) = 0 (20)



119889119876119903

lowast

119889119901= minus1205972119864119903(119876119903

lowast 119901) 120597119876120597119901

1205972119864119903(119876119903

lowast 119901) 1205971198762

=1 minus 119865 (119876

119903

lowast)

(119901 minus 119907 + 119906) 119891 (119876119903

lowast)

(21)



119903


119864119903(119876119903

lowast 119901)

= (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876119903


119876119903

lowast

0

119865 (120585) 119889120585

+ 119906(119876119903


119876119903

lowast

0

119865 (120585) 119889120585)

(22)

From (22) 119864119903(119876119903




119889119864119903(119876119903

lowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585

(23)

1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876

119903

lowast))119889119876119903

lowast

119889119901 (24)


1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876119903

lowast))2

(119901 minus 119907 + 119906) 119891 (119876119903

lowast) (25)


lowast 119901) is


lowast 119901)119889119901|

119901=119908minus119906= (119886 minus

119896(119908 minus 119906)) + int119876119903

lowast

(119908minus119906)


119901119903






119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585

(26)

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876) (27)



119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (28)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (29)







119864119903(119876 119901) = (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876


119876

0

119865 (120585) 119889120585 minus 119866

(30)

119881119903(119876 119901) = 119864(prod

119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

120578 (119876)

(31)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (32)

119886 minus 119896 (2119901119903

lowastminus 119908) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (33)



119903

lowast 119901119903









119876




in 119901



lowast 119901119903119898119907

lowast) satisfies

119864119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903

119876119903119898119907

lowastle 119876119903

lowast

119901119903119898119907

lowastle 119901119903

lowast

(34)





119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903


119903

lowast 119901 le

119901119903

lowast) (119876 gt 119876

119903

lowast 119901 le 119901

119903

lowast) (119876 le 119876

119903

lowast 119901 gt 119901

119903

lowast) and

(119876 gt 119876119903

lowast 119901 gt 119901

119903





lowast 119901119903119898119907


119876119903

lowast 119901 le 119901

119903


119903119898119907

lowast 119901119903119898119907



lowastle 119876119903

lowast and 119901119903119898119907

lowastle 119901119903

lowast



119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc





119903 (2) 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0


119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc



119903 On the other



119903(119876 119901)






119903(119876 119901)ge119896

119903and 119881

119903(119876 119901)lt119881





If (119876119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc




lowast Since 119864119903(119876 119901) is a


lowast]


lowast we have 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119903(119876 119901sc

lowast)120597119876|119876=119876sc





119864119903(119876 119901sc

lowast) and 119881

119903(119876 119901sc

lowast) lt 119881





119901119903



120597119864119903(119876sclowast 119901)120597119901|


119903(119876 119901) is strictly



119903(119876sclowast 119901) and

119881119903(119876sclowast 119901) lt 119881




lowast)120597119876|119876=119876sc

lowast ge

0 and 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0








119903 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



119903(119876sclowast 119901)120597119901|




0119865(120585)119889120585 ge 0 And from (7) it













119876sclowast

0119865(120585)119889120585)119902sc

lowast From120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


lowastminus



lowastminus


120597119864119903(119876sclowast 119901)120597119901|










lowastminus(119896119903+ (119901sc


119876sclowast

0119865(120585)119889120585 +

119866)119902sclowast From 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc




120597119864119903(119876sclowast 119901)120597119901|



lowastminus

int119876sclowast



lowastminus


0119865(120585)119889120585 + 119866)119902sc











prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879) (35)

119864119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 + 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(36)

119881119903(119876 119901) = [120582 (119901 minus 119907) + 119906]

2

120578 (119876) (37)


119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) + 119906] 119865 (119876

119903

lowast) = 0 (38)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908 + 119906) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (39)



119903

lowast 119901119903

lowast) = (119876sc




119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585) minus 119866

119881119903(119876 119901) = (119901 minus 119907 + 119906)

2

120578 (119876)

(40)


119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (41)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (42)



119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876)

(43)




119903(119876 119901) is strictly



decision (119876119903119898119907

lowast 119901119903119898119907






119903 120597119864119903(119876

119901sclowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



120597119864119903(119876sclowast 119901)120597119901|











119896119903


1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363












119903lt


119903= 1000 2000 3000 4000











119903lt 119864sc we




119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= (119901 minus 119907)2

(int

119876

0

(119876 minus 120585)2

119891 (120585) 119889120585 minus (int

119876

0

119865 (120585) 119889120585)

2

)

+ 1199062(int

119876

0

(120585 minus 119879)2

119891 (120585) 119889120585 + int

infin

119876

(119876 minus 119879)2119891 (120585) 119889120585


119876

0

119865 (120585) 119889120585)

2

)

+ 2119906 (119901 minus 119907) (int

119876

0

119865 (120585) 119889120585 (119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

minusint

119876

0

(119876 minus 120585) (120585 minus 119879) 119891 (120585) 119889120585)

= (119901 minus 119907 + 119906)2

(2119876int

119876

0

119865 (120585) 119889120585 minus 2int

119876

0

120585119865 (120585) 119889120585

minus(int

119876

0

119865 (120585) 119889120585)

2

)

= (119901 minus 119907 + 119906)2

120578 (119876)

(17)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (18)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (19)




and 1205972119864119903(119876 119901)120597119876


119903(119876 119901)




lowast) = 0 (20)



119889119876119903

lowast

119889119901= minus1205972119864119903(119876119903

lowast 119901) 120597119876120597119901

1205972119864119903(119876119903

lowast 119901) 1205971198762

=1 minus 119865 (119876

119903

lowast)

(119901 minus 119907 + 119906) 119891 (119876119903

lowast)

(21)



119903


119864119903(119876119903

lowast 119901)

= (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876119903


119876119903

lowast

0

119865 (120585) 119889120585

+ 119906(119876119903


119876119903

lowast

0

119865 (120585) 119889120585)

(22)

From (22) 119864119903(119876119903




119889119864119903(119876119903

lowast 119901)

119889119901= 119886 minus 119896 (2119901 minus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585

(23)

1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 + (1 minus 119865 (119876

119903

lowast))119889119876119903

lowast

119889119901 (24)


1198892119864119903(119876119903

lowast 119901)

1198891199012= minus2119896 +

(1 minus 119865 (119876119903

lowast))2

(119901 minus 119907 + 119906) 119891 (119876119903

lowast) (25)


lowast 119901) is


lowast 119901)119889119901|

119901=119908minus119906= (119886 minus

119896(119908 minus 119906)) + int119876119903

lowast

(119908minus119906)


119901119903






119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585

(26)

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876) (27)



119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (28)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (29)







119864119903(119876 119901) = (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876


119876

0

119865 (120585) 119889120585 minus 119866

(30)

119881119903(119876 119901) = 119864(prod

119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

120578 (119876)

(31)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (32)

119886 minus 119896 (2119901119903

lowastminus 119908) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (33)



119903

lowast 119901119903









119876




in 119901



lowast 119901119903119898119907

lowast) satisfies

119864119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903

119876119903119898119907

lowastle 119876119903

lowast

119901119903119898119907

lowastle 119901119903

lowast

(34)





119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903


119903

lowast 119901 le

119901119903

lowast) (119876 gt 119876

119903

lowast 119901 le 119901

119903

lowast) (119876 le 119876

119903

lowast 119901 gt 119901

119903

lowast) and

(119876 gt 119876119903

lowast 119901 gt 119901

119903





lowast 119901119903119898119907


119876119903

lowast 119901 le 119901

119903


119903119898119907

lowast 119901119903119898119907



lowastle 119876119903

lowast and 119901119903119898119907

lowastle 119901119903

lowast



119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc





119903 (2) 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0


119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc



119903 On the other



119903(119876 119901)






119903(119876 119901)ge119896

119903and 119881

119903(119876 119901)lt119881





If (119876119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc




lowast Since 119864119903(119876 119901) is a


lowast]


lowast we have 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119903(119876 119901sc

lowast)120597119876|119876=119876sc





119864119903(119876 119901sc

lowast) and 119881

119903(119876 119901sc

lowast) lt 119881





119901119903



120597119864119903(119876sclowast 119901)120597119901|


119903(119876 119901) is strictly



119903(119876sclowast 119901) and

119881119903(119876sclowast 119901) lt 119881




lowast)120597119876|119876=119876sc

lowast ge

0 and 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0








119903 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



119903(119876sclowast 119901)120597119901|




0119865(120585)119889120585 ge 0 And from (7) it













119876sclowast

0119865(120585)119889120585)119902sc

lowast From120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


lowastminus



lowastminus


120597119864119903(119876sclowast 119901)120597119901|










lowastminus(119896119903+ (119901sc


119876sclowast

0119865(120585)119889120585 +

119866)119902sclowast From 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc




120597119864119903(119876sclowast 119901)120597119901|



lowastminus

int119876sclowast



lowastminus


0119865(120585)119889120585 + 119866)119902sc











prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879) (35)

119864119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 + 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(36)

119881119903(119876 119901) = [120582 (119901 minus 119907) + 119906]

2

120578 (119876) (37)


119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) + 119906] 119865 (119876

119903

lowast) = 0 (38)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908 + 119906) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (39)



119903

lowast 119901119903

lowast) = (119876sc




119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585) minus 119866

119881119903(119876 119901) = (119901 minus 119907 + 119906)

2

120578 (119876)

(40)


119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (41)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (42)



119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876)

(43)




119903(119876 119901) is strictly



decision (119876119903119898119907

lowast 119901119903119898119907






119903 120597119864119903(119876

119901sclowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



120597119864119903(119876sclowast 119901)120597119901|











119896119903


1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363












119903lt


119903= 1000 2000 3000 4000











119903lt 119864sc we




119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119864119903(119876 119901) = (119901 minus 119908)119863 (119901) + (119901 minus 119908)119876


119876

0

119865 (120585) 119889120585 minus 119866

(30)

119881119903(119876 119901) = 119864(prod

119903(119876 119901) minus 119864

119903(119876 119901))

2

= (119901 minus 119907)2

120578 (119876)

(31)



119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907) 119865 (119876

119903

lowast) = 0 (32)

119886 minus 119896 (2119901119903

lowastminus 119908) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (33)



119903

lowast 119901119903









119876




in 119901



lowast 119901119903119898119907

lowast) satisfies

119864119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903

119876119903119898119907

lowastle 119876119903

lowast

119901119903119898119907

lowastle 119901119903

lowast

(34)





119903(119876119903119898119907

lowast 119901119903119898119907

lowast) = 119896119903


119903

lowast 119901 le

119901119903

lowast) (119876 gt 119876

119903

lowast 119901 le 119901

119903

lowast) (119876 le 119876

119903

lowast 119901 gt 119901

119903

lowast) and

(119876 gt 119876119903

lowast 119901 gt 119901

119903





lowast 119901119903119898119907


119876119903

lowast 119901 le 119901

119903


119903119898119907

lowast 119901119903119898119907



lowastle 119876119903

lowast and 119901119903119898119907

lowastle 119901119903

lowast



119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc





119903 (2) 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0


119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc



119903 On the other



119903(119876 119901)






119903(119876 119901)ge119896

119903and 119881

119903(119876 119901)lt119881





If (119876119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc




lowast Since 119864119903(119876 119901) is a


lowast]


lowast we have 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119903(119876 119901sc

lowast)120597119876|119876=119876sc





119864119903(119876 119901sc

lowast) and 119881

119903(119876 119901sc

lowast) lt 119881





119901119903



120597119864119903(119876sclowast 119901)120597119901|


119903(119876 119901) is strictly



119903(119876sclowast 119901) and

119881119903(119876sclowast 119901) lt 119881




lowast)120597119876|119876=119876sc

lowast ge

0 and 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0








119903 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



119903(119876sclowast 119901)120597119901|




0119865(120585)119889120585 ge 0 And from (7) it













119876sclowast

0119865(120585)119889120585)119902sc

lowast From120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


lowastminus



lowastminus


120597119864119903(119876sclowast 119901)120597119901|










lowastminus(119896119903+ (119901sc


119876sclowast

0119865(120585)119889120585 +

119866)119902sclowast From 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc




120597119864119903(119876sclowast 119901)120597119901|



lowastminus

int119876sclowast



lowastminus


0119865(120585)119889120585 + 119866)119902sc











prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879) (35)

119864119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 + 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(36)

119881119903(119876 119901) = [120582 (119901 minus 119907) + 119906]

2

120578 (119876) (37)


119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) + 119906] 119865 (119876

119903

lowast) = 0 (38)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908 + 119906) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (39)



119903

lowast 119901119903

lowast) = (119876sc




119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585) minus 119866

119881119903(119876 119901) = (119901 minus 119907 + 119906)

2

120578 (119876)

(40)


119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (41)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (42)



119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876)

(43)




119903(119876 119901) is strictly



decision (119876119903119898119907

lowast 119901119903119898119907






119903 120597119864119903(119876

119901sclowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



120597119864119903(119876sclowast 119901)120597119901|











119896119903


1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363












119903lt


119903= 1000 2000 3000 4000











119903lt 119864sc we




119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119903(119876 119901)ge119896

119903and 119881

119903(119876 119901)lt119881





If (119876119903119898119907

lowast 119901119903119898119907

lowast) = (119876sc




lowast Since 119864119903(119876 119901) is a


lowast]


lowast we have 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119903(119876 119901sc

lowast)120597119876|119876=119876sc





119864119903(119876 119901sc

lowast) and 119881

119903(119876 119901sc

lowast) lt 119881





119901119903



120597119864119903(119876sclowast 119901)120597119901|


119903(119876 119901) is strictly



119903(119876sclowast 119901) and

119881119903(119876sclowast 119901) lt 119881




lowast)120597119876|119876=119876sc

lowast ge

0 and 120597119864119903(119876sclowast 119901)120597119901|

119901=119901sclowast ge 0








119903 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



119903(119876sclowast 119901)120597119901|




0119865(120585)119889120585 ge 0 And from (7) it













119876sclowast

0119865(120585)119889120585)119902sc

lowast From120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc


lowastminus



lowastminus


120597119864119903(119876sclowast 119901)120597119901|










lowastminus(119896119903+ (119901sc


119876sclowast

0119865(120585)119889120585 +

119866)119902sclowast From 120597119864

119903(119876 119901sc

lowast)120597119876|119876=119876sc




120597119864119903(119876sclowast 119901)120597119901|



lowastminus

int119876sclowast



lowastminus


0119865(120585)119889120585 + 119866)119902sc











prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879) (35)

119864119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 + 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(36)

119881119903(119876 119901) = [120582 (119901 minus 119907) + 119906]

2

120578 (119876) (37)


119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) + 119906] 119865 (119876

119903

lowast) = 0 (38)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908 + 119906) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (39)



119903

lowast 119901119903

lowast) = (119876sc




119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585) minus 119866

119881119903(119876 119901) = (119901 minus 119907 + 119906)

2

120578 (119876)

(40)


119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (41)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (42)



119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876)

(43)




119903(119876 119901) is strictly



decision (119876119903119898119907

lowast 119901119903119898119907






119903 120597119864119903(119876

119901sclowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



120597119864119903(119876sclowast 119901)120597119901|











119896119903


1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363












119903lt


119903= 1000 2000 3000 4000











119903lt 119864sc we




119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











prod119903(119876 119901)


+ 119906 (min (120585 119876) minus 119879) (35)

119864119903(119876 119901)

= (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 + 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585)

(36)

119881119903(119876 119901) = [120582 (119901 minus 119907) + 119906]

2

120578 (119876) (37)


119903

lowast 119901119903

lowast) is given by

(120582119901119903


119903

lowastminus 119907) + 119906] 119865 (119876

119903

lowast) = 0 (38)

120582119886 minus 119896 (2120582119901119903

lowastminus 119908 + 119906) + 120582119876

119903


119876119903

lowast

0

119865 (120585) 119889120585 = 0 (39)



119903

lowast 119901119903

lowast) = (119876sc




119864119903(119876 119901)


119876

0

119865 (120585) 119889120585

+ 119906(119876 minus 119879 minus int

119876

0

119865 (120585) 119889120585) minus 119866

119881119903(119876 119901) = (119901 minus 119907 + 119906)

2

120578 (119876)

(40)


119903

lowast 119901119903

lowast) is given by

(119901119903


119903

lowastminus 119907 + 119906) 119865 (119876

119903

lowast) = 0 (41)

119886 minus 119896 (2119901119903

lowastminus 119908 + 119906) + 119876

119903

lowastminus int

119876119903

lowast

0

119865 (120585) 119889120585 = 0 (42)



119864119903(119876 119901) = (120582119901 minus 119908)119863 (119901) + (120582119901 minus 119908)119876

minus 120582 (119901 minus 119907)int

119876

0

119865 (120585) 119889120585 minus 119866

119881119903(119876 119901) = 120582

2(119901 minus 119907)

2

120578 (119876)

(43)




119903(119876 119901) is strictly



decision (119876119903119898119907

lowast 119901119903119898119907






119903 120597119864119903(119876

119901sclowast)120597119876|119876=119876sc


119901=119901sclowast ge 0

From 120597119864119903(119876 119901sc

lowast)120597119876|119876=119876sc



lowast) + 119906119865(119876sc



120597119864119903(119876sclowast 119901)120597119901|











119896119903


1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363












119903lt


119903= 1000 2000 3000 4000











119903lt 119864sc we




119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896119903


1000 237 016 15 532670 02 3 265342000 474 032 15 432670 04 6 530683000 711 047 15 332670 05 75 163354000 948 063 15 232670 07 105 428695000 1185 079 15 132670 08 12 61366000 1423 095 15 32670 096 144 7363












119903lt


119903= 1000 2000 3000 4000











119903lt 119864sc we




119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119902sclowast

119901sclowast

119896119903

119908 120582 119864sc 119881sc 119881119903

119886

400 8747 18069 3068 2000 1397 093 214718 67652401 58695445500 9867 23954 3591 3000 1131 075 397772 123693208 70358994600 10674 29618 4106 5000 1185 079 636270 193126113 120622067700 11289 35136 4615 8000 1306 087 918750 275120256 208596718800 11774 40553 5122 10000 1195 080 1255640 369156492 234142445900 12167 45897 5627 14000 1278 085 1643110 474891261 3447602581000 12493 51185 6131 20000 1442 096 2081009 592087315 546887088

119896

10 10674 29618 4106 200 047 003 636270 193126113 19299515 8457 24096 2957 200 104 007 289546 57822379 27587920 6453 18877 2379 200 218 015 137539 18540079 39203325 4629 13899 2029 200 489 033 61305 5413263 5761430 2957 9115 1795 200 1333 089 22507 1180057 931777

CV

15 11825 29064 4276 4000 815 053 753751 26995109 760237335 25540 38392 4715 4000 630 042 952290 693542341 12236423955 29677 42447 4723 4000 662 044 905938 1860423072 36268930275 9484 29140 4034 4000 1012 067 593002 215447637 9802783395 4588 25858 3873 4000 1121 075 535443 64994919 36232117








119903subject to








119908 119866 119881119903

119886

400 2000 15 14718 67652401500 3000 15 97772 123693208600 5000 15 132670 193126113700 8000 15 118750 275120256800 10000 15 255640 369156492900 14000 15 243111 4748912611000 20000 15 81009 592087315

119896

10 200 15 612670 19312611315 200 15 265946 5782237920 200 15 117539 1854007925 200 15 41305 541326330 200 15 2507 1180057

CV

15 4000 15 353751 2699510935 4000 15 552290 69354234155 4000 15 505938 186042307275 4000 15 193002 21544763795 4000 15 135443 64994919









120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120582 119908 119866 119881119903

119886

400 2000 095 1425 3983 61056293500 3000 08 12 18218 79163653600 5000 08 12 6136 123600712700 8000 09 135 26875 222847407800 10000 085 1275 67294 266715565900 14000 09 135 78799 3846619221000 20000 098 147 39389 568640657

119896

10 200 01 15 43267 193126115 200 02 3 37909 231289520 200 04 6 35015 296641325 200 07 105 22913 265249930 200 09 135 256 955846

CV

15 4000 07 105 127626 1322760335 4000 06 9 171374 24967524355 4000 05 75 52969 46510576875 4000 08 12 74402 13788648895 4000 09 135 81899 52645884


119903in revenue







Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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