research article revenue-sharing contract models...

Research ArticleRevenue-Sharing Contract Models for Logistics Service SupplyChains with Mass Customization Service

Weihua Liu Xuan Zhao and Runze Wu

College of Management and Economics Tianjin University Tianjin 300072 China

Correspondence should be addressed to Weihua Liu lwhliu888163com

Received 13 April 2015 Revised 14 August 2015 Accepted 24 August 2015

Academic Editor Young Hae Lee

Copyright copy 2015 Weihua Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The revenue-sharing contract is one of the most important supply chain coordination contracts it has been applied in varioussupply chains However studies related to service supply chains with mass customization (MC) are lacking Considering the equityof benefit distribution between themembers of service supply chains in this paper we designed two revenue-sharing contractsThefirst contract for the maximum equity of a single logistics service integrator (LSI) and single functional logistics service provider(FLSP) in a two-echelon logistics service supply chain was designed by introducing the fair entropy function (ldquoone to onerdquo model)Furthermore the method is extended to a more complex supply chain which consists of a single LSI and multiple FLSPs A newcontract was designed not only for considering the equity of an LSI and each FLSP but also for the equity between each FLSP (ldquooneto 119873rdquo model) The ldquoone to onerdquo model in three-echelon LSSC is also provided The result exemplifies that whether in the ldquoone toonerdquo model or ldquoone to 119873rdquo model there exists a best interval of customized level when the revenue-sharing coefficient reaches itsmaximum

1 Introduction

The revenue-sharing coefficient is the key to designing arevenue-sharing contract which is a form of supply chaincoordination The problem of designing a revenue-sharingmechanism for the product supply chain has been researchedby many scholars for a long time [1 2] With the rise ofthe logistics service industry studies on revenue-sharingmechanisms of service supply chain have appeared in recentyears [3] But whether a revenue-sharing contract can beapplied to a service supply chain has not been reportedunder MC background Taking a logistics service supplychain for example through the integration of service capacityof upstream FLSPs the LSI establishes the logistics servicesupply chain to provide customers with mass customizationlogistic service (MCLS) [4 5] In MC the customized levelthat was asked for by customers will affect the flexibility ofservice directly But there is no exploration about whetherthe customized level will affect the application of the logisticsservice supply chain revenue-sharing contract

In this paper themotivation of research comes fromprac-tice and theory From the practical level as a new method to

attract customersmass customized service has obtainedwideattention in recent years Because of the capacity of providingscale service for multiple customers and providing cus-tomized service for individuals mass customized service hasdual characteristics of a scale effect and a customized effectIt has been applied in many industries such as insuranceservices [6] logistics services [7] and air services [8] Underthe environment of MCLS in order to meet the customizeddemand of clients downstream LSIs and upstream FLSPshave to address the coordination and profit distribution issueConsidering the customized level and the equity of profits itis necessary to design a reasonable revenue-sharing contractto guarantee the stability of a logistics service supply chainOn the other hand most of the existing revenue-sharingcontract research mainly focuses on the coordination ofa common product or service supply chain [3 9ndash11] butspecial cases such as the supply chain revenue-sharing con-tract research under the MCLS environment have not beenreported before The most closely related research to thispaper is the one by Liu et al [3] they studied the method ofdetermining the optimal revenue-sharing coefficient of two-echelon and three-echelon logistics service supply chains but

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 572624 21 pageshttpdxdoiorg1011552015572624

2 Mathematical Problems in Engineering

the study did not involve MCLS In fact MCLS and com-mon services have two significant discrepancies the first isthat the customized level will affect the design of supplychain contracts It is urgent to introduce the customized levelinto revenue-sharing contract design from the perspectiveof theory The second is that MCLS is provided by an LSIvia integrating FLSPsrsquo service capacity therefore FLSPs haveto evaluate the equity of profit distribution So a way tointroduce a fairness factor into a revenue-sharing contract isalso necessary to be considered

This paper expands the research content on the basis ofLiu et al [3] First this paper will take theMCLS environmentinto consideration and introduce the customized level asan important factor for the design of a revenue-sharingcontract mechanism Second this paper is based on thelogistics service supply chain of two echelons and establishesa revenue-sharing mechanism model in two relations byexpanding the ldquoone to onerdquo relationship between the LSIand FLSP to ldquoone to 119873rdquo Finally this paper aims at themaximum of fair entropy and setting improved fair entropyas the objective function of the model and then explores theinfluence of customized level on supply chain fair entropyand some interesting findings are obtained For examplewhether in the case of ldquoone to onerdquo or ldquoone to 119873rdquo revenue-sharing coefficients of the LSI will first increase and thendecrease with the increase of the customized level that isthere is an optimal interval of customized level that let therevenue-sharing coefficient of the LSI reach its maximum

This paper is organized as follows the second chapteris a literature review in which we systematically review andsummarize the studies of MC mode service supply chainand revenue-sharing contract The third and fourth chaptersintroduce the procedure of model establishing We willrespectively illuminate the procedure of building ldquoone toonerdquo and ldquoone to 119873rdquo service supply chain revenue-sharingcontract models under the MC environment The fifthchapter will conduct an example analysis LINGO 110 willbe used for the numerical simulation of the two modelsestablished in this paper In last chapter the conclusionsome important conclusions in this paper and managementinsights are summarized and finally the limitation of thispaper and future research directions are put forward

2 Literature Review

Our research mainly concerns the order allocation decisionunder the situation of order insertion in the MCLS environ-ment Thus the literature review is mainly related to the MCmode order allocation and order insertion Our researchdirection will be proposed after summarizing the progressionof developments in the literature as well as its current gaps

21 MC Mode and Customized Level In 1993 Pine II andStan proposed that MC is a manufacturing mode to widelyprovide personalized products and services which would bethe frontier of commercial competition [12] With 20 yearsrsquodevelopment and application MC mode has become themain operation mode and the key to improving commercial

competition Including the autoindustry clothing industryand computer industry many major economic industriesbenefit from this productionmode [13] Overall most studiesof MC mainly discuss the production modes including theMC approach and its product design [14 15] MC produc-tion planning and control technology [16 17] MC cost ofproduction [18] and factors and conditions that affect MC[19] With the increase of research on MC many scholarsreview the literature in the field Fogliatto made a detailedarrangement and summary of the literature from the 1980sHe indicated many unsolved problems with MC like supplychain coordination and quality control issues which wouldbe the new fields of MC research [20]

The customized level is a crucial factor in MC A highlevel of customization leads to a huge cost of production [21]Enterprises will not produce the customized products tomeetcustomers demand when the customized level is too high[22]Therefore it is important to define a rational customizedlevel Through a mathematical optimization model somescholars have made specialized exploration From the per-spective of customer satisfaction and corporate customizedcost Liang and Zhou worked out the way to determinethe optimal degree under the condition of fixed customizedproducts supply capacity [23] By building a mathematicalmodel on the relationship between customized level marketdemand and corporate profits Zhou et al provided guidancefor manufacturers to determine a rational customized level[24]

With more attention paid to service supply chains somescholars began to research the customized level in MCservice Liu researched the method of determining an opti-mal customized level in a logistics service supply chainThey proposed three different decision-making modes fora customized level (the LSI decides the customized levelthe customer decides the customized level customized leveldecision of centralized supply chain) and these modelsdiscussed the reaction of profits of the LSI customers and thewhole supply chain under different decision-making modeswith the change of customized level respectively [7] Overallmost of MC research on customized levels is focused onproduct supply chain currently but there is little research onservice supply chains

22 Logistics Service Supply Chain With the development ofthe logistics industry the logistics service supply chain hasbecome the new hotspot in the supply chain field Currentlyresearch on logistics service supply chains mainly focuses onlogistics service supply chain connotation [4 25] logisticsservice supply chain coordinatemechanism [3 26] empiricalstudy of logistics service supply chain [27 28] and selectionand performance evaluation of FLSPs [5]

In the 21st century with the rise in MC logistics servicemode research results show that logistics service supplychains under the background of MC are fruitful becauseof the endeavors of many scholars For example Rhondaet al built an MC supply chain service model to meet thecustomersrsquo demand of personality [29] Liu et al carried outresearch on the time scheduling problem under the modeof MC logistics service and constructed the time scheduling

Mathematical Problems in Engineering 3

model of logistics service supply chain [30] Subsequently Liuet al explored the order allocation problemof logistics servicesupply chains under MC background [3] In addition thedecision-making problem of the customer order decouplingpoint (CODP) under MC service was also studied [31]

23 Revenue-Sharing Contract and Its Coefficient A supplychain contract is an important means to promote the coordi-nation of supply chain members As a main form of supplychain contract the revenue-sharing contract has become thehot topic in the supply chain coordination field because ofits characteristics of effectively constraining supply chainmembersrsquo behavior The revenue-sharing contract plays asignificant role in supply chainmanagement especially whenconfronting the uncertainty of customer demand [32]There-fore most research about revenue-sharing contracts assumescustomer demand uncertainty and is mainly for two-echelonsupply chains [33ndash36] In recent years revenue-sharing con-tract researches extended to more complex situations moreresearch concerning multiechelon supply chains and moreconstraints Xiang proposed a revenue-sharing contractmodel with an assumption that supply chainmembers shouldshare the cost as well as the revenue which guarantees therationality and fairness of the contract [37] Considering thelimitation of the two-echelon supply chain van der Rhee etal designed a revenue-sharing contract model for extendedsupply chain under stochastic demand (the extended supplychain is the multiechelon global supply chain that fits reality)[38]

As the key element when designing supply chain revenue-sharing contracts the revenue-sharing coefficient has beena concern for many years For example Giannoccaro estab-lished a revenue-sharing contract model in view of a three-echelon supply chain and presented a revenue-sharing coef-ficient range that can escalate the profits of supply chainmembers [39] Pang designed a supply chain revenue-sharingmodel for stochastic demandAccording to themodelrsquos resultwith the increase of retailerrsquos waste-averse decision bias thecoefficient of profit distribution of the retailer to the supplierwill be decreased [10] Qin found that if the revenue-sharingcontract defines a coefficient inconsistent with reality it willnot be sustained But the literatures above mainly focus onproduct supply chains besides they only gave the revenue-sharing coefficient range instead of an accurate value [34] Liuet al put forward a method to confirm an optimal sharingcoefficient of a two-echelon logistics service supply chainunder stochastic-customer demand [3] They also extendedit to a three-echelon logistics service supply chain containingan LSI an FLSP and a logistics service subcontractor But theresearch only contains a single LSI and single FLSP It doesnot consider the MC environment and fairness factor whenthere are multiple FLSPs

24 Summary of Literature Review By combining the lit-erature of three areas we can find that the research onsupply chain revenue-sharing contracts under theMC serviceenvironment is deficient Moreover equity between multipleFLSPs has not been considered in existing supply chainrevenue-sharing contract research Due to the shortage of

Q

P

Customer(manufacturing

retailer)

Logisticsservice

integrator R

Functional logisticsservice provider

S

(D t)

(w 120593)

Figure 1 The two-echelon logistics service supply chain of a singleLSI and single FLSP

previous research this paper will focus on the revenue-sharing distribution mechanism problem of logistics servicesupply chains and then will consider an equity factor underMC mode From the perspective of maximizing the equityperceived by supply chain members this paper will set uptwo revenue-sharing contract models and investigate theinfluence of the customized level on supply chain revenue-sharing coefficient and fair entropy

3 The (One to One) Model

This paper focuses on the revenue-sharing contract mecha-nism design of two stages of logistics service supply chainwith consideration of customer customization demand onthe background of MC service This chapter will exhibit therevenue-sharing contract model of logistics service supplychains considering customization demand In Section 31we will describe the problem and list basic assumptionsof this model In Section 32 the revenue-sharing contractmodel of two-echelon logistics service supply chains underconsideration of a single LSI and a single FLSP is presented

31 Problem Description It is assumed that a two-echelonlogistics service supply chain is composed of a single LSI 119877

and single FLSP 119878 which we called the ldquoone to onerdquo modelFigure 1 shows the procedure

Figure 1 shows that the LSI will buy logistics servicecapacity from the FLSP to satisfy the customized demandwhen customersrsquo orders contain a certain level of customiza-tion to LSI

The notation of parameters and variables in this model isdefined as follows

Notations for the ldquoOne to Onerdquo Model

Decision Variables

119876 logistics capacity that 119877 needs to buy from 119878119905 customized level of logistics capacity that customerneeds120593 119877rsquos share of the total sales revenue under revenue-sharing contract1 minus 120593 119878rsquos share of the total sales revenue under rev-enue-sharing contract

Other Parameters

119862119877 marginal cost of LSI 119877

119862119906(119875) the unit price of 119877 paying to customer when

logistics service capability is lacking


119863(119905) the total demand of customer to logistics serviceunder MC119867 the fair entropy measuring revenue-sharing fair-ness of 119877 and 1198781198670 threshold of fair entropy

119875 unit price of service customer buys from 119877119878(119876) expectation of the logistics capacity of 119877119880 the unit price when 119877 returns extra logisticscapacity to 119878119908 the unit price when 119877 purchased services from 119878120587119877 total revenue of logistics service 119877 with 120587

1015840

119877indi-

cating the total revenue of 119877 under revenue-sharingcontract and 120587

10158401015840

119877indicating the total revenue of 119877

under game situation120587119878 total revenue of 119878 with 120587

1015840

119878indicating the total

revenue under revenue-sharing contract of 119878 and 12058710158401015840

119878

indicating the total revenue under game situation of119878120587119879 total revenue of whole logistics service supply

chain with 1205871015840

119879indicating the total revenue of supply

chain under revenue-sharing contract and 12058710158401015840

119879indi-

cating the total revenue of supply chain under gamesituation120583 the average value of total customer service demand1198631205902 the variance of total customer service demand 119863

120574 the weight of 119877 in supply chain relationship1 minus 120574 the weight of 119878 in supply chain relationship

Other assumptions of the ldquoone to onerdquo model are as fol-lows

Assumption 1 In order to explore the effect of differentcustomerrsquos customized level on a revenue-sharing contractas in Liu et alrsquos study [7] assuming that customized level isa variable 119905 (119905 gt 0) 119905 follows a kind of distribution which isnot sure and its distribution function is 119865(119905) and probabilitydensity function is 119891(119905) Since the customized level 119905 is arandom variable and it will change with different customersso referring to Liu et al [7] we assume that 119905 is subject to acertain distribution

Assumption 2 It is supposed that there is a correlationbetween customer demand and customized level [7 24]which is assumed as 119863(119905) = 119863

0+ 119896119905 minus (12)ℎ119905

2 Weassume the average value of119863(119905) is 120583 and variance is 1205902 Thisassumption is proposed to make it clear that the customerdemand is related to the customized level In practice highercustomized level may not lead to the increase of the customerdemand because too high customized level will increase theservice price and finally will reduce the customer satisfactionReferring to [7 24] the demand function is set as 119863(119905) =

1198630+ 119896119905 minus (12)ℎ119905

2

Assumption 3 It is assumed that the LSIrsquos marginal cost andthe unit price when the LSI purchase service from FLSP arerelated to customized level similar to Liu et al [7] It canbe assumed that 119862

119877= 1198620+ 119911119905 and 119908 = 119908

0+ 119910119905 Since

the customized level will affect the cost of LSI the higherthe customized level is the higher the cost of LSI will be soreferring to Liu et al [7] the customized level will affect thewholesale price and the marginal cost of LSI

Assumption 4 When the LSI purchases extra logistics servicecapacity it is permitted to return the extra capacity to theFLSP and refund a unit price 119880 for the extra service capacitylater 119880 is related to the price 119908 for which the LSI purchaseslogistics capacity from the FLSP and variance 120590

2 of demandIt is assumed that 119880 = 119908

21198971205902 [3] 119897 indicates the sensitivity

coefficient of FLSP to demand variance This assumption isproposed to guarantee that extra logistics service capacity willnot be wasted and it should return to the FLSP in a properway the return price is referring to Liu et al [3]

Assumption 5 In order to simplify the model it is assumedthat the FLSP has sufficient supply capacity of logistics ser-vice namely the FLSP does not lack any capacity when theLSI purchases service capacity from it

Assumption 6 Under the revenue-sharing contract the LSIand FLSP are in compliance with the principles of revenue-sharing and loss sharing It is assumed that the LSI andFLSP follow the ratio of 120593 and 1 minus 120593 to distribute the totalrevenue of supply chain [3 40] When the logistics servicecapacity purchased by the LSI from FLSP is not able to satisfythe customer demand the LSI and FLSP should undertakecompensation to the customer at a ratio of 120593 and 1 minus 120593 Inrevenue-sharing contract all the members should obey therule of ldquorevenue and loss sharing togetherrdquo this assumptionis proposed to restrain the behavior of all the members

32 The ldquoOne to Onerdquo Revenue-Sharing Contract ModelReferring to [3 41] in the case of ldquoone to onerdquo the expectationof logistics service capacity purchased by the LSI is

119878 (119876) = 119864 (min (119876119863 (119905)))

= int

infin

0

min (119876119863 (119905)) 119891 (119905) 119889119905

= 119876 minus int

119876

0

[119876 minus 119863 (119905)] 119891 (119905) 119889119905

= 119876 minus int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(1)

The expectation of the LSIrsquos excess capacity is

119864 (119876 minus 119863 (119905))+= 119876 minus 119878 (119876)

= int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(2)


The capacity deficit expectation of the LSI is

119864 (119863 (119905) minus 119876)+= 120583 minus 119878 (119876)

= 120583 minus 119876

+ int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(3)

TheLSI revenue function under revenue-sharing contractis

120587119877

= 120593119875119878 (119876) minus 119908119876 minus 119862119877119876 minus 120593119862

119906 (119875) [120583 minus 119878 (119876)]

minus 119880 [119876 minus 119878 (119876)]

(4)

The revenue function of the FLSP is120587119878= (1 minus 120593) 119875119878 (119876) + 119908119876 minus 119862

119878119876

minus (1 minus 120593)119862119906 (119875) [120583 minus 119878 (119876)] + 119880 [119876 minus 119878 (119876)]

(5)

The revenue of the whole supply chain is

120587119879

= 120587119877+ 120587119878

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119906 (119875) [120583 minus 119878 (119876)]

(6)

In logistics service supply chains usually the LSI is theleader and the FLSP is the follower [3] In general the LSIand FLSP coexist in two kinds of relationship one is a gamerelationship namely the LSI and FLSP proceed with theStackelberg game to maximize their own profits the otheris the revenue-sharing contract which will be studied in thispaper Under the revenue-sharing contract supposing thatthere exists win-win cooperation between the LSI and FLSPthe two units can be treated as a whole which generate acentralized decision-making model Then maximizing therevenue of the whole supply chain is the final target Thecondition of adopting the revenue-sharing contract is thefact that the profits gained by the LSI and FLSP under therevenue-sharing contract should be no less than the profitsin Stackelberg game

321 Model under Centralized Decision-Making Under cen-tralized decision-making for obtaining the maximum rev-enue of a supply chain the first-order condition should besatisfied

120597119864 (120587119879)

120597119876= 0 (7)

Then the second derivative should be less than 0 that is1205972119864(120587119879)1205971198762lt 0

Simplify the first-order condition

119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878

119875 + 119862119906 (119875)

(8)

We can calculate the optimal value of purchasing capacity1198761015840 and optimal expectation of supply chain total revenue

119864(1205871015840

119879)

322 Model under Decentralized Decision-Making For en-suring the implementation of the revenue-sharing contractwe need to investigate the Stackelberg game model betweenthe LSI and FLSP namely a decentralized decision-makingmodel

For better understanding we define that 119887 = int119876

0[119876minus119863

0minus

119896119905 + (12)ℎ1199052]119891(119905)119889119905 thus 119878(119876) = 119876 minus 119887

Another definition is 119904 = 120597119887120597119876 thus 119904 = 119865(119876) +

[(12)ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891(119876) and 120597119878(119876)120597119876 = 1 minus 119904

Respective expectation revenue of the LSI and FLSP is

119864 (12058710158401015840

119877) = 119875 (119876 minus 119887) minus 119908119876 minus 119862

119877119876 minus 119862

119906 (119875) (120583 minus 119876 + 119887)

minus1199082

1198971205902119887

119864 (12058710158401015840

119878) = [119908 minus 119862

119878] 119876 +

1199082

1198971205902119887

(9)

First to maximize the profits of FLSP 119878 the first-ordercondition should be satisfied

120597119864 (12058710158401015840

119878)

120597119908= 119876 +

2119908

1198971205902119887 = 0 (10)

Then 119908 = minus11987611989712059022119887

Next to maximize the profits of the LSI the first-ordercondition should be satisfied

120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862

119906 (119875)] (1 minus 119904) minus 119862119877minus 119908 minus

1199041199082

1198971205902= 0 (11)

while satisfying the fact that second derivative is less than 0that is 1205972119864(120587

10158401015840

119877)1205971198762lt 0

We can obtain the optimal value of purchasing the capac-ity 11987610158401015840 under the Stackelberg game And by substituting

11987610158401015840 into the expression of 119908 the corresponding 119908 can be

calculatedUnder the constraints of rationality the LSI and FLSP

first consider their own profits Only if their own profitsare satisfied will the maximization of whole supply chainrsquosrevenue be considered So the condition of supply chainmembers to accept the revenue-sharing contract is that therevenue obtained in this contract should be no less thanthat in the decentralized decision-making model Thus therestraint is as follows

120593119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 120593) 119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)

le 120593 le 1 minus

119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

(12)

Formula (12) indicates that the constraints of the revenue-sharing contract are transformed into constraints for the rev-enue-sharing coefficient Namely the revenue-sharing con-tract can be applied only when the revenue-sharing coeffi-cient is in the interval mentioned above


323 The Objective Function and Constraints of OptimalRevenue-Sharing Coefficient

(1) Establishment of Objective Function After satisfying thecondition of revenue-sharing contract implementation anoptimal revenue-sharing coefficient should be confirmedto achieve fair distribution of profits between the LSI andFLSP thereby maintaining the revenue-sharing relationshipbetween them The core idea to establish the objectivefunction of supply chain optimal revenue-sharing coefficientis that the profit growth with unit-weight resource of eachsupply chain member is equal [3 42] When the revenue-sharing coefficient of the LSI is 120593 the respective profit growthof the LSI and FLSP is

Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

(13)

Considering the different weights of the LSI and FLSP inthe supply chain suppose the weights of each are given It isassumed that the weight of the LSI is 120574 thus the weight ofFLSP is 1 minus 120574 then the profit growth on unit-weight resourceof the LSI and FLSP is

1205851=

Δ120579119877

120593120574119879119877

1205852=

Δ120579119878

(1 minus 120593) (1 minus 120574) 119879119878

(14)

The total cost of the LSI is119879119877

= 119908119876 + 119862119877119876 + 120593119862

119906 (119875) [120583 minus 119878 (119876)]

+ 119881 [119876 minus 119878 (119876)]

(15)

The total cost of the FLSP is

119879119878= 119862119878119876 + (1 minus 120593)119862

119906 (119875) [120583 minus 119878 (119876)] (16)

Then we introduce the concept of entropy making thefollowing standardized transformation to 120576

1and 1205762[43]

1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

(17)

Among them

120585 =1

2

2

sum

119894=1

120585119894

120590 = radic1

2

2

sum

119894=1

(120585119894minus 120585)2

(18)

They are the average value and the standard deviation of1205761015840

119894

In order to eliminate the negative situation of 12058510158401and 1205851015840

2

coordinate translation can be applied [44] After translationaltransformation 1205851015840

119894turned into 120585

10158401015840

119894 12058510158401015840119894

= 119870+ 1205851015840

119894119870 is the range

of coordinate translationThen calculating the ratio 120582119894of 12058510158401015840119894

set

1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2

(19)

After obtaining 1205821and 120582

2 the fair entropy of whole

supply chain is

119867 = minus1

ln119898

119898

sum

119894=1

120582119894ln 120582119894 (20)

The objective function of the optimal revenue-sharingcoefficient in this model is as follows

119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822) (21)

(2) Constraint Condition The constraint conditions of themodel are as follows

First a certain constraint condition exists between theweight and revenue-sharing coefficient of the LSI and FLSPit is shown as follows

10038161003816100381610038161003816100381610038161003816

120593119877

120593119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576 (22)

Namely the absolute value of the difference between thespecific value of the FLSPrsquos weight and the specific value ofrevenue-sharing coefficient cannot exceed the critical valueThe constraint condition mentioned above can be simplifiedas follows

10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576 (23)

Second threshold value 1198670can be set in reality set 119867 ge

1198670 It indicates that the revenue distribution fairness of both

sides of the supply chain should be greater than a certainthreshold value

Then the optimal revenue-sharing coefficient modelunder the condition of ldquoone to onerdquo is shown in the followingformula

max 119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(24)


Functional logistics



retailer)

Logisticsservice

integrator RP


service provider S1

service provider Si

SN

middotmiddotmiddot

(Q1 wi 120593i)

(Qi wi 120593i)

(QN wN 120593N)

(D t)

Figure 2 The two-echelon logistics service supply chain of ldquoone to119873rdquo

This model is a single goal programming model Underthe condition of certain notations that are given by makinguse of LINGO 110 we can program the objective functionof the customized level and its constraint conditions andcalculate the optimal customized level

4 The (One to 119873) Model

41 Problem Description Another supply chain modelresearched in this paper is a two-echelon logistics servicesupply chain model which is composed of a single LSI and119873 FLSPs (in a short form of one to 119873) The model operationschematic diagram is shown in Figure 2

It is assumed that customers have 119873 kinds of demand119873 FLSPs are needed to satisfy the various demands Thesedemands have a certain similarity in that they belong to thesame broad heading of logistics service And MC is providedto customers by integrating these119873 kinds of logistics servicedemand As the number of optional FLSPs in reality is alwaysgreater than 119873 the LSI needs to choose 119873 from numerousFLSPs to meet the multiple customization demands and thenprovide service The process of choosing FLSPs will not beconsidered in this paper thus assuming that the119873FLSPs havebeen chosen to accomplish the customization service In thispaper research is focused on the revenue-sharing problembetween the LSI and the decided 119873 FLSPs The operationprocess of the ldquoone to 119873rdquo model is as follows

Firstly the LSI will analyze the customization demand oflogistics service asked by customers then different FLSPs willbe chosen by the LSI to purchase different logistics servicecapacity Based on the given contract parameter (119908

119894 120593119894) each

FLSP will provide the optimal service capacity 119876119894to the LSI

and both sides realize their profits In the case of ldquoone to 119873rdquothe LSI to each FLSP is same as ldquoone to onerdquo but comparedwith the ldquoone to onerdquo model three discrepancies shouldbe considered in the ldquoone to 119873rdquo model first the fairnessbetween the LSI and each FLSP second the fairness factorof distribution between 119873 FLSP and third maximizing thetotal fair entropy of all FLSPs

The same as in the ldquoone to onerdquo assumptions the conno-tation of notations and variables in this model are as follows

Notations for the ldquoOne to 119873rdquo Model

Decision Variables

119876119894 logistics capacity that 119877 needs to buy from 119878

119894

119905119894 customized level of 119894 logistics service capacity that

customer needs120593119894 the revenue-sharing coefficient of 119877 between the

revenue-sharing contract of 119877 and 119878119894

1 minus 120593119894 the revenue-sharing coefficient of 119878

119894between

the revenue-sharing contract of 119877 and 119878119894

Other Parameters

119862119877119894 marginal cost of LSI 119877 aiming at the type 119894 of

Customized service119862119878119894 Production cost of unit logistics service of FLSPs

aiming at the type 119894 of customized service119862119906(119875119894) the unit price of 119877 paying to customer when

logistics service capability is lacking119863 the total demand of customer to logistics serviceunder MC119863(119905119894) customization service demand faced by 119878

119894

1198671119894 the fair entropy to measure the revenue-distri-

bution fairness between 119877 and 119878119894under revenue-

sharing contract1198672119894 the fair entropy to measure the relative equity

between 119878119894and all of the other FLSPs

119867 the total of all entropy1198670 threshold of fair entropy

119894 FLSP 119878119894 119894 = 1 2 119873

119873 the number of all FLSPs119875119894 unit price of service customer purchasing type 119894 of

customized service from 119877119878(119876119894) expectation of the logistics capacity of119877 aiming

at type 119894 of customized service119878(119876) expectation of the total logistics capacity of 119877with 119878(119876) = sum

119873

119894=1119878(119876119894)

119880119894 the unit price when 119877 returns extra logistics

capacity to 119878119894

119908119894 the unit price when 119877 purchased services from 119878

119894

for type 119894 of customized service120587119877119894 the profits of 119877 for customization service 119894 with

1205871015840

119877119894indicating the profits of 119877 under revenue-sharing

contract and 12058710158401015840

119877119894indicating the profits of 119877 under

game situation

120587119877 the total revenue of 119877 with 120587

119877= sum119873

119894=1120587119877119894

120587119878119894 the total revenue of 119878

119894 with 120587

1015840

119878indicating the

total revenue of 119878119894under revenue-sharing contract

and 12058710158401015840


119894under game

situation


120587119878 the total revenue of all the FLSPs

120587119879119894 the total revenue of the supply chain composed of

119878119894and 119877

120587119879 the total revenue of the whole logistics service

supply chain

120574119894 the weight of LSI in the supply chain composed of

119877 and 119878119868

1 minus 120574119894 the weight of 119878

119894in the supply chain composed

of 119877 and 119878119894

Specific assumptions of the ldquoone to 119873rdquo model are asfollows the practical basis of Assumptions 1 2 and 4 is thesame as ldquoone to onerdquo model

Assumption 1 Assuming that the logistics capacity demandsof customers exhibit diversity and assuming a customizedlevel of each demand as 119905

119894 the same as Liu et al [7] set 119905

119894

obeys a certain distribution the distribution function is119865(119905119894)

and the probability density function is 119891(119905119894)

Assumption 2 It is assumed that there is a correlationbetween customer demand and customized level [7 24]setting the sum of customer customization demand faced bythe LSI as119863 the demand for each FLSP distributed by the LSIis119863(119905

119894) = 119863

1198940+119896119905119894minus (12)119905

119894

2 We assume the average value of119863(119905119894) is 120583119894and variance is 120590

119894

2

Assumption 3 Similar to [5] the capacity supplied by eachFLSP is independent mutual utilization of different FLSPsrsquocapacities are forbidden forbidden It will be a very compli-cated problem if their capacities are related mutually So thisassumption is proposed to guarantee the independence ofeach FLSP

Assumption 4 Under a revenue-sharing contract assumethat the LSI and each FLSP follow the principle of revenue-sharing and loss-sharing According to the ratio of 120593

119894and

1 minus 120593119894 the LSI and FLSP will respectively distribute the

total revenue of the supply chain or undertake the loss ofinsufficient supply capacity

Assumption 5 From the view of [45] assume that the rev-enue-sharing coefficients between the LSI and each FLSP aremutually visible in 119873 FLSPs

The supply chain members perceive fairness by compar-ing the revenue-sharing coefficients in a proper way If thecoefficients are invisible it will be difficult for each FLSP toassess relatively fairness

42 Revenue-Sharing Model under Condition of ldquoOne to 119873rdquoIn the case of multiple FLSPs each FLSP is still cooperatingwith the LSI one to one then the capacity offered to customeris the total capacity purchased from all FLSPs When the LSI

cooperates with the FLSP 119894 the logistics capacity expectationof the LSI is

119878 (119876119894) = 119864 (min (119876

119894 119863 (119905119894)))

= int

infin

0

min (119876119894 119863 (119905119894)) 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 119863 (119905

119894)] 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(25)

Expectation of the LSIrsquos excess capacity is

119864119894(119876119894minus 119863 (119905

119894))+= 119876119894minus 119878 (119876

119894)

= int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(26)

Expectation of the LSIrsquos insufficient capacity is

119864119894(119863 (119905119894) minus 119876119894)+= 120583119894minus 119878 (119876

119894)

= 120583119894minus 119876119894+ int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(27)

The revenue function of the LSI under a revenue-sharingcontract is

120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(28)

The function of the LSIrsquos total revenue is

120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(29)

The revenue function of the FLSP 119894 is

120587119878119894

= (1 minus 120593119894) 119875119894119878 (119876119894) + 119908119894119876119894minus 119862119878119894119876119894

minus (1 minus 120593119894) 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

+ 119880 [119876119894minus 119878 (119876

119894)]

(30)

The revenue function of a supply chain branch composedof the LSI and the FLSP 119894 is

120587119879119894

= 120587119877119894

+ 120587119878119894

= 119875119894119878 (119876119894) minus 119862119877119894119876119894minus 119862119878119894119876119894

minus 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

(31)

There is a relationship of a cooperative game existingbetween the LSI 119877 and each FLSP and the process of thecooperative game is similar to that mentioned in the ldquoone toonerdquo model of third chapter


421 Centralized Decision-Making Model Being similar toldquoone to onerdquo situation when the LSI obtains the optimal valueof logistics capacity 119876

119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)

Satisfy the condition that second variance is less than 01205972119864(120587119879119894)120597119876119894

2lt 0

Simplify the condition of first order

119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)

From formula (33) we can calculate the optimal purchasevolumes of logistics service 119876

1015840

119894from the FLSP 119894 and the opti-

mal expectation of profits119864(1205871015840

119879119894) of the 119894 supply chain branch

The last formula is suitable for every supply chain branchso the optimal purchase volumes of logistics capacity pur-chased from all FLSPs by the LSI can be calculated 1198761015840 =

sum119873

119894=11198761015840

119894

And the optimal expectation of supply chain total revenueis 119864(120587

1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)

422 Stackelberg GameModel For the Stackelberg game therevenue expectation functions of 119877 is

119864 (120587119877119894) = 119875119894(119876119894minus 119887119894) minus 119908119894119876119894minus 119862119877119876119894

minus 119862119906(119875119894) (120583119894minus 119876119894+ 119887119894) minus

119908119894

2

1198961198941205901198942119887119894

(34)

The revenue expectation functions of 119878119894is

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)

Before 119877 adopts the value of 119876119894 the reaction of 119908

119894to 119878119894

should be consideredThe calculation details of119908119894and119876

119894are

shown in Appendix AFor each supply chain branch the condition of supply

chain members who receive the revenue-sharing contract isthat the revenue they obtain under this contract should not beless than that under decentralized decision-making Consider

120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)

Under the ldquoone to 119873rdquo condition in order to applythe revenue-sharing contract revenue-sharing coefficients

between the LSI and each FLSP should satisfy the conditionmentioned above

423 The Objective Function and Constraint Conditions ofOptimal Revenue-Sharing Coefficient

(1) Establishment of Objective Function Under the case ofmultiple FLSPs apart from the revenue-sharing fairnessbetween the LSI and each FLSP the fairness of the revenue-sharing coefficient between multiple FLSPs should also beconsidered So a new fair entropy is defined which is com-posed of two parts one is the fair entropy between the LSIand FLSP 119894 and the other is the fair entropy between the FLSP119894 and other FLSPs

(i) The Fair Entropy between the LSI and FLSP 119894 Similar tothe establishment of fair entropy in the situation of ldquoone toonerdquo the core idea is that the profitsrsquo growth on unit-weightresource of each supply chain member is equal [4 44] Thecalculation details of the fair entropy between the LSI and theFLSP 119894 are shown in Appendix B

So the fair entropy between the LSI and FLSP 119894 in thismodel is

1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)

(ii) The Fair Entropy between FLSP 119894 and Other FLSPs As therevenue-sharing coefficient between the LSI and each FLSP isvisible each FLSP will perceive the fairness of the revenue-sharing coefficient So it is significant to take this part offair entropy into consideration The core idea of establishingthe fair entropy is the relative equity of the revenue-sharingcoefficient

Assuming that the weight of each FLSP is 120596119894 and as the

revenue-sharing coefficient of FLSP 119894 is (1 minus 120593119894) set 120578

119894= (1 minus

120593119894)120596119894

Introducing the concept of entropy makes a transfor-mation of 120578

119894 1205781015840119894

= ((120578119894minus 120578119894)120590120578)120578119894is the average value of

120578119894 120590120578is the standard deviation of 120578

119894 For eliminating the

negative value make a coordinate translation 12057810158401015840

119894= 1205781015840

119894+

1198702 1198702is the range of coordinate translation Then proceed

with the normalization 120588119894= 12057810158401015840

119894sum119873

119894=112057810158401015840

119894 Finally calculate

the fair entropy value between the FLSP and others 1198672119894

=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)

(iii) Objective Function of Total Fair Entropy In the case ofldquoone to 119873rdquo maximizing the total fair entropy is the con-notation of objective function in this model For FLSP 119894 twoparts are combined in fair entropy they are the fair entropybetween the LSI and FLSP 119894 and the fair entropy betweenFLSP 119894 and other FLSPs So the summation of 119873 FLSPsrsquo fairentropies is

119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S

The logisticssubcontractor

I

(wR e1) (D t)(wS e2)

Figure 3 The model of three-echelon LSSC

(2) Establishing the Constrains To be similar to the ldquoone toonerdquomodel the ldquoone to119873rdquomodel restrains the absolute valueof the difference between the ratio of the revenue-sharingcoefficient of the LSI and that of FLSP to be not greater than

the critical value For equity we assume the threshold valueof total fair entropy is 119867

0 set 119867 gt 119867

0 Under the condition

of ldquoone to 119873rdquo the model of the optimal revenue-sharingcoefficient is shown in the following formula

max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)

This model is a single objective programming modeland is aimed at calculating 120593

119894(119894 = 1 2 119873) Under the

condition of given notations the model can be programmedusing LINGO 110 to find the optimal customized level

5 Model Extension The (One to One)Model of Three-Echelon Logistics ServiceSupply Chains

Now we extend the ldquoone to onerdquo and ldquoone to 119873rdquo modelsto three-echelon logistics service supply chains Since theway to extend the two models is the same the ldquoone to onerdquomodel of three-echelon LSSC is chosen to be described indetail In Section 51 we will describe the problem and listbasic assumptions of this model In Section 52 the revenue-sharing contract model of three-echelon logistics servicesupply chains under consideration of a single LSI a singleFLSP and a single subcontractor is presented

51 Problem Description In practice the supply chain mem-bers always contain more than the LSP and the FLSP it isassumed that the FLSP subcontracts its logistics capacity tothe upstream logistics subcontractor 119868 so there are threeparticipants in LSSC The model of the three-echelon LSSCis shown in Figure 3 Notations for the ldquoOne to Onerdquo Modelof Three-Echelon LSSC are given as follows

Notations for the ldquoOne to Onerdquo Model of Three-Echelon LSSC

Decision Variables119876 logistics capacity that 119877 needs to buy from 119878119905 customized level of logistics capacity that customerneeds

1198901 119877rsquos share of the sales revenue under revenue-shar-

ing contract of 119877 and 1198781 minus 1198901 119878rsquos share of the sales revenue under revenue-

sharing contract of 119877 and 1198781198902 119878rsquos share of the sales revenue under revenue-shar-


sharing contract of 119878 and 119868

Other Parameters



logistics service capability is lacking119863(119905) the total demand of customer to logistics serviceunder MC119867 the fair entropy measuring revenue-sharing fair-ness of 119877 and 1198781198670 threshold of fair entropy

119875 unit price of service customer buys from 119877119878(119876) expectation of the logistics capacity of 119877119880 the unit price when 119877 returns extra logisticscapacity to 119878119908119877 The unit price when 119877 purchased services from 119878

119908119878 The unit price when 119878 purchased services from 119868

120587119877 total revenue of logistics service 119877 with 120587

1198771

indicating the total revenue of 119877 under revenue-shar-ing contract and 120587


under game situation


120587119878 total revenue of 119878 with 120587



indicating the total revenue under game situation of119878120587119868 total revenue of 119868 with 120587




chain with 1205871198791

indicating the total revenue of supplychain under revenue-sharing contract and 120587

1198792indi-

cating the total revenue of supply chain under gamesituation120583 the average value of total customer service demand1198631198801 unit price paid to 119878when119877has extra capacity with

1198801= 119908119877

211989711205902

1198802 unit price paid to 119868when 119878 has extra capacity with

1198802= 119908119878

211989721205902

1205902 the variance of total customer service demand 119863

1205741015840 the weight of 119877 in the relationship of 119878 and 119877

1 minus 1205741015840 the weight of 119878 in the relationship of 119878 and 119877



52 The ldquoOne to Onerdquo Model in Three-Echelon LSSC In therevenue-sharing contract of the ldquoone to onerdquo model in three-echelonLSSC the revenue expectation functions of LSI FLSPsubcontractor and the entire LSSC are as follows

The revenue of the integrator 119877 is

120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)

The revenue of the functional service provider 119878 is

120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)

The revenue of the subcontractor 119868 is

120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876

minus (1 minus 1198902) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)

Under centralized control the revenue of the entire LSSCis

120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)

521 Centralized Decision-Making Model Being similar toldquoone to onerdquo situation in two-echelon supply chain when theLSI obtains the optimal value of logistics capacity it mustsatisfy

120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)

We can calculate the optimal value of purchasing capacity1198761and optimal expectation of supply chain total revenue120587

1198791

522 Stackelberg GameModel For the Stackelberg game therevenue expectation functions of 119877 are

119864 (1205871198772

) = 119875 (119876 minus 119887) minus 119908119877119876 minus 119862

119877119876

minus 119862119906 (119875) (120583 minus 119876 + 119887) minus

119908119877

2

11989711205902

119887

(46)

The revenue expectation functions of 119878 are

119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)

Before 119877 adopts the value of 119876 the reaction of 119908119877to

119878 should be considered while 119878 make decision on 119868 Thecalculation details are shown in Appendix C

Under the constraints of rationality the LSI FLSP andsubcontractor will first consider their own profits Only iftheir own profits are satisfied will the maximization of wholesupply chainrsquos revenue be considered So the condition ofsupply chainmembers to accept the revenue-sharing contractis that the revenue obtained in this contract should be no less


than that in the decentralized decision-making model Thusthe restraint is listed as follows

1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)

le 1198901le 1 minus

119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)


(1) Objective Function Similar to the establishment of fairentropy in the situation of ldquoone to onerdquo the core idea is thatthe profitsrsquo growth on unit-weight resource of each supplychain member is equal The calculation details of the fairentropy between 119877 119878 and 119868 are shown in Appendix D

So the fair entropy in this model is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)

(2) Constraint Condition The constraint conditions of themodel are listed as follows

First a certain constraint condition exists between theweight and revenue-sharing coefficient of the LSI and theFLSP and the FLSP and the subcontractor 1205741015840 represents theweight of LSI in the whole LSSC 12057410158401015840 represents the weightof FLSP in the relationship of FLSP and subcontractor it isshown as follows

10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)

The constraint conditionmentioned above can be simpli-fied as follows

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)


1198670 It indicates that the revenue distribution fairness of each

side of the supply chain should be greater than a certainthreshold value


max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)

This model is a single goal programming model Underthe condition of certain notations that are given by makinguse of LINGO 110 we can program the objective functionof the customized level and its constraints and calculate theoptimal customized level

6 The Numerical Analysis

This chapter will illustrate the validity of themodel by specificexample and explore the effect that the customized levelhas on the optimal revenue-sharing coefficient of the supplychain and fair entropy finally giving a specific suggestionfor applying the revenue-sharing contract For all parametersused in numerical example we have already made themdimensionless The example analysis is conducted with a PCwith 24GHzdual-core processor 2 gigabytes ofmemory andwindows 7 system we programmed and simulated numericalresults using LINGO 11 software The content of this chapteris arranged as follows In Section 61 numerical analysis ofldquoone to onerdquo model is presented in Section 62 numericalanalysis of ldquoone to 119873rdquo model is presented in Section 63 thenumerical analysis of the ldquoone to onerdquomodel in three-echelonLSSC is provided In Section 64 a discussion based on theresults of Sections 61 to 63 is presented

61 The Numerical Analysis of the ldquoOne to Onerdquo ModelThe notations and formulas involved in the logistics servicesupply chain model composed of a single LSI and a singleFLSP are as follows

It is assumed that customized level 119905 follows the uniformdistribution in interval of [0 8] that is 119905 sim 119880(0 8) 119863(119905) =

6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670

= 06 Most of the data used here are referring to Liu etal [3] others are assumed according to the practical businessdata

By changing the value of the customized level the vari-ation trend of the corresponding revenue-sharing coefficient


057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast

Figure 4 120593 value curve under a different customized level

0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t


and overall fair entropy can be worked out The customizedlevel 119905 obeys the uniform distribution in [0 8] From theapplication result when 119905 is greater than 2 then the fairentropy will be lower than 05 as shown by Figure 4 whichindicates that the unfair level is high So the case of 119905 gt 2 willnot be considered

As shown by Figures 4 and 5 according to the results thegraphs of the optimal revenue-sharing coefficient 120593 and fairentropy 119867 changing with customized level are presented

As shown in Figure 4 with an increase in customizedlevel the optimal revenue-sharing coefficient first increasedand then decreased and the speed of increase is greater thanthe speed of decrease The optimal revenue-sharing coeffi-cient reaches its maximum at 119867 = 02 when 120593

lowast= 05874

As shown by Figure 5 with the value of 119905 in the intervalof [0 015] the fair entropy can reach the maximum that is119867lowast

= 1 When 119905 gt 15 with the increase of the customizedlevel fair entropy decreases gradually with a low speedWhen119905 = 2 and119867 = 0517 the fair entropy reaches a very low level

In this model in order to discuss the influence of finalservice price on revenue and fair entropy we choose 119875 = 146

and119875 = 154 to compare with the initial price119875 = 15 and thechanges of revenue-sharing coefficients under different pricesare shown in Figure 6 the changes of fair entropy underdifferent prices are shown in Figure 7

As shown in Figure 6 for a certain customized levelwith an increase of the price the optimal revenue-sharingcoefficient is increased whichmeans the increase of price willbenefit the integrator

As shown in Figure 7 the fair entropy can reach themaximum under different prices that is 119867lowast = 1 and thenthe fair entropy decreases gradually with a low speed Itshould be noticed that with an increase of the price the fairentropy will decrease under the certain customized level

0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

Figure 6 120593 value curve under different prices

040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t


62 The Numerical Analysis of the ldquoOne to 119873rdquo Model Facingthe ldquoone to 119873rdquo condition this paper chose a typical exampleof a single LSI and three FLSPs to analyze The notations ofeach FLSP are different and are given as follows

FLSP 1 it is assumed that customized level 1199051follows

uniform distribution on [0 85] that is 1199051sim 119880(0 8) 119863(119905

1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and

1205741= 07FLSP 2 it is assumed that customized level 119905

2follows

uniform distribution on [0 8] that is 1199051

sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065


uniform distribution on[0 75] that is 1199053sim 119880(0 8) 119863(119905

3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065

Most of the data used here are referring to Liu et al [3]others are assumed according to the practical business situa-tion

As the arbitrary value of the uniform distribution intervalcan be chosen by the customized level of three FLSPs forfinding the effect of different customized level combinationson the optimal revenue-sharing coefficient we assume that


Table 1 Example analysis result of one to 119873 model

119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382

the customized level of each demand of three FLSPs has aninitial value of 119905

1= 01 119905

2= 02 and 119905

3= 04 respectively

then zoom in (or out) the three customized levels to a specificratio 119872 at the same time and study the variation in theoptimal revenue-sharing coefficient value under different 119872with 119872 being the independent variable The data result isshown in Table 1

Based on the result shown in Table 1 trend charts of119872 to three absolutely fair entropies (fair entropy betweenthe LSI and three FLSPs indicated by 119867

11 11986712 and 119867

13

resp) three revenue-sharing coefficients (revenue-sharingcoefficient between the LSI and three FLSPs indicated by 120593

1

1205932 and 120593

3 resp) and total fair entropy (119867) can be severally

drawn they are shown in Figures 8 9 and 10Figure 8 shows that fair entropies between the LSI and

each FLSP are less than 1 but can be infinitely close to 1 inthe case of ldquoone to threerdquo The maximum of each entropy is119867lowast

11= 119867lowast

12= 119867lowast

13= 09999 and shows a declining trend with

the increase of the customized levelSimilar to Figure 4 Figure 9 shows that all three revenue-

sharing coefficients show a trend of first increasing and thendecreasing with the increase of 119872 The maximum of threerevenue-sharing coefficients can be found 120593lowast

1= 05872 120593lowast

2=

05990 and 120593lowast

3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi

Figure 8Three absolutely fair entropy value curves under different119872

05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1

Figure 9 Three revenue-sharing coefficient value curves underdifferent 119872

093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

Figure 10 Total fair entropy value curve under different 119872

Figure 10 indicates the trend of total fair entropy changingwith 119872 which is decrease 119889 and has its maximum at 119867lowast =

09988 By observing Figure 10 the total fair entropy can beinfinitely close to 1 and when 119872 lt 14 it stays above 0998with a tiny drop These numbers indicate that there exists abetter interval of customized level that canmake entropy stayat a high level

Similar to the ldquoone to onerdquo model in this ldquoone to 119873rdquomodel the price of final service product is constant so inorder to discuss the influence of final service price on revenueand fair entropy we choose119875 = 146 and119875 = 154 to comparewith the initial price 119875 = 15 and the changes of the threerevenue-sharing coefficients under different prices are shownin Figures 11 12 and 13 the change of fair entropy underdifferent price is shown in Figure 14


P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06

Figure 11 Value curves of the revenue-sharing coefficient 1205931under

different prices4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices

Figures 10 11 and 12 indicate that under certain119872 withthe increase of price the three revenue-sharing coefficientswill also increase which means that the increase of price willbenefit the integrator

Figure 13 shows that under certain 119872 with the increaseof price the total fair entropy will decrease which means the

092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

Figure 14 Total fair entropy value curve under different prices

21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1

Figure 15 1198901under a different customized level

increase of price will lower the fairness of the whole supplychain

63 The Numerical Analysis of the ldquoOne to Onerdquo Model inThree-Echelon LSSC The notations and formulas involvedin the logistics service supply chain model composed of asingle LSI and a single FLSP and a single subcontractor areas follows


81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055

Most of the data used here are referring to Liu et al [3] othersare assumed according to the practical business data

By changing the value of the customized level the vari-ation trend of the corresponding revenue-sharing coefficientand overall fair entropy can be found

As shown in Figure 15 with the increase of the cus-tomized level 119890

1first increases and then decreases which is

the same as Figure 4 But 1198902keeps increasingwith the increase

of the customized level as shown in Figure 16That is becausebefore 119890

lowast

1the revenue-sharing ratio of119877will increase with the

increase of the customized level which means the revenue-sharing ratio of 119878 and 119868 will decrease so FLSP 119878 will improvethe revenue-sharing ratio of 119878 between 119878 and 119868 to maximizeits own profit then after 119890

lowast

1the revenue-sharing ratio of 119877


01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t

Figure 17 119867 under a different customized level

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices

will decrease with the increase of the customized level whichmeans the revenue-sharing ratio of 119878 and 119868 will increase butFigure 16 shows that the revenue-sharing ratio of 119878 between119878 and 119868 is very low FLSP 119878 will try to share more benefit sovalue curve of 119890

2will keep increasing

Figure 17 is similar to Figure 5 which indicates thatin three-echelon LSSC the fair entropy also can reach themaximum that is 119867

lowast= 1 With the increase of the cus-

tomized level fair entropy decreases gradually with a lowspeed

Similar to the ldquoone to onerdquomodel in two-echelon LSSC inthis three-echelon model the price of final service product isconstant so in sensitivity analysis of service price we choose119875 = 146 and 119875 = 154 to compare with the initial price 119875 =

15 and the changes of the two kinds of revenue-sharing coef-ficients under different prices are shown in Figures 18 and 19

e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t


the change of fair entropy under different price is shown inFigure 20

As shown in Figure 18 which is similar to Figure 6 underthe certain customized level with an increase of the pricethe optimal revenue-sharing coefficient of 119877 is increasedFigure 19 indicates that under the certain customized levelwith an increase of the price the optimal revenue-sharingcoefficient of 119878 between 119878 and 119868 also increases

As shown in Figure 20 with three different prices all thefair entropies can reach themaximumwhichmeans the LSSCcan reach absolutely fairness similar to Figure 7 with theincrease of price the total fair entropy will decrease

64 Example Result Analysis By analyzing the exampleresults and Figures 1ndash20 the conclusions can be drawn asfollows

(1) In the case of a single LSI and single FLSP fairentropy of the LSI and FLSP in a revenue-sharing contract canreach 1 (that means absolute fair) under a specific interval ofcustomized level (such as [0 015] in this example) But withan increase in the customized level the fair entropy betweenthe LSI and FLSP shows a trend of descending

(2)Whether in the case of ldquoone to onerdquo or ldquoone to119873rdquo therevenue-sharing coefficient of the LSI always shows a trend


of first increasing and then decreasing with the increase inthe customized level namely an optimal customized levelmaximizes the revenue-sharing coefficient

(3) As shown in Figure 5 with the increase of zoomingin (or out) of 119905 namely 119872 under the case of ldquoone to 119873rdquofair entropy between three FLSPs and an LSI shows a trendof decrease It suggests that fair entropy between each FLSPand LSI cannot reach absolute fair under the case of ldquoone to119873rdquo due to the relatively fair factors between FLSPs

(4) In the case of ldquoone to 119873rdquo total fair entropy decreaseswith the increase of119872 although it cannot reach the absolutemaximum 1 there exists an interval that keeps the total fairentropy at a high level which is close to 1

(5) Whether in the case of ldquoone to onerdquo or ldquoone to 119873rdquowhen the customized level is certain the revenue-sharingcoefficient of the LSI will increase with the increase of pricebut the total fair entropy will decrease

(6) In the case of ldquoone to onerdquo model in three-echelonLSSC there are two kinds of revenue-sharing coefficientsthe revenue-sharing coefficient of the LSI will first increaseand then decrease with the increase of the customized levelwhen the customized level is certain the revenue-sharingcoefficient of the LSI will increase with the increase of pricethe revenue-sharing coefficient of the FLSP between FLSPand subcontractor will keep increasing with the increase ofthe customized level When the customized level is certainthe revenue-sharing coefficient of the FLSP will also increasewith the increase of price For the ldquoone to onerdquo model inthree-echelon LSSC the total fair entropy also can reachthe maximum and with the increase of price the total fairentropy will decrease

7 Conclusions and Management Insights

71 Conclusions Based on MC service mode this paperconsidered the profit fairness factor of supply chain membersand constructed a revenue-sharing contractmodel of logisticsservice supply chain Combined with examples to analyzethe effect of service customized level on optimal revenue-sharing coefficient and fair entropy this paper also raised anintuitional conclusion and unforeseen result In particularthe intuitional conclusion has the following several aspects

(1) Similar to the conclusion of Liu et al [3] there existsan optimal customized level range that makes theLSI and FLSP get absolutely fair in a revenue-sharingcontract between only a single LSI and a single FLSPBut under the ldquoone to 119873rdquo condition there exists aninterval of customized level value that maximizes thefair entropy provided by the LSI and 119873 FLSPs butcannot guarantee that the fair entropy is equal to 1

(2) Under the situation of ldquoone to onerdquo and ldquoone to 119873rdquowith the increase of the customized level the revenue-sharing coefficient of the LSI showed a trend of firstincreasing and then decreasing This illuminates thatthere exists an optimal customized level that makesrevenue-sharing coefficient reach the maximum

(3) For ldquoone to onerdquo model in three-echelon LSSC thereexists an interval of customized level value that max-imizes the fair entropywhichmakes the LSI the FLSPand the subcontractor get absolutely fair in a revenue-sharing contract

In addition two unforeseen conclusions are put forwardin this paper

(1) In the case of ldquoone to onerdquo the interval of customizedlevel that can realize absolute fair is limited Supplychain fair entropy will decrease as the customizedlevel increases when the degree surpasses the maxi-mum of the interval

(2) In the case of ldquoone to119873rdquo considering the fairness fac-tor between FLSPs the revenue distribution betweenthe LSI and each FLSP will not reach absolute fair Butoverall fair entropy will approach absolute fairnesswhen the customized level is in a specified range

(3) Whether in the case of ldquoone to onerdquo or ldquoone to 119873rdquo intwo-echelon LSSC the increase of price will benefitthe integrator but will reduce the fairness of the wholesupply chain

(4) Under the situation of ldquoone to onerdquo model in three-echelon LSSC the revenue-sharing coefficient of theLSI showed a trend of first increasing and thendecreasing which means there is an optimal cus-tomized level that makes the revenue-sharing coef-ficient of the LSI reach its maximum The revenue-sharing coefficient of the FLSP between FLSP andsubcontractor showed a trend of increasing

(5) Under the situation of ldquoone to onerdquo model in three-echelon LSSC the increase of price will benefit theintegrator but will reduce the fairness of the wholesupply chain

72 Management Insights The research conclusions of thispaper have important significance for the LSI First whetherin ldquoone to onerdquo or ldquoone to 119873rdquo when operating in realitythe customized level can be restricted in a rational range byconsulting with the customer This will impel the fairnessbetween the LSI and FLSP to the maximum and sustain therevenue-sharing contract relationship Second in the case ofldquoone to 119873rdquo profits between the LSI and each FSLP cannotachieve absolute fair therefore it is not necessary for theLSI to achieve absolute fair with all FLSPs Third the LSIcan try to choose the FLSP with greater flexibility when thecustomized level changes For example if the customer needstime to be customized the LSI can choose the FLSP thathas flexibility in time preferentially Not only will it meet theneed of customers but also it is good for obtaining largersupply chain total fair entropy between the LSI and FLSP andfor realizing the long-term coordination contract Fourth toguarantee the fairness of all the members in LSSC since theincrease of price will obviously benefit the integrator FLSPand subcontractor can participate into the price setting beforethe revenue-sharing contract is signed


73 Research Limitation and Future Research DirectionsThere are some defects in the revenue-sharing contractsestablished in this paper For example this paper assumedthat the final price of the LSIrsquos service products for thecustomer is a constant value but in fact the higher thecustomized level is the higher themarket pricemay be Futureresearch can assume that the final price of service is related tothe level of customization

Otherwise in order to simplify themodel this paper onlyconsidered two stages of logistics service supply chain butthere have always been three in reality Future research cantry to extend the numbers of stages to three for enhancingthe utility of the model

Appendices

A The Calculation Details of StackelbergGame in (One to 119873) Model

This appendix contains the calculation details of119908119894and119876

119894in

the ldquoone to 119873rdquo modelFor the Stackelberg game the revenue expectation func-

tions of 119877 and 119878119894are as follows



119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894

First to maximize the profits the following first-ordercondition should be satisfied

120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)

Obtaining the result 119908119894= minus119876119894119897119894120590119894

22119887119894

Tomaximize the profits of eachmembers of supply chainthe following first-order condition should be satisfied

120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)

Then the second derivative is less than 0 namely1205972119864(120587119877119894)120597119876119894

2lt 0

From formula (A3) under the Stackelberg game we canobtain the optimal purchase volumes of capacity 119876

10158401015840

119894and 119908

10158401015840

119894

and the optimal profits expectation of 119877 is 119864(12058710158401015840

119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)

Then the optimal logistics capacity volumes that the LSIpurchases from all FLSPs can be calculated 11987610158401015840 = sum

119873

119894=111987610158401015840

119894

And the optimal expected total profits of 119877 are 119864(12058710158401015840

119877) =

sum119873

119894=1119864(12058710158401015840

119877119894) The expectation total profits of all FLSPs are

119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)

B The Calculation Details of the FairEntropy between the LSI and the FLSP 119894 in(One to 119873) Model

This appendix contains the calculation details of the fairentropy between the LSI and FLSP 119894 Consider

120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879

119878119894 respectively indicate the total

cost of the LSI and that of FLSP in supply chain branchIntroducing the entropy makes the following standard-

ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)

120585119894is average value and 120590

119894is standardized deviation

To eliminate the negative value make a coordinate trans-lation [44] 12058510158401015840

119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range

of coordinate translation and processes the normalization120582119877119894

= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582

119878119894and substitute them into the function

to calculate the fair entropy of the whole supply chain 1198671119894

=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867

1119894le 1 the fair entropy

between the LSI and FLSP 119894 in this model is

1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)

C The Calculation Details of StackelbergGame in (One to One) Model of Three-Echelon Logistics Service Supply Chains

First to maximize the profits of subcontractor 119868 the first-order condition should be satisfied

120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887

Next to maximize the profits of FLSP 119878 the first-ordercondition should be satisfied

120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887


Finally to maximize the profits of the LSI the first-ordercondition should be satisfied

120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862

119906 (119875)] (1 minus 119904) minus 119862119877minus 119908119877minus

119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0

We can obtain the optimal value of purchasing thecapacity119876

10158401015840 under the Stackelberg game And by substituting11987610158401015840 into the expression of 119908

119877and 119908

119878 the corresponding 119908

10158401015840

119877

and 11990810158401015840

119878can be calculated

D The Calculation Details of the FairEntropy between 119877 119878 and 119868 in (One toOne) Model of Three-Echelon LogisticsService Supply Chains

Since the revenue-sharing coefficients are 1198901and 119890

2 the

respective profit growth of the LSI the FLSP and subcontrac-tor is

Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)

Considering the different weights of the LSI and FLSPand the FLSP and subcontractor in the supply chain supposethe weights of each are given It is assumed that the weightof the LSI is 120574

1in the relationship of LSP and FLSP thus

the weight of FLSP is 1 minus 1205741in the relationship of LSP and

FLSP the weight of the FLSP is 1205742in the relationship of FLSP

and subcontractor thus the weight of the subcontractor is1minus1205742in the relationship of FLSP and subcontractor then the

profit growth on unit-weight resource of the LSI the FLSPand subcontractor is

1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)

The total cost of the LSI is

119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)

The total cost of the FLSP is119879119878= 119862119878119876 + 119908

119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)

The total cost of the subcontractor is119879119868= 119862119868119876 + (1 minus 119890

2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)

Then we introduce the concept of entropy

1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)


119894Similar to the ldquoone to onerdquo model in two-echelon LSSC

then calculating the ratio 120582119894of 12058510158401015840119894 set

1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)

After obtaining 1205821 1205822 and 120582


supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This research is supported by the National Natural ScienceFoundation of China (Grant no 71372156) and supportedby Humanity and Social Science Youth foundation of Min-istry of Education of China (Grant no 13YJC630098) andsponsored by China State Scholarship Fund and IndependentInnovation Foundation of Tianjin University The reviewersrsquocomments are also highly appreciated


References

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

[2] S Li Z Zhu and L Huang ldquoSupply chain coordination anddecision making under consignment contract with revenuesharingrdquo International Journal of Production Economics vol120 no 1 pp 88ndash99 2009

[3] W-H Liu X-C Xu and A Kouhpaenejad ldquoDeterministicapproach to the fairest revenue-sharing coefficient in logisticsservice supply chain under the stochastic demand conditionrdquoComputers amp Industrial Engineering vol 66 no 1 pp 41ndash522013

[4] K L Choy C-L Li S C K So H Lau S K Kwok andDW KLeung ldquoManaging uncertainty in logistics service supply chainrdquoInternational Journal of Risk Assessment amp Management vol 7no 1 pp 19ndash43 2007

[5] W H Liu X C Xu Z X Ren and Y Peng ldquoAn emergencyorder allocationmodel based onmulti-provider in two-echelonlogistics service supply chainrdquo Supply Chain Management vol16 no 6 pp 391ndash400 2011

[6] H F Martin ldquoMass customization at personal lines insurancecenterrdquo Planning Review vol 21 no 4 pp 27ndash56 1993

[7] W H Liu W Qian D L Zhu and Y Liu ldquoA determi-nation method of optimal customization degree of logisticsservice supply chain with mass customization servicerdquo DiscreteDynamics in Nature and Society vol 2014 Article ID 212574 14pages 2014

[8] J Liou and L Yen ldquoUsing decision rules to achieveMCof airlineservicesrdquoEuropean Journal of Operational Research vol 205 no3 pp 690ndash686 2010

[9] S S Chauhan and J-M Proth ldquoAnalysis of a supply chainpartnership with revenue sharingrdquo International Journal of Pro-duction Economics vol 97 no 1 pp 44ndash51 2005

[10] H Krishnan and R A Winter ldquoOn the role of revenue-sharingcontracts in supply chainsrdquo Operations Research Letters vol 39no 1 pp 28ndash31 2011

[11] Q Pang ldquoRevenue-sharing contract of supply chain with waste-averse and stockout-averse preferencesrdquo in Proceedings of theIEEE International Conference on Service Operations and Logis-tics and Informatics (IEEESOLI rsquo08) pp 2147ndash2150 October2008

[12] B J Pine II and D Stan MC The New Frontier in BusinessCompetition Harvard Business School Press Boston MassUSA 1993

[13] F Salvador P M Holan and F Piller ldquoCracking the code ofMCrdquo MIT Sloan Management Review vol 50 no 3 pp 71ndash782009

[14] Y X Feng B Zheng J R Tan andZWei ldquoAn exploratory studyof the general requirement representation model for productconfiguration in mass customization moderdquo The InternationalJournal of Advanced Manufacturing Technology vol 40 no 7pp 785ndash796 2009

[15] D Yang M Dong and X-K Chang ldquoA dynamic constraintsatisfaction approach for configuring structural products undermass customizationrdquo Engineering Applications of Artificial Intel-ligence vol 25 no 8 pp 1723ndash1737 2012

[16] C Welborn ldquoCustomization index evaluating the flexibility ofoperations in aMC environmentrdquoThe ICFAI University Journalof Operations Management vol 8 no 2 pp 6ndash15 2009

[17] X-F Shao ldquoIntegrated product and channel decision in masscustomizationrdquo IEEETransactions on EngineeringManagementvol 60 no 1 pp 30ndash45 2013

[18] H Wang ldquoDefects tracking in mass customisation productionusing defects tracking matrix combined with principal compo-nent analysisrdquo International Journal of Production Research vol51 no 6 pp 1852ndash1868 2013

[19] D-C Li F M Chang and S-C Chang ldquoThe relationshipbetween affecting factors and mass-customisation level thecase of a pigment company in Taiwanrdquo International Journal ofProduction Research vol 48 no 18 pp 5385ndash5395 2010

[20] F S Fogliatto G J C Da Silveira and D Borenstein ldquoThemass customization decade an updated review of the literaturerdquoInternational Journal of Production Economics vol 138 no 1 pp14ndash25 2012

[21] A Bardakci and J Whitelock ldquoMass-customisation in market-ing the consumer perspectiverdquo Journal of ConsumerMarketingvol 20 no 5 pp 463ndash479 2003

[22] H Cavusoglu H Cavusoglu and S Raghunathan ldquoSelecting acustomization strategy under competitionmass customizationtargeted mass customization and product proliferationrdquo IEEETransactions on Engineering Management vol 54 no 1 pp 12ndash28 2007

[23] L Liang and J Zhou ldquoThe optimization of customized levelbased on MCrdquo Chinese Journal of Management Science vol 10no 6 pp 59ndash65 2002 (Chinese)

[24] L Zhou H J Lian and J H Ji ldquoAn analysis of customized levelon MCrdquo Chinese Journal of Systems amp Management vol 16 no6 pp 685ndash689 2007 (Chinese)

[25] P Goran and V Helge ldquoGrowth strategies for logistics serviceproviders a case studyrdquo The International Journal of LogisticsManagement vol 12 no 1 pp 53ndash64 2001

[26] J Korpela K Kylaheiko A Lehmusvaara and M TuominenldquoAn analytic approach to production capacity allocation andsupply chain designrdquo International Journal of Production Eco-nomics vol 78 no 2 pp 187ndash195 2002

[27] A Henk and V Bart ldquoAmplification in service supply chain anexploratory case studyrdquo Production amp Operations Managementvol 12 no 2 pp 204ndash223 2003

[28] A Martikainen P Niemi and P Pekkanen ldquoDeveloping a ser-vice offering for a logistical service providermdashcase of local foodsupply chainrdquo International Journal of Production Economicsvol 157 no 1 pp 318ndash326 2014

[29] R L Rhonda K D Leslie and J V Robert ldquoSupply chainflexibility building ganew modelrdquo Global Journal of FlexibleSystems Management vol 4 no 4 pp 1ndash14 2003

[30] W Liu Y Yang X Li H Xu and D Xie ldquoA time schedulingmodel of logistics service supply chain with mass customizedlogistics servicerdquo Discrete Dynamics in Nature and Society vol2012 Article ID 482978 18 pages 2012

[31] W H Liu Y Mo Y Yang and Z Ye ldquoDecision model of cus-tomer order decoupling point onmultiple customer demands inlogistics service supply chainrdquo Production Planning amp Controlvol 26 no 3 pp 178ndash202 2015

[32] I Giannoccaro and P Pontrandolfo ldquoNegotiation of the revenuesharing contract an agent-based systems approachrdquo Interna-tional Journal of Production Economics vol 122 no 2 pp 558ndash566 2009

[33] A Z Zeng J Hou and L D Zhao ldquoCoordination throughrevenue sharing and bargaining in a two-stage supply chainrdquoin Proceedings of the IEEE International Conference on Service


Operations and Logistics and Informatics (SOLI rsquo07) pp 38ndash43IEEE Philadelphia Pa USA August 2007

[34] Z Qin ldquoTowards integration a revenue-sharing contract in asupply chainrdquo IMA Journal of Management Mathematics vol19 no 1 pp 3ndash15 2008

[35] J Hou A Z Zeng and L Zhao ldquoAchieving better coordinationthrough revenue-sharing and bargaining in a two-stage supplychainrdquo Computers and Industrial Engineering vol 57 no 1 pp383ndash394 2009

[36] F El Ouardighi ldquoSupply quality management with optimalwholesale price and revenue sharing contracts a two-stagegame approachrdquo International Journal of Production Economicsvol 156 pp 260ndash268 2014

[37] S Xiang W You and C Yang ldquoStudy of revenue-sharing con-tracts based on effort cost sharing in supply chainrdquo in Pro-ceedings of the 5th International Conference on Innovation andManagement vol I-II pp 1341ndash1345 2008

[38] B van der Rhee G Schmidt J A A van der Veen andV Venugopal ldquoRevenue-sharing contracts across an extendedsupply chainrdquo Business Horizons vol 57 no 4 pp 473ndash4822014

[39] I Giannoccaro and P Pontrandolfo ldquoSupply chain coordinationby revenue sharing contractsrdquo International Journal of Produc-tion Economics vol 89 no 2 pp 131ndash139 2004

[40] SWKang andH S Yang ldquoThe effect of unobservable efforts oncontractual efficiency wholesale contract vs revenue-sharingcontractrdquo Management Science and Financial Engineering vol19 no 2 pp 1ndash11 2013

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

[42] J-C Hennet and S Mahjoub ldquoToward the fair sharing ofprofit in a supply network formationrdquo International Journal ofProduction Economics vol 127 no 1 pp 112ndash120 2010

[43] Z-H Zou Y Yun and J-N Sun ldquoEntropymethod for determi-nation of weight of evaluating indicators in fuzzy synthetic eval-uation for water quality assessmentrdquo Journal of EnvironmentalSciences vol 18 no 5 pp 1020ndash1023 2006

[44] X Q Zhang C B Wang E K Li and C D Xu ldquoAssessmentmodel of ecoenvironmental vulnerability based on improvedentropy weight methodrdquoThe Scientific World Journal vol 2014Article ID 797814 7 pages 2014

[45] C ErbaoWCan andMY 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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the study did not involve MCLS In fact MCLS and com-mon services have two significant discrepancies the first isthat the customized level will affect the design of supplychain contracts It is urgent to introduce the customized levelinto revenue-sharing contract design from the perspectiveof theory The second is that MCLS is provided by an LSIvia integrating FLSPsrsquo service capacity therefore FLSPs haveto evaluate the equity of profit distribution So a way tointroduce a fairness factor into a revenue-sharing contract isalso necessary to be considered

This paper expands the research content on the basis ofLiu et al [3] First this paper will take theMCLS environmentinto consideration and introduce the customized level asan important factor for the design of a revenue-sharingcontract mechanism Second this paper is based on thelogistics service supply chain of two echelons and establishesa revenue-sharing mechanism model in two relations byexpanding the ldquoone to onerdquo relationship between the LSIand FLSP to ldquoone to 119873rdquo Finally this paper aims at themaximum of fair entropy and setting improved fair entropyas the objective function of the model and then explores theinfluence of customized level on supply chain fair entropyand some interesting findings are obtained For examplewhether in the case of ldquoone to onerdquo or ldquoone to 119873rdquo revenue-sharing coefficients of the LSI will first increase and thendecrease with the increase of the customized level that isthere is an optimal interval of customized level that let therevenue-sharing coefficient of the LSI reach its maximum

This paper is organized as follows the second chapteris a literature review in which we systematically review andsummarize the studies of MC mode service supply chainand revenue-sharing contract The third and fourth chaptersintroduce the procedure of model establishing We willrespectively illuminate the procedure of building ldquoone toonerdquo and ldquoone to 119873rdquo service supply chain revenue-sharingcontract models under the MC environment The fifthchapter will conduct an example analysis LINGO 110 willbe used for the numerical simulation of the two modelsestablished in this paper In last chapter the conclusionsome important conclusions in this paper and managementinsights are summarized and finally the limitation of thispaper and future research directions are put forward

2 Literature Review

Our research mainly concerns the order allocation decisionunder the situation of order insertion in the MCLS environ-ment Thus the literature review is mainly related to the MCmode order allocation and order insertion Our researchdirection will be proposed after summarizing the progressionof developments in the literature as well as its current gaps

21 MC Mode and Customized Level In 1993 Pine II andStan proposed that MC is a manufacturing mode to widelyprovide personalized products and services which would bethe frontier of commercial competition [12] With 20 yearsrsquodevelopment and application MC mode has become themain operation mode and the key to improving commercial

competition Including the autoindustry clothing industryand computer industry many major economic industriesbenefit from this productionmode [13] Overall most studiesof MC mainly discuss the production modes including theMC approach and its product design [14 15] MC produc-tion planning and control technology [16 17] MC cost ofproduction [18] and factors and conditions that affect MC[19] With the increase of research on MC many scholarsreview the literature in the field Fogliatto made a detailedarrangement and summary of the literature from the 1980sHe indicated many unsolved problems with MC like supplychain coordination and quality control issues which wouldbe the new fields of MC research [20]

The customized level is a crucial factor in MC A highlevel of customization leads to a huge cost of production [21]Enterprises will not produce the customized products tomeetcustomers demand when the customized level is too high[22]Therefore it is important to define a rational customizedlevel Through a mathematical optimization model somescholars have made specialized exploration From the per-spective of customer satisfaction and corporate customizedcost Liang and Zhou worked out the way to determinethe optimal degree under the condition of fixed customizedproducts supply capacity [23] By building a mathematicalmodel on the relationship between customized level marketdemand and corporate profits Zhou et al provided guidancefor manufacturers to determine a rational customized level[24]

With more attention paid to service supply chains somescholars began to research the customized level in MCservice Liu researched the method of determining an opti-mal customized level in a logistics service supply chainThey proposed three different decision-making modes fora customized level (the LSI decides the customized levelthe customer decides the customized level customized leveldecision of centralized supply chain) and these modelsdiscussed the reaction of profits of the LSI customers and thewhole supply chain under different decision-making modeswith the change of customized level respectively [7] Overallmost of MC research on customized levels is focused onproduct supply chain currently but there is little research onservice supply chains

22 Logistics Service Supply Chain With the development ofthe logistics industry the logistics service supply chain hasbecome the new hotspot in the supply chain field Currentlyresearch on logistics service supply chains mainly focuses onlogistics service supply chain connotation [4 25] logisticsservice supply chain coordinatemechanism [3 26] empiricalstudy of logistics service supply chain [27 28] and selectionand performance evaluation of FLSPs [5]

In the 21st century with the rise in MC logistics servicemode research results show that logistics service supplychains under the background of MC are fruitful becauseof the endeavors of many scholars For example Rhondaet al built an MC supply chain service model to meet thecustomersrsquo demand of personality [29] Liu et al carried outresearch on the time scheduling problem under the modeof MC logistics service and constructed the time scheduling






Q

P


retailer)

Logisticsservice

integrator R


S

(D t)

(w 120593)










Decision Variables


Other Parameters







1015840

119877indi-


10158401015840



1015840



119878


chain with 1205871015840



119879indi-






0+ 119896119905 minus (12)ℎ119905


1198630+ 119896119905 minus (12)ℎ119905

2


119877= 1198620+ 119911119905 and 119908 = 119908

0+ 119910119905 Since









119878 (119876) = 119864 (min (119876119863 (119905)))

= int

infin

0

min (119876119863 (119905)) 119891 (119905) 119889119905

= 119876 minus int

119876

0

[119876 minus 119863 (119905)] 119891 (119905) 119889119905

= 119876 minus int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(1)


119864 (119876 minus 119863 (119905))+= 119876 minus 119878 (119876)

= int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(2)



119864 (119863 (119905) minus 119876)+= 120583 minus 119878 (119876)

= 120583 minus 119876

+ int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(3)


120587119877


119906 (119875) [120583 minus 119878 (119876)]

minus 119880 [119876 minus 119878 (119876)]

(4)


119878119876


(5)


120587119879

= 120587119877+ 120587119878

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119906 (119875) [120583 minus 119878 (119876)]

(6)



120597119864 (120587119879)

120597119876= 0 (7)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878

119875 + 119862119906 (119875)

(8)


119864(1205871015840

119879)



0[119876minus119863

0minus

119896119905 + (12)ℎ1199052]119891(119905)119889119905 thus 119878(119876) = 119876 minus 119887


[(12)ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891(119876) and 120597119878(119876)120597119876 = 1 minus 119904


119864 (12058710158401015840


119877119876 minus 119862

119906 (119875) (120583 minus 119876 + 119887)

minus1199082

1198971205902119887

119864 (12058710158401015840

119878) = [119908 minus 119862

119878] 119876 +

1199082

1198971205902119887

(9)


120597119864 (12058710158401015840

119878)

120597119908= 119876 +

2119908

1198971205902119887 = 0 (10)

Then 119908 = minus11987611989712059022119887


120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


1199041199082

1198971205902= 0 (11)


10158401015840

119877)1205971198762lt 0





120593119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 120593) 119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

(12)





Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

(13)


1205851=

Δ120579119877

120593120574119879119877

1205852=

Δ120579119878

(1 minus 120593) (1 minus 120574) 119879119878

(14)


= 119908119876 + 119862119877119876 + 120593119862

119906 (119875) [120583 minus 119878 (119876)]

+ 119881 [119876 minus 119878 (119876)]

(15)


119879119878= 119862119878119876 + (1 minus 120593)119862

119906 (119875) [120583 minus 119878 (119876)] (16)


1and 1205762[43]

1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

(17)

Among them

120585 =1

2

2

sum

119894=1

120585119894

120590 = radic1

2

2

sum

119894=1

(120585119894minus 120585)2

(18)


119894


2



10158401015840

119894 12058510158401015840119894

= 119870+ 1205851015840



set

1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2

(19)



supply chain is

119867 = minus1

ln119898

119898

sum

119894=1

120582119894ln 120582119894 (20)


119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822) (21)



10038161003816100381610038161003816100381610038161003816

120593119877

120593119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576 (22)


10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576 (23)





max 119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(24)





retailer)

Logisticsservice

integrator RP


service provider S1

service provider Si

SN

middotmiddotmiddot

(Q1 wi 120593i)

(Qi wi 120593i)

(QN wN 120593N)

(D t)







119894 120593119894) each





Decision Variables


119894





119894between


Other Parameters





119894






119894 FLSP 119878119894 119894 = 1 2 119873




119873

119894=1119878(119876119894)




119894


1205871015840


contract and 12058710158401015840


game situation


119877= sum119873

119894=1120587119877119894


119894 with 120587

1015840



and 12058710158401015840


119894under game

situation




119878119894and 119877


supply chain


119877 and 119878119868



of 119877 and 119878119894




119894




119894) = 119863

1198940+119896119905119894minus (12)119905

119894


119894

2



119894and







119878 (119876119894) = 119864 (min (119876

119894 119863 (119905119894)))

= int

infin

0

min (119876119894 119863 (119905119894)) 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 119863 (119905

119894)] 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(25)


119864119894(119876119894minus 119863 (119905

119894))+= 119876119894minus 119878 (119876

119894)

= int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(26)


119864119894(119863 (119905119894) minus 119876119894)+= 120583119894minus 119878 (119876

119894)

= 120583119894minus 119876119894+ int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(27)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(28)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(29)


120587119878119894

= (1 minus 120593119894) 119875119894119878 (119876119894) + 119908119894119876119894minus 119862119878119894119876119894


119894)]

+ 119880 [119876119894minus 119878 (119876

119894)]

(30)


120587119879119894

= 120587119877119894

+ 120587119878119894

= 119875119894119878 (119876119894) minus 119862119877119894119876119894minus 119862119878119894119876119894

minus 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

(31)




119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)


2lt 0


119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)


1015840





sum119873

119894=11198761015840

119894


1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)




119908119894

2

1198961198941205901198942119887119894

(34)


119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)


119894to 119878119894


119894are



120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)







1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)




119894= (1 minus

120593119894)120596119894


119894 1205781015840119894





119894= 1205781015840

119894+



119894sum119873

119894=112057810158401015840



=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)


119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Q

P


retailer)

Logisticsservice

integrator R


S

(D t)

(w 120593)










Decision Variables


Other Parameters







1015840

119877indi-


10158401015840



1015840



119878


chain with 1205871015840



119879indi-






0+ 119896119905 minus (12)ℎ119905


1198630+ 119896119905 minus (12)ℎ119905

2


119877= 1198620+ 119911119905 and 119908 = 119908

0+ 119910119905 Since









119878 (119876) = 119864 (min (119876119863 (119905)))

= int

infin

0

min (119876119863 (119905)) 119891 (119905) 119889119905

= 119876 minus int

119876

0

[119876 minus 119863 (119905)] 119891 (119905) 119889119905

= 119876 minus int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(1)


119864 (119876 minus 119863 (119905))+= 119876 minus 119878 (119876)

= int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(2)



119864 (119863 (119905) minus 119876)+= 120583 minus 119878 (119876)

= 120583 minus 119876

+ int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(3)


120587119877


119906 (119875) [120583 minus 119878 (119876)]

minus 119880 [119876 minus 119878 (119876)]

(4)


119878119876


(5)


120587119879

= 120587119877+ 120587119878

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119906 (119875) [120583 minus 119878 (119876)]

(6)



120597119864 (120587119879)

120597119876= 0 (7)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878

119875 + 119862119906 (119875)

(8)


119864(1205871015840

119879)



0[119876minus119863

0minus

119896119905 + (12)ℎ1199052]119891(119905)119889119905 thus 119878(119876) = 119876 minus 119887


[(12)ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891(119876) and 120597119878(119876)120597119876 = 1 minus 119904


119864 (12058710158401015840


119877119876 minus 119862

119906 (119875) (120583 minus 119876 + 119887)

minus1199082

1198971205902119887

119864 (12058710158401015840

119878) = [119908 minus 119862

119878] 119876 +

1199082

1198971205902119887

(9)


120597119864 (12058710158401015840

119878)

120597119908= 119876 +

2119908

1198971205902119887 = 0 (10)

Then 119908 = minus11987611989712059022119887


120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


1199041199082

1198971205902= 0 (11)


10158401015840

119877)1205971198762lt 0





120593119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 120593) 119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

(12)





Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

(13)


1205851=

Δ120579119877

120593120574119879119877

1205852=

Δ120579119878

(1 minus 120593) (1 minus 120574) 119879119878

(14)


= 119908119876 + 119862119877119876 + 120593119862

119906 (119875) [120583 minus 119878 (119876)]

+ 119881 [119876 minus 119878 (119876)]

(15)


119879119878= 119862119878119876 + (1 minus 120593)119862

119906 (119875) [120583 minus 119878 (119876)] (16)


1and 1205762[43]

1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

(17)

Among them

120585 =1

2

2

sum

119894=1

120585119894

120590 = radic1

2

2

sum

119894=1

(120585119894minus 120585)2

(18)


119894


2



10158401015840

119894 12058510158401015840119894

= 119870+ 1205851015840



set

1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2

(19)



supply chain is

119867 = minus1

ln119898

119898

sum

119894=1

120582119894ln 120582119894 (20)


119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822) (21)



10038161003816100381610038161003816100381610038161003816

120593119877

120593119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576 (22)


10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576 (23)





max 119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(24)





retailer)

Logisticsservice

integrator RP


service provider S1

service provider Si

SN

middotmiddotmiddot

(Q1 wi 120593i)

(Qi wi 120593i)

(QN wN 120593N)

(D t)







119894 120593119894) each





Decision Variables


119894





119894between


Other Parameters





119894






119894 FLSP 119878119894 119894 = 1 2 119873




119873

119894=1119878(119876119894)




119894


1205871015840


contract and 12058710158401015840


game situation


119877= sum119873

119894=1120587119877119894


119894 with 120587

1015840



and 12058710158401015840


119894under game

situation




119878119894and 119877


supply chain


119877 and 119878119868



of 119877 and 119878119894




119894




119894) = 119863

1198940+119896119905119894minus (12)119905

119894


119894

2



119894and







119878 (119876119894) = 119864 (min (119876

119894 119863 (119905119894)))

= int

infin

0

min (119876119894 119863 (119905119894)) 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 119863 (119905

119894)] 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(25)


119864119894(119876119894minus 119863 (119905

119894))+= 119876119894minus 119878 (119876

119894)

= int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(26)


119864119894(119863 (119905119894) minus 119876119894)+= 120583119894minus 119878 (119876

119894)

= 120583119894minus 119876119894+ int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(27)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(28)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(29)


120587119878119894

= (1 minus 120593119894) 119875119894119878 (119876119894) + 119908119894119876119894minus 119862119878119894119876119894


119894)]

+ 119880 [119876119894minus 119878 (119876

119894)]

(30)


120587119879119894

= 120587119877119894

+ 120587119878119894

= 119875119894119878 (119876119894) minus 119862119877119894119876119894minus 119862119878119894119876119894

minus 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

(31)




119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)


2lt 0


119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)


1015840





sum119873

119894=11198761015840

119894


1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)




119908119894

2

1198961198941205901198942119887119894

(34)


119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)


119894to 119878119894


119894are



120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)







1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)




119894= (1 minus

120593119894)120596119894


119894 1205781015840119894





119894= 1205781015840

119894+



119894sum119873

119894=112057810158401015840



=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)


119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1015840

119877indi-


10158401015840



1015840



119878


chain with 1205871015840



119879indi-






0+ 119896119905 minus (12)ℎ119905


1198630+ 119896119905 minus (12)ℎ119905

2


119877= 1198620+ 119911119905 and 119908 = 119908

0+ 119910119905 Since









119878 (119876) = 119864 (min (119876119863 (119905)))

= int

infin

0

min (119876119863 (119905)) 119891 (119905) 119889119905

= 119876 minus int

119876

0

[119876 minus 119863 (119905)] 119891 (119905) 119889119905

= 119876 minus int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(1)


119864 (119876 minus 119863 (119905))+= 119876 minus 119878 (119876)

= int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(2)



119864 (119863 (119905) minus 119876)+= 120583 minus 119878 (119876)

= 120583 minus 119876

+ int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(3)


120587119877


119906 (119875) [120583 minus 119878 (119876)]

minus 119880 [119876 minus 119878 (119876)]

(4)


119878119876


(5)


120587119879

= 120587119877+ 120587119878

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119906 (119875) [120583 minus 119878 (119876)]

(6)



120597119864 (120587119879)

120597119876= 0 (7)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878

119875 + 119862119906 (119875)

(8)


119864(1205871015840

119879)



0[119876minus119863

0minus

119896119905 + (12)ℎ1199052]119891(119905)119889119905 thus 119878(119876) = 119876 minus 119887


[(12)ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891(119876) and 120597119878(119876)120597119876 = 1 minus 119904


119864 (12058710158401015840


119877119876 minus 119862

119906 (119875) (120583 minus 119876 + 119887)

minus1199082

1198971205902119887

119864 (12058710158401015840

119878) = [119908 minus 119862

119878] 119876 +

1199082

1198971205902119887

(9)


120597119864 (12058710158401015840

119878)

120597119908= 119876 +

2119908

1198971205902119887 = 0 (10)

Then 119908 = minus11987611989712059022119887


120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


1199041199082

1198971205902= 0 (11)


10158401015840

119877)1205971198762lt 0





120593119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 120593) 119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

(12)





Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

(13)


1205851=

Δ120579119877

120593120574119879119877

1205852=

Δ120579119878

(1 minus 120593) (1 minus 120574) 119879119878

(14)


= 119908119876 + 119862119877119876 + 120593119862

119906 (119875) [120583 minus 119878 (119876)]

+ 119881 [119876 minus 119878 (119876)]

(15)


119879119878= 119862119878119876 + (1 minus 120593)119862

119906 (119875) [120583 minus 119878 (119876)] (16)


1and 1205762[43]

1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

(17)

Among them

120585 =1

2

2

sum

119894=1

120585119894

120590 = radic1

2

2

sum

119894=1

(120585119894minus 120585)2

(18)


119894


2



10158401015840

119894 12058510158401015840119894

= 119870+ 1205851015840



set

1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2

(19)



supply chain is

119867 = minus1

ln119898

119898

sum

119894=1

120582119894ln 120582119894 (20)


119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822) (21)



10038161003816100381610038161003816100381610038161003816

120593119877

120593119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576 (22)


10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576 (23)





max 119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(24)





retailer)

Logisticsservice

integrator RP


service provider S1

service provider Si

SN

middotmiddotmiddot

(Q1 wi 120593i)

(Qi wi 120593i)

(QN wN 120593N)

(D t)







119894 120593119894) each





Decision Variables


119894





119894between


Other Parameters





119894






119894 FLSP 119878119894 119894 = 1 2 119873




119873

119894=1119878(119876119894)




119894


1205871015840


contract and 12058710158401015840


game situation


119877= sum119873

119894=1120587119877119894


119894 with 120587

1015840



and 12058710158401015840


119894under game

situation




119878119894and 119877


supply chain


119877 and 119878119868



of 119877 and 119878119894




119894




119894) = 119863

1198940+119896119905119894minus (12)119905

119894


119894

2



119894and







119878 (119876119894) = 119864 (min (119876

119894 119863 (119905119894)))

= int

infin

0

min (119876119894 119863 (119905119894)) 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 119863 (119905

119894)] 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(25)


119864119894(119876119894minus 119863 (119905

119894))+= 119876119894minus 119878 (119876

119894)

= int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(26)


119864119894(119863 (119905119894) minus 119876119894)+= 120583119894minus 119878 (119876

119894)

= 120583119894minus 119876119894+ int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(27)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(28)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(29)


120587119878119894

= (1 minus 120593119894) 119875119894119878 (119876119894) + 119908119894119876119894minus 119862119878119894119876119894


119894)]

+ 119880 [119876119894minus 119878 (119876

119894)]

(30)


120587119879119894

= 120587119877119894

+ 120587119878119894

= 119875119894119878 (119876119894) minus 119862119877119894119876119894minus 119862119878119894119876119894

minus 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

(31)




119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)


2lt 0


119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)


1015840





sum119873

119894=11198761015840

119894


1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)




119908119894

2

1198961198941205901198942119887119894

(34)


119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)


119894to 119878119894


119894are



120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)







1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)




119894= (1 minus

120593119894)120596119894


119894 1205781015840119894





119894= 1205781015840

119894+



119894sum119873

119894=112057810158401015840



=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)


119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119864 (119863 (119905) minus 119876)+= 120583 minus 119878 (119876)

= 120583 minus 119876

+ int

119876

0

[119876 minus 1198630minus 119896119905 +

1

2ℎ1199052]119891 (119905) 119889119905

(3)


120587119877


119906 (119875) [120583 minus 119878 (119876)]

minus 119880 [119876 minus 119878 (119876)]

(4)


119878119876


(5)


120587119879

= 120587119877+ 120587119878

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119906 (119875) [120583 minus 119878 (119876)]

(6)



120597119864 (120587119879)

120597119876= 0 (7)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878

119875 + 119862119906 (119875)

(8)


119864(1205871015840

119879)



0[119876minus119863

0minus

119896119905 + (12)ℎ1199052]119891(119905)119889119905 thus 119878(119876) = 119876 minus 119887


[(12)ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891(119876) and 120597119878(119876)120597119876 = 1 minus 119904


119864 (12058710158401015840


119877119876 minus 119862

119906 (119875) (120583 minus 119876 + 119887)

minus1199082

1198971205902119887

119864 (12058710158401015840

119878) = [119908 minus 119862

119878] 119876 +

1199082

1198971205902119887

(9)


120597119864 (12058710158401015840

119878)

120597119908= 119876 +

2119908

1198971205902119887 = 0 (10)

Then 119908 = minus11987611989712059022119887


120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


1199041199082

1198971205902= 0 (11)


10158401015840

119877)1205971198762lt 0





120593119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 120593) 119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

(12)





Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

(13)


1205851=

Δ120579119877

120593120574119879119877

1205852=

Δ120579119878

(1 minus 120593) (1 minus 120574) 119879119878

(14)


= 119908119876 + 119862119877119876 + 120593119862

119906 (119875) [120583 minus 119878 (119876)]

+ 119881 [119876 minus 119878 (119876)]

(15)


119879119878= 119862119878119876 + (1 minus 120593)119862

119906 (119875) [120583 minus 119878 (119876)] (16)


1and 1205762[43]

1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

(17)

Among them

120585 =1

2

2

sum

119894=1

120585119894

120590 = radic1

2

2

sum

119894=1

(120585119894minus 120585)2

(18)


119894


2



10158401015840

119894 12058510158401015840119894

= 119870+ 1205851015840



set

1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2

(19)



supply chain is

119867 = minus1

ln119898

119898

sum

119894=1

120582119894ln 120582119894 (20)


119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822) (21)



10038161003816100381610038161003816100381610038161003816

120593119877

120593119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576 (22)


10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576 (23)





max 119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(24)





retailer)

Logisticsservice

integrator RP


service provider S1

service provider Si

SN

middotmiddotmiddot

(Q1 wi 120593i)

(Qi wi 120593i)

(QN wN 120593N)

(D t)







119894 120593119894) each





Decision Variables


119894





119894between


Other Parameters





119894






119894 FLSP 119878119894 119894 = 1 2 119873




119873

119894=1119878(119876119894)




119894


1205871015840


contract and 12058710158401015840


game situation


119877= sum119873

119894=1120587119877119894


119894 with 120587

1015840



and 12058710158401015840


119894under game

situation




119878119894and 119877


supply chain


119877 and 119878119868



of 119877 and 119878119894




119894




119894) = 119863

1198940+119896119905119894minus (12)119905

119894


119894

2



119894and







119878 (119876119894) = 119864 (min (119876

119894 119863 (119905119894)))

= int

infin

0

min (119876119894 119863 (119905119894)) 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 119863 (119905

119894)] 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(25)


119864119894(119876119894minus 119863 (119905

119894))+= 119876119894minus 119878 (119876

119894)

= int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(26)


119864119894(119863 (119905119894) minus 119876119894)+= 120583119894minus 119878 (119876

119894)

= 120583119894minus 119876119894+ int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(27)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(28)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(29)


120587119878119894

= (1 minus 120593119894) 119875119894119878 (119876119894) + 119908119894119876119894minus 119862119878119894119876119894


119894)]

+ 119880 [119876119894minus 119878 (119876

119894)]

(30)


120587119879119894

= 120587119877119894

+ 120587119878119894

= 119875119894119878 (119876119894) minus 119862119877119894119876119894minus 119862119878119894119876119894

minus 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

(31)




119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)


2lt 0


119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)


1015840





sum119873

119894=11198761015840

119894


1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)




119908119894

2

1198961198941205901198942119887119894

(34)


119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)


119894to 119878119894


119894are



120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)







1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)




119894= (1 minus

120593119894)120596119894


119894 1205781015840119894





119894= 1205781015840

119894+



119894sum119873

119894=112057810158401015840



=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)


119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

(13)


1205851=

Δ120579119877

120593120574119879119877

1205852=

Δ120579119878

(1 minus 120593) (1 minus 120574) 119879119878

(14)


= 119908119876 + 119862119877119876 + 120593119862

119906 (119875) [120583 minus 119878 (119876)]

+ 119881 [119876 minus 119878 (119876)]

(15)


119879119878= 119862119878119876 + (1 minus 120593)119862

119906 (119875) [120583 minus 119878 (119876)] (16)


1and 1205762[43]

1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

(17)

Among them

120585 =1

2

2

sum

119894=1

120585119894

120590 = radic1

2

2

sum

119894=1

(120585119894minus 120585)2

(18)


119894


2



10158401015840

119894 12058510158401015840119894

= 119870+ 1205851015840



set

1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2

(19)



supply chain is

119867 = minus1

ln119898

119898

sum

119894=1

120582119894ln 120582119894 (20)


119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822) (21)



10038161003816100381610038161003816100381610038161003816

120593119877

120593119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576 (22)


10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576 (23)





max 119867 = minus1

ln 2(1205821ln 1205821+ 1205822ln 1205822)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878)

119864 (1205871015840

119879)

10038161003816100381610038161003816100381610038161003816

120593

1 minus 120593minus

120574

1 minus 120574

10038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(24)





retailer)

Logisticsservice

integrator RP


service provider S1

service provider Si

SN

middotmiddotmiddot

(Q1 wi 120593i)

(Qi wi 120593i)

(QN wN 120593N)

(D t)







119894 120593119894) each





Decision Variables


119894





119894between


Other Parameters





119894






119894 FLSP 119878119894 119894 = 1 2 119873




119873

119894=1119878(119876119894)




119894


1205871015840


contract and 12058710158401015840


game situation


119877= sum119873

119894=1120587119877119894


119894 with 120587

1015840



and 12058710158401015840


119894under game

situation




119878119894and 119877


supply chain


119877 and 119878119868



of 119877 and 119878119894




119894




119894) = 119863

1198940+119896119905119894minus (12)119905

119894


119894

2



119894and







119878 (119876119894) = 119864 (min (119876

119894 119863 (119905119894)))

= int

infin

0

min (119876119894 119863 (119905119894)) 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 119863 (119905

119894)] 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(25)


119864119894(119876119894minus 119863 (119905

119894))+= 119876119894minus 119878 (119876

119894)

= int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(26)


119864119894(119863 (119905119894) minus 119876119894)+= 120583119894minus 119878 (119876

119894)

= 120583119894minus 119876119894+ int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(27)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(28)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(29)


120587119878119894

= (1 minus 120593119894) 119875119894119878 (119876119894) + 119908119894119876119894minus 119862119878119894119876119894


119894)]

+ 119880 [119876119894minus 119878 (119876

119894)]

(30)


120587119879119894

= 120587119877119894

+ 120587119878119894

= 119875119894119878 (119876119894) minus 119862119877119894119876119894minus 119862119878119894119876119894

minus 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

(31)




119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)


2lt 0


119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)


1015840





sum119873

119894=11198761015840

119894


1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)




119908119894

2

1198961198941205901198942119887119894

(34)


119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)


119894to 119878119894


119894are



120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)







1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)




119894= (1 minus

120593119894)120596119894


119894 1205781015840119894





119894= 1205781015840

119894+



119894sum119873

119894=112057810158401015840



=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)


119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













retailer)

Logisticsservice

integrator RP


service provider S1

service provider Si

SN

middotmiddotmiddot

(Q1 wi 120593i)

(Qi wi 120593i)

(QN wN 120593N)

(D t)







119894 120593119894) each





Decision Variables


119894





119894between


Other Parameters





119894






119894 FLSP 119878119894 119894 = 1 2 119873




119873

119894=1119878(119876119894)




119894


1205871015840


contract and 12058710158401015840


game situation


119877= sum119873

119894=1120587119877119894


119894 with 120587

1015840



and 12058710158401015840


119894under game

situation




119878119894and 119877


supply chain


119877 and 119878119868



of 119877 and 119878119894




119894




119894) = 119863

1198940+119896119905119894minus (12)119905

119894


119894

2



119894and







119878 (119876119894) = 119864 (min (119876

119894 119863 (119905119894)))

= int

infin

0

min (119876119894 119863 (119905119894)) 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 119863 (119905

119894)] 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(25)


119864119894(119876119894minus 119863 (119905

119894))+= 119876119894minus 119878 (119876

119894)

= int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(26)


119864119894(119863 (119905119894) minus 119876119894)+= 120583119894minus 119878 (119876

119894)

= 120583119894minus 119876119894+ int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(27)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(28)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(29)


120587119878119894

= (1 minus 120593119894) 119875119894119878 (119876119894) + 119908119894119876119894minus 119862119878119894119876119894


119894)]

+ 119880 [119876119894minus 119878 (119876

119894)]

(30)


120587119879119894

= 120587119877119894

+ 120587119878119894

= 119875119894119878 (119876119894) minus 119862119877119894119876119894minus 119862119878119894119876119894

minus 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

(31)




119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)


2lt 0


119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)


1015840





sum119873

119894=11198761015840

119894


1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)




119908119894

2

1198961198941205901198942119887119894

(34)


119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)


119894to 119878119894


119894are



120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)







1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)




119894= (1 minus

120593119894)120596119894


119894 1205781015840119894





119894= 1205781015840

119894+



119894sum119873

119894=112057810158401015840



=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)


119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119878119894and 119877


supply chain


119877 and 119878119868



of 119877 and 119878119894




119894




119894) = 119863

1198940+119896119905119894minus (12)119905

119894


119894

2



119894and







119878 (119876119894) = 119864 (min (119876

119894 119863 (119905119894)))

= int

infin

0

min (119876119894 119863 (119905119894)) 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 119863 (119905

119894)] 119891 (119905

119894) 119889119905

= 119876119894minus int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(25)


119864119894(119876119894minus 119863 (119905

119894))+= 119876119894minus 119878 (119876

119894)

= int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(26)


119864119894(119863 (119905119894) minus 119876119894)+= 120583119894minus 119878 (119876

119894)

= 120583119894minus 119876119894+ int

119876119894

0

[119876119894minus 1198630119894

minus 119896119905119894+

1

2ℎ119905119894

2]119891 (119905119894) 119889119905

(27)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(28)


120587119877119894

= 120593119894119875119894119878 (119876119894) minus 119908119894119876119894minus 119862119877119894119876119894

minus 120593119894119862119906(119875119894) [120583119894minus 119878 (119876

119894)] minus 119880 [119876

119894minus 119878 (119876

119894)]

(29)


120587119878119894

= (1 minus 120593119894) 119875119894119878 (119876119894) + 119908119894119876119894minus 119862119878119894119876119894


119894)]

+ 119880 [119876119894minus 119878 (119876

119894)]

(30)


120587119879119894

= 120587119877119894

+ 120587119878119894

= 119875119894119878 (119876119894) minus 119862119877119894119876119894minus 119862119878119894119876119894

minus 119862119906(119875119894) [120583119894minus 119878 (119876

119894)]

(31)




119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)


2lt 0


119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)


1015840





sum119873

119894=11198761015840

119894


1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)




119908119894

2

1198961198941205901198942119887119894

(34)


119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)


119894to 119878119894


119894are



120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)







1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)




119894= (1 minus

120593119894)120596119894


119894 1205781015840119894





119894= 1205781015840

119894+



119894sum119873

119894=112057810158401015840



=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)


119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894 it must be

120597119864 (120587119879119894)

120597119876119894

= 0 (32)


2lt 0


119865 (1198761015840

119894) + [

1

2ℎ11987610158402

119894+ (1 minus 119896)119876

1015840

119894minus 1198630]119891 (119876

1015840

119894)

=119875119894minus 119862119877119894

minus 119862119878119894

+ 119862119906(119875119894)

119875119894+ 119862119906(119875119894)

= 1 minus119862119877119894

+ 119862119878119894

119875 + 119862119906(119875119894)

(33)


1015840





sum119873

119894=11198761015840

119894


1015840

119879) = sum

119873

119894=1119864(1205871015840

119879119894)




119908119894

2

1198961198941205901198942119887119894

(34)


119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894 (35)


119894to 119878119894


119894are



120593119894119864 (1205871015840

119879119894) ge 119864 (120587

10158401015840

119877119894)

(1 minus 120593119894) 119864 (120587

1015840

119879119894) ge 119864 (120587

10158401015840

119878119894)

119894 = 1 2 119873

dArr

119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

119894 = 1 2 119873

(36)







1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (37)




119894= (1 minus

120593119894)120596119894


119894 1205781015840119894





119894= 1205781015840

119894+



119894sum119873

119894=112057810158401015840



=

minus(1 ln119873)(sum119873

119894=1120588119894ln 120588119894)


119867 =

119873

sum

119894=1

119867119894= minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894)

+1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

(38)



retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











retailer)

Logistics service

integrator R


S


I





0 set 119867 gt 119867



max 119867 = minus

119873

sum

119894=1

[1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) +

1

ln119873(

119873

sum

119894=1

120588119894ln 120588119894)]

st119864 (12058710158401015840

119877119894)

119864 (1205871015840

119879119894)

le 120593119894le 1 minus

119864 (12058710158401015840

119878119894)

119864 (1205871015840

119879119894)

10038161003816100381610038161003816100381610038161003816

120593119894

1 minus 120593119894

minus120574119894

1 minus 120574119894

10038161003816100381610038161003816100381610038161003816

le 120576119894

119867 ge 1198670

(39)


119894(119894 = 1 2 119873) Under the












Other Parameters







1198771












chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















chain with 1205871198791


1198792indi-


1198801= 119908119877

211989711205902


1198802= 119908119878

211989721205902








120587119877

= 1198901119875119878 (119876) minus 119908

119877119876 minus 119862

119877119876 minus 1198901119862119906 (119875) [120583 minus 119878 (119876)]

minus 1198801 [119876 minus 119878 (119876)]

(40)


120587119878= 1198902[(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] minus 119862

119878119876 minus 119908

119878119876

minus 1198902(1 minus 120593

2) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198801 [119876 minus 119878 (119876)] minus 119880

2 [119876 minus 119878 (119876)]

(41)


120587119868= (1 minus 119890

2) [(1 minus 119890

1) 119875119878 (119876) + 119908

119877119876] + 119908

119877119876 minus 119862

119868119876


1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198802 [119876 minus 119878 (119876)]

(42)


120587119879

= 120587119877+ 120587119878+ 120587119868

= 119875119878 (119876) minus 119862119877119876 minus 119862

119878119876 minus 119862

119868119876

minus 119862119906 (119875) [120583 minus 119878 (119876)]

(43)


120597119864 (120587119879)

120597119876= 0 (44)



119865 (119876) + [1

2ℎ1198762+ (1 minus 119896)119876 minus 119863

0]119891 (119876)

=119875 minus 119862

119877minus 119862119878minus 119862119868+ 119862119906 (119875)

119875 + 119862119906 (119875)

= 1 minus119862119877+ 119862119878+ 119862119868

119875 + 119862119906 (119875)

(45)


1198791


119864 (1205871198772


119877119876


119908119877

2

11989711205902

119887

(46)


119864 (1205871198782) = [119908

119877minus 119862119878minus 119862119868] 119876 +

119908119877

2

11989711205902

119887 minus119908119878

2

11989721205902

119887 (47)


119864 (1205871198682) = [119908

119878minus 119862119868] 119876 +

119908119878

2

11989721205902

119887 (48)






1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198901119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119877)

(1 minus 1198901) 1198902119864 (1205871015840

119879) ge 119864 (120587

10158401015840

119878)

(1 minus 1198901) (1 minus 119890

2) 119864 (120587

1015840

119879) ge 119864 (120587

10158401015840

119868)

dArr

119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

(49)




119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (50)



10038161003816100381610038161003816100381610038161003816

119890119877

119890119878

minus120574119877

120574119878

10038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816

119890119878

119890119868

minus120574119878

120574119868

10038161003816100381610038161003816100381610038161003816

le 120576

(51)


100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

(52)





max 119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823)

st119864 (12058710158401015840

119877)

119864 (1205871015840

119879)


119864 (12058710158401015840

119878) + 119864 (120587

10158401015840

119868)

119864 (1205871015840

119879)

119864 (12058710158401015840

119878)

(1 minus 1198901) 119864 (120587

1015840

119879)


119864 (12058710158401015840

119868)

(1 minus 1198901) 119864 (120587

1015840

119879)

100381610038161003816100381610038161003816100381610038161003816

1198901

(1 minus 1198901) 1198902

minus1205741015840

(1 minus 1205741015840) 12057410158401015840

100381610038161003816100381610038161003816100381610038161003816

le 120576

10038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 1198901) 1198902

(1 minus 1198901) (1 minus 119890

2)minus

(1 minus 1205741015840) 12057410158401015840

(1 minus 1205741015840) (1 minus 12057410158401015840)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 120576

119867 ge 1198670

(53)






6 + 18119905 minus 051199052 119862119877

= 06 + 025119905 119908 = 04 + 05119905 119875 = 15119862119906(119875) = 4 119862

119878= 04 119897 = 2 120583 = 10 1205902 = 8 120574 = 06 and

1198670




057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










057305750577057905810583058505870589

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593

120593lowast


0506070809

111

H

Hlowast Hlowast

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t





lowast= 05874







0560565

0570575

0580585

0590595

06

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03

t

120593


040506070809

111

H

P = 154

P = 15

P = 146

1 20

04

05

06

07

08

09

02

11

12

13

14

15

16

17

18

19

01

03t





1) =

6 + 181199051minus 05119905

1

2 1198621198771

= 06 + 13 lowast 1199051 1199081= 04 + 13 lowast 119905

1

1198751= 15 119862

119906(1198751) = 4 119862

1198781= 04 119897

1= 2 120583

1= 10 120590

1

2= 8 and


2follows


sim 119880(0 8) 119863(1199052) =

6 + 191199052minus 05119905

2

2 1198621198772

= 06 + 14 lowast 1199052 1199082= 04 + 13 lowast 119905

2

1198752= 15 119862

119906(1198752) = 38 119862

1198782= 04 119897

2= 2 120583

2= 101 120590

2

2= 81

and 1205742= 065



3) =

6+ 1851199053minus025119905

3

2 1198621198773

= 055 + 13lowast 11990531199083= 04 + 13lowast 119905

3

1198753= 15 119862

119906(1198753) = 41 119862

1198783= 04 119897

3= 2 120583

3= 98 120590

3

2= 8

and 1205743= 065





119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119872 1205931

1205932

1205933

11986711

11986712

11986713

119867

0 05796 05886 05682 09999 09999 09999 0998802 05805 05903 05720 09999 09999 09999 0998804 05814 05919 05757 09999 09999 09999 0998806 05823 05935 05794 09999 09999 09999 0998708 05831 05951 05830 09999 09999 09999 099871 05840 05967 05866 09999 09999 09999 0998612 05849 05983 05900 09999 09998 09999 0998514 05857 05990 05922 09999 09997 09997 0998416 05865 05989 05917 09999 09987 09972 0997918 05872 05987 05912 09999 09970 09923 099682 05871 05985 05907 10000 09946 09854 0995322 05870 05983 05902 09999 09917 09766 0993424 05868 05981 05897 09995 09881 09663 0991026 05866 05979 05891 09985 09840 09546 0988228 05865 05978 05886 09977 09795 09417 098513 05864 05976 05881 09966 09745 09276 0981832 05862 05974 05876 09953 09691 09127 0978234 05862 05972 05870 09946 09632 08970 0974536 05860 05970 05865 09931 09571 08806 0970538 05859 05968 05860 09914 09506 08635 096634 05858 05966 05854 09895 09438 08460 0962042 05856 05965 05849 09875 09368 08281 0957544 05855 05963 05844 09853 09295 08098 0952846 05854 05961 05838 09830 09219 07912 0948148 05852 05959 05833 09806 09142 07724 094335 05852 05957 05827 09793 09043 07533 09382


1= 01 119905

2= 02 and 119905

3= 04 respectively



11 11986712 and 119867

13


1

1205932 and 120593




11= 119867lowast

12= 119867lowast




1= 05872 120593lowast

2=


3= 05922

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

H11

H12

H13

075

080

085

090

095

100

Hi


05650570057505800585059005950600

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

1205932

1205931 1205933

120593

120593lowast2

120593lowast3

120593lowast1


093094095096097098099100

H

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M






P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










P = 154

P = 15

P = 146

43 51 200

2

44

42

38

22

24

26

28

36

32

34

46

48

04

08

18

16

06

14

12

M

1205931

0560565

0570575

0580585

0590595

06



14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

1205932

0570575

0580585

0590595

060605

061


different prices

1205933

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M

P = 154

P = 15

P = 146

054

055

056

057

058

059

06

061


different prices



092093094095096097098099

1101

H

P = 154

P = 15

P = 146

4321 50

14

16

18

08

22

24

26

28

06

32

34

36

38

04

42

44

46

48

02

12

M


21 30

05

06

07

08

09

03

11

12

13

14

15

16

17

18

19

02

21

22

23

24

25

26

27

28

29

01

04

t

e 1

048049

05051052053054055

elowast1





81 + 18119905 minus 051199052 119862119877

= 05 + 025119905 119908119877

= 03 + 13 lowast 119905 119908119878=

03+025lowast119905119875 = 15119862119906(119875) = 39119862

119878= 04119862

119868= 03 119897

1= 35

1198972= 32 120583 = 13 1205902 = 6 119867

0= 06 1205741015840 = 06 and 120574

10158401015840= 055







lowast



lowast



01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










01011012013014015016017018019

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 2


06065

07075

08085

09095

1105

H

Hlowast

21 30

06

04

02

16

18

12

22

24

26

28

14

08

t


P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

e 1

046047048049

05051052053054055056


different prices








e 2

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t

01012014016018

02022024026028


different prices

06507

07508

08509

0951

105

H

P = 154

P = 15

P = 146

21 30

06

12

14

16

18

04

22

24

26

28

02

08

t





























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Appendices



119894in





119908119894

2

1198961198941205901198942119887119894

119864 (120587119878119894) = [119908

119894minus 119862119878119894] 119876119894+

119908119894

2

1198971198941205901198942119887119894

(A1)

Among them 119887119894= int119876119894

0[119876119894minus 1198630119894

minus 119896119905119894+ (12)ℎ119905

119894

2]119891(119905119894)119889119905

and 119904119894= 120597119887119894120597119876119894

So 119904119894= 119865(119876

119894) + [(12)ℎ119876

119894

2+ (1 minus 119896)119876

119894minus 1198630119894]119891(119876119894) and

120597119878(119876119894)120597119876119894= 1 minus 119904

119894


120597119864 (120587119878119894)

120597119908119894

= 119876119894+

2119908119894

1198971198941205901198942119887119894= 0 (A2)


22119887119894


120597119864 (120587119877119894)

120597119876119894

= [119875119894+ 119862119906(119875119894)] (1 minus 119904

119894) minus 119862119877119894

minus 119908119894

minus119904119894119908119894

2

1198971198941205901198942

= 0

(A3)


2lt 0


10158401015840

119894and 119908

10158401015840

119894


119877119894) and that of

119878119894is 119864(120587

10158401015840

119878119894)


119873

119894=111987610158401015840

119894


119877) =

sum119873

119894=1119864(12058710158401015840


119864(12058710158401015840

119878) = sum119873

119894=1119864(12058710158401015840

119878119894)



120585119877119894

=Δ120579119877119894

120593119894120574119894119879119877119894

120585119878119894

=Δ120579119878119894

(1 minus 120593119894) (1 minus 120574

119894) 119879119878119894

(B1)

Among them 119879119877119894and 119879



ized transformation

1205851015840

119877119894=

(120585119877119894

minus 120585119894)

120590119894

1205851015840

119878119894=

(120585119878119894

minus 120585119894)

120590119894

(B2)




119877119894= 1198701+ 1205851015840

119877119894 12058510158401015840119878119894

= 1198701+ 1205851015840

119878119894 1198701is the range


= 12058510158401015840

119877119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894) and 120582

119878119894= 12058510158401015840

119878119894(12058510158401015840

119877119894+ 12058510158401015840

119878119894)

Obtain 120582119877119894and 120582



=

minus(1 ln119898)sum119898

119894=1120582119894ln 120582119894 For 0 le 119867



1198671119894

= minus1

ln 2(120582119877119894ln 120582119877119894

+ 120582119878119894ln 120582119878119894) (B3)



120597119864 (12058710158401015840

119868)

1205971199082

= 119876 +21199082

11989721205902

119887 = 0 (C1)

Then 119908119878= minus119876119897

212059022119887


120597119864 (12058710158401015840

119878)

120597119908119877

= 119876 +2119908119877

11989711205902

119887 = 0 (C2)

Then 119908119877

= minus119876119897112059022119887



120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119864 (12058710158401015840

119877)

120597119876= [119875 + 119862


119904119908119877

2

11989711205902

= 0

(C3)


10158401015840

119877)1205971198762lt 0



119877and 119908


10158401015840

119877

and 11990810158401015840




2 the


Δ120579119877

=

119864 (1205871015840

119877) minus 119864 (120587

10158401015840

119877)

119864 (12058710158401015840

119877)

Δ120579119878=

119864 (1205871015840

119878) minus 119864 (120587

10158401015840

119878)

119864 (12058710158401015840

119878)

Δ120579119868=

119864 (1205871015840

119868) minus 119864 (120587

10158401015840

119868)

119864 (12058710158401015840

119868)

(D1)







1205851=

Δ120579119877

11989011205741119879119877

1205852=

Δ120579119878

(1 minus 1198901) (1 minus 120574

1) 11989021205742119879119878

1205853=

Δ120579119868

(1 minus 1198901) (1 minus 120574

1) (1 minus 119890

2) (1 minus 120574

2) 119879119868

(D2)


119879119877

= 119908119877119876 + 119862

119877119876 + 1198901119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198811 [119876 minus 119878 (119876)]

(D3)


119878119876 + 1198902(1 minus 120593

1) 119862119906 (119875) [120583 minus 119878 (119876)]

+ 1198812 [119876 minus 119878 (119876)]

(D4)


2) (1 minus 119890

1) 119862119906 (119875) [120583 minus 119878 (119876)] (D5)


1205851015840

1=

(1205851minus 120585)

120590

1205851015840

2=

(1205852minus 120585)

120590

1205851015840

3=

(1205853minus 120585)

120590

(D6)

Among them

120585 =1

3

3

sum

119894=1

120585119894

120590 = radic1

3

3

sum

119894=1

(120585119894minus 120585)2

(D7)




1205821=

12058510158401015840

1

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205822=

12058510158401015840

2

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

1205823=

12058510158401015840

3

12058510158401015840

1+ 12058510158401015840

2+ 12058510158401015840

3

(D8)



supply chain is

119867 = minus1

ln 3(1205821ln 1205821+ 1205822ln 1205822+ 1205823ln 1205823) (D9)



Acknowledgment



References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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