The Impact of RFID Investment on Complex Product in Three-Level Assembly Supply Chain

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 271496 11 pageshttpdxdoiorg1011552013271496

Research ArticleThe Impact of RFID Investment on Complex Product inThree-Level Assembly Supply Chain

Wei Xu1 Zhaotong Lian1 and Xifan Yao2

1 Faculty of Business Administration University of Macau Macau2 School of Mechanical and Automotive Engineering South China University of Technology Guangzhou China

Correspondence should be addressed to Zhaotong Lian lianztumacmo

Received 15 March 2013 Accepted 20 May 2013

Academic Editor Qingsong Xu

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

Motivated by the complex product with the feature about error-prone assembly system and supply chain inventory inaccuracythis paper elaborates on the impact of information technology investment on complex product by establishing a three-stagesupply chain model involving two suppliers one manufacturer and retailer which carried out Stackelberg games In additionit not only compares the manufacturer and the retailerrsquos optimal decision and maximum profit under the situation of theinformation asymmetry and free information sharing but also analyzes their market behavior and changes inmarket performanceMeanwhile it points out that the downstream in supply chain masters more information about market demands compared tothe upstream one The optimal cost threshold values of technology investment are also examined both for the centralized andthe decentralized scenarios utilizing quantitative and modeling methods By analyzing and comparing the optimal profit with orwithout investment on information technology it establishes a supply chain coordination model which boosts the applicationof information technology At the same time it offers the conditions on which the upstream and downstream enterprises cancoordinate with one another The results of this paper have contributed significantly to making the price and ordering decisionson whether RFID should be adopted among members of the supply chain Finally we present numerical analyses and severalextensions of the model are considered as well

1 Introduction

Represented typically by autoindustry complex productassembly process is characterized by typical discrete manu-facturing It involves numerous components and parts mis-cellaneous technology and high-accuracy requirement Theassembly system has the characteristics of mixed productionHowever these complex product assembly operations aremainly completed manually with error-prone steps Due tothe limits of technical investment the range of assemblymonitoring is only restricted to the executive process itselfand it neglects logical relationship between the assembly taskand logistics task so that the assembly and logistics cannot bemanaged uniformly thus causing the disconnection betweenmaterial flow and information flow Since it is easy to getwrong and timeconsuming the information technology is

required to ensure that the architectural decisions will beimplemented correctly

To solve the problems above the model of assemblysystem involving the assembly agent nodes and logistics agentnodesmust be established so as to realize uniformmonitoringand management Additionally RFID can identify itemsconveniently by virtue of the characteristics of noncontactremote distance and read-write so as to improve the realtime of assembly monitoring and promote synchronizationof material flow and information flow What is more thechanges in assembly states and logistics states can be reflectedthrough generation destruction and update of RFID tagIn that case if assembly agent and logistics agent can bemodeled uniformly to establish themapping relation betweenRFID and assembly executive state the feasibility of assemblyexecutive process can be enhanced

2 Mathematical Problems in Engineering

Obviously the wide application of information technol-ogy will free people from the heavy labour Some large-scale retail enterprises such as Wal-Mart and METRO haveobtained obvious effects by trying out RFID technology inthe aspects of clothes and daily necessities However thehigh cost of RFID equipment limits its wider popularizationand application to more supply chains and enterprisesTherefore how to coordinate the upstream and downstreamfirms in the assembly system to use the RFID technologyto improve the performance of the supply chain is of greatsignificance

According to the report of American Shipper the averageoccurrence rate of demand exceeding supply and out-of-stock situations is 8 in global retail business Surprisinglyfor a 8000m2 retail store with the annual sales volume ofRMB 150 million it will lose RMB 84 million sales volumeonly due to the stockout per year The replenishment speedis 3 times faster than that of goods without the tags AfterWal-Mart adopts electronic product code (EPC) supportedby RFID technology to monitor goods the stockout ratereduces 16 For the goods tagged with RFID labels manualordering reduces by 10 RFID can decrease the quantity ofsafety inventory effectively and make quick response to thestockout [1] After BestBuy adopts RFID goods availabilityhas been added to 93 from 80 At present a famousBritish retailer TESCO introduces a device called smart shelfwhich can record each move of all products on the shelvesThis is also one of applications of RFID technology At thesame time RFID antennae are installed at the entrance andexit of the warehouse which are connected with warehousemanagement system Thus when the goods enter and leavethe warehouse electronic label information can be readpromptly Meanwhile the most accurate goods data can begained easily Additionally the accurate information aboutthe inventory quantity can be obtained as well The studyof American organs concerned shows that the accuracyrate of the inventory can reach 995 for the retail enter-prises applying information technology to inventory man-agement while other enterprises can only reach 80ndash85[2 3]

With the help of information technology enterpriseinformatization can be improved meanwhile industrial in-formation sharing can be promoted Real-time dynamic indi-cators in the supply chain such as inventory analysis marketand customer analysis and other data need monitoring byRFID sensor and are transmitted to the server throughinternet network Thus data information from differentchannels is gathered through cloud calculating data center ofthe internet of things so as to draw information strategiesigniting the profit point

Although complex product can be traced more accu-rately in real time with RFID technology one of the majorobstacles to adoption is the cost of RFID implementationRFID technology incurs huge cost including not only theinstallation investment but the costs of tags and readers aswell Motivated by the RFID investment issue we aim toexplore the prospects by mainly focusing on the optimalinvestment strategies in order to maximize the profits under

conditions with or without the perishable products valuetracing when the demand is stochastic in a two-level supplychain involving a manufacturer and a retailer We considerboth the centralized and decentralized scenario

Particularly the main research questions that this paperaddresses are as follows

(i) What are the levels of the optimal investment thatmake RFID adoption economically feasible in three-level centralized and decentralized assembly supplychains

(ii) How to combine the actual supply chain problemsuch as the shelf shrinkage errors in assemblyprocess and misplaced products to implement thetechnical innovation investment to be able to realizegreater investment returns

The remainder of this paper is organized as follows In thefollowing section we give a quick idea of the topics coveredin the literature review In addition literatures related tothis study will be reviewed In Section 3 we introduce thesystem architecture and benefit evaluation mainly involvingassembly manufacturing system based on multi-agent andRFIDThemodel analysis and themain results of this researchare presented in detail in Sections 4 to 6 The numericalanalysis is examined in Section 7 and we offer a summarytogether with a critique of the model and a direction forfuture research in the final section

2 Literature Review

RFID is a term that has been around for several years Inrecent years the analysis of RFID technology investment onvarious industries has attracted the attention of more andmore scholars As an exploratory research in essence morespecially the investment analysis based on mathematicalmodels is extremely scarce This section mainly builds onthree streams of literature the potential benefits and mostcommon application of RFID technology RFID investmenton supply chain with contracts and two- or three-levelsupply chain coordination as well as the development ofan RFID and multi-agent system based on manufacturingcontrol for industrial applications and dynamic logisticsprocess

The transaction and transportation errors misplacementdelay delivery and shrinkage errors are the main causes ofinventory inaccuracy all of which have been discussed asareas where RFID can increase accuracyThere are numerouspapers in the literature that have considered the advantagesof RFID technology The benefits under different categoriesare presented as follows Lee and Ozer [4] study a single-item periodic-review inventory system with replenishmentpolicy(sS) They integrate RFID technology with uncertaindemands and random distribution of transaction errors andobserve the inventory cost related to transaction errorsAtali et al [5] extend the abovementioned models theydevelop two models based on shrinkage such as theftsand damages misplacement and transaction or scanningerrors They consider that RFID can effectively reduce

Mathematical Problems in Engineering 3

Software agentarchitecture

Portable devicefor inventory

checking

Interactivesystem

Wirelesscommunication

RFID tagsreader

antenna

Informationextraction agent

Error detectingagent

Goods schedulingagent

Informationintegration agent

Stockshelfallocation

Real-timereplenishment

Marketinganalysis

Customerbehaviour

Phys

ical

laye

rA

gent

laye

rAp

plic

atio

nla

yer

Assembly systemarchitecture

Subprocess (1)

Subprocess (2)

Process (2)

Process (1)

Agent node

Agent nodeAgent node Process (i)

Three-levelsupply chain

RFID tag RFID tag RFID tag

Suppliers Manufacturer Retailer Customer

RFID

+ ag

ent

Figure 1 Architecture of the information technology in complex product assembly system

the mistake rate and improve inventory visibility Tajima [6]first mentions that the shrinkage errors include employeetheft shoplifting administration and paperwork errors ven-dor fraud and unavailable products for sale Besides RFIDhas been discussed as a solution to black market saleswhich could also improve inventory records by reducinghuman errors in material handling (as discussed elsewhere[7 8]) Ngai et al [9] analyze a case study on mobilecommerce system based on RFID technology with strongfunction on locating tracking and managing the containersGaukler et al [10] focus on the investment cost of RFIDtechnology and assume that the RFID tag cost can beshared by all supply chain members Besides they arguethat RFID can improve stock control policies as well asinventory replenishment policies They extend the previousmodel to analyze an item-level RFID application in two-level supply chain They propose two cases of centralizedand decentralized decision with or without RFID and mainlydiscuss how the manufacturer and the retailer can optimizetheir ownprofitswithout cooperationHeese [11] indicate thatRFID technology is more beneficial in decentralized supplychains Although RFID cannot eliminate all errors it canbe quickly detected and effectively handled Several authorssuch as Dutta et al [12] Rekik et al [13] Szmerekovsky andZhang [14] Bottani and Rizzi [15] Tu et al [16] Wang etal [17] and Whitaker et al [18] were attracted by its useof eliminating these errors They all pay close attention to

the impact of RFID on inventory inaccuracy due to variety oferrors

3 Overview of System Architecture andBenefit Evaluation

In this section we mainly discuss the system platformarchitecture based on multi-agent and RFID Combiningthese two new technologies together it has become one ofthe potential candidates for all assembly manufacturing andsupply chain systems in the future

As is shown in Figure 1 all the assembly transition andlogistics distribution are guided and controlled respectivelythrough multi-agent and it is like the human brain whichnot only improves agility of manufacturing system but alsomakesmanufacturing systemcharacterized by heterogeneousand geographically distributed In addition in order toimprove the efficiency of raw material identification thematerial can be identified with RFID tag When the partsset needed by some assemblies are taken out from thewarehouse or some assembly tasks are completed RFIDtags are immediately adopted to identify the parts set ornew assembly When the materials are transported to thedestination according to the design route the content of tagshould be updated The construction and update of RFIDtag are unified with events in assembly executive process


Supplier 1Q1

120572

Supplier 2Q2

120572Qi

qm

qm

Manufacturermin(120572Qi q)

Retailermin(120572120573Qi q D)

D(x)

120572120573120573

Customer

Figure 2 The schematic diagram of system model

The assembly executive state can be reflected through tagstate In that case all transitions can be regarded as mutual-lycoordinated andmutuallyindependent equal entity throughcertain protocol with certain limits of authority Besides eachentity has certain structural patterns and completes differentwork driven by respective local data

The control system may be realized through a multi-agent system mainly including assembly agent and logisticsagent Figure 1 shows functional models of the two typesof agent It can be seen that assembly agent firstly readsRFID tag information gives assembly task promptly thenreads task scheduling and assembly technology respectivelyfrom assembly database and conducts guide and processinspection control of assembly operation After the assem-bly is finished task information in the database shouldbe updated Then the logistics agent identifies RFID andobtains material information from assembly database andthen guides the transport process By using distributeddatabase to realize distributed ontology we can ensure thedistributivity of the system the real-time updating anddata consistency of the system As mentioned before theautomotive assembly management is based on multi-agentand RFID which includes application layer agent layer andphysical layThe bottommodule is mainly responsible for theprocessing of various businesses in the whole system and thetransmission of various data into upper-layer managementsystem after saving them in the database besides the datacollection and processing mainly aim to collect various datain field production The intermediate layer is responsiblefor monitoring the adaptive configuration of field collectionnodes (intelligent terminal) to realize the collection andfiltration of RFIDdata tomonitor collection event of primarylayer and information event of upper-layer system and tocommand parsing and sending the data into business logiclayer

4 Notations and Model Scenarios

This paper targets the assembly system and supply chaininvolving two suppliers one manufacturer and one retailerThe schematic diagram of system model is shown as illus-trated Figure 2The theoretical framework of assembly supplychain is established on a hypothesis that information issymmetric and complete we also assume the partners arecompletely rational and themarket is completely competitiveThe following summary of notations shown at the end of thepaper are used in this paper

In order to explore the description and notation of thissystem there are two different scenarios for our model thecentralized scenario and the decentralized one Under thefirst one it is assumed that a central decision maker deter-mineswhether or not to invest in information technology andthe order quantity to maximize the total profit In the otherscenario there is no cooperation or information sharingamong the suppliers the manufacturer and the retailer Eachmember tries to pursue his own maximum profits

Under the above-discussed structure there are two subsi-tuations whether the information technology station is in theassembly supply chain or not Once the information systemis adopted it will contribute to optimizing the goods storedin warehouse and ensure inventory accuracy and safety

Therefore we can consider using real-time monitoringtechnology which is based on multi-agent and RFID to helpenterprises monitor inventory accurately and avoid losses ofenterprises However within the shrinkage of the error-proneassembly system and supply chain inventory inaccuracy thecost from technology investment has risen up dramaticallyIn this paper we only consider the variable cost resultedfrom the application of themulti-agent and RFID technologythat is the cost of multi-agent public software platform andRFID tag 119888

119905 When the retailer employs the technology that is

120573 = 1 the complex production assembly system of upstreamsuppliers and the manufacturer will not be improved bythese so that the maximum effect of the technology in thesupply chain could not be exerted When the suppliers adoptmulti-agent and RFID that is 120572 = 1 the downstreamretailers can use the RFID labels tagged on commodities bythe manufacturer which is beneficial to the retailer from this(120573 = 1) At this time the suppliers or the manufacturerundertake the technology investment alone however theretailer can free-ride all theseThus the supplierrsquos enthusiasmfor implementing multi-agent and RFID is totally weakenedTo make the upstream have a growing interest in investingin information technology reduce the losses resulted frominaccurate inventory and increase the performance of thesupply chain the downstream in supply chain can encouragethe suppliers and themanufacturer with some compensationsand reduce the impacts of the free-ride effect Based on thisthis paper will also study the supply chain coordination whenmulti-agent and RFID technology are invested among thethree members

5 Optimal Strategy in Centralized Scenario

In order to achieve the highest performance of the wholeassembly supply chain all the members make their owndecision together Therefore it is necessary to analyze theconditions on which the optimal technology investment isimplemented in centralized scenario

51 Optimal Decision in Centralized Supply Chain withoutMulti-Agent and RFID Technology

511 Two Suppliers with the Same Yields When the orderquantity is 119876 and the actual inventory which can be used to


0 2 4 6 8 1070

80

90

100

110

120

130

140

150

160

170

ct

Qclowast sc

3

(a)

0 2 4 6 8 10200

400

600

800

1000

1200

1400

1600

ct

Πclowast sc

3

(b)

Figure 3 Variation of 119876119888lowast

sc3 and Π119888lowast

sc3 with 119888119905

satisfy consumersrsquo demands by the normal sales links is 120572120573119876the expected sales of whole supply chain are 119864min(119863 120572120573119876)meanwhile the surplus is 119864(120572120573119876 minus 119863)+ Then we can easilyget the expected profit of the whole supply chain

Π119888

sc1 = 119901119898min (119863 120572120573119876) + V[120572120573119876 minus 119863]+ minus 2119888119876 (1)

Using the following equalities

119864min (119863 120572120573119876) = 120572120573119876 minus int120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

119864(120572120573119876 minus 119863)+

= 120572120573119876 minus 119864min (119909 120572120573119876)

= int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

(2)

the expected profit function associated with the same yieldsof two suppliers is given by

Π119888

sc1 = 119901119898 int120572

120572

int

120573

120573

[120572120573119876 minus int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572

+ 119904int

120572

120572

int

120573

120573

[int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572 minus 2119888119876

(3)

Such an assumption which is explored by several authorssuch as Heese [11] enables us to assume that the customerdemand is subject to uniformly distribution Therefore infact the above equation can be simplified further to

Π119888

sc1 = 119901119898120583120572120583120573119876 minus119901119898minus 119904119898

4120583119909

sdot (1205902

120572+ 1205832

120572)

times (1205902

120573+ 1205832

120573)1198762minus 2119888119876

(4)

Next we calculate the second derivative about 119876 andknow that 1205972Π119888sc1120597119876

2lt 0 therefore Π119888sc1 is a concave

function about 119876 In the particular case of a uniformlydistributed demand the optimal order quantity and optimalexpected profit of the whole supply chain can be written asfollows

119876119888lowast

sc1 =2120583119909(119901119898120583120572120583120573minus 119888)

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

Π119888lowast

sc1 =(119901119898120583120572120583120573minus 119888)2

120583119909

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

(5)

The Figures 3(a) and 3(b) represent the variation of 119876119888lowastsc3and Π119888lowastsc3 with RIFD investment cost 119888

119905respectively

512 Two Suppliers with Different Yields There are someproblems within the current operational process which makethe internal production not well synchronized so the twosuppliers with different yields are also discussed as follows

Π119888

sc2 = 119901119898 sdotmin (1205721205731198761 120572120573119876

2 119863)

minus 119904119898[119863 minusmin (120572120573119876

1 120572120573119876

2)]+

+ V[min (1205721205731198761 120572120573119876

2) minus 119863]

+

minus 119888 (1198761+ 1198762)

(6)


For the different yield problem of two suppliers in orderto derive the optimal order decision we must distinguish the

different positions of the distributions of 119863(119909) and 120572120573119876119894to

get the expression of the expected profit Then the aboveequation for each configuration is shown in

Π119888

sc2=

Π119888

sc2 = 1199011198981205721205731198761 minus 119904119898 (119863 minus 1205721205731198761) minus 119888 (1198761 + 1198762) 1205721205731198761 lt 1205721205731198762 lt 119863 or 1205721205731198761lt 119863 lt 120572120573119876

2

Π119888


1

Π119888

sc2 = 119901119898119863 + V (1205721205731198762 minus 119863) minus 119888 (1198761 + 1198762) 119863 lt 1205721205731198762lt 120572120573119876

1

Π119888


2

(7)

To refresh the equation set we obtain the optimal expect-ed profit of the whole supply chain

Π119888lowast

sc2 = (119901119898 + 119904119898 minus V) sdotmin (1205721205731198761 120572120573119876

2 119863)

minus 119904119898119863 + V sdotmin (120572120573119876

1 120572120573119876

2) minus 119888 (119876

1+ 1198762)

(8)

52 Optimal Decision in Centralized Supply Chain withMulti-Agent and RFID Technology Next we investigate the central-ized systemwith technology input in detailThe RFID systemcan identify items conveniently by virtue of the characteristicsof noncontact remote distance and read-write so as toimprove the real time of assembly field monitoring andpromote synchronization of logistic flow and informationflow What is more multi-agent system including agentassembly nodes and agent logistics nodes can be modeleduniformly to establish the mapping relation between RFIDtag and assembly executive state Moreover the feasibilityof assembly executive process can be enhanced so as to getreal time and accuracy of assembly manufacturer processmonitoring Under such circumstances all synchronizationprogress status and any errors in assembly supply chainprocess can be reflected through the updated information ofRFID tag

Above all as multi-agent and RFID system provides anaccurate information of the current inventory it contributesto the monitoring and management of the products inrealtime Meanwhile there is also the technology investmentcost 119888

119905 Therefore the unit purchasing cost is 119888 + 119888

119905 The

excepted profits are as follows

Π119888

sc3 = 119901119898min (119863119876) + 119904119898[119876 minus 119863]

+minus 2 (119888 + 119888

119905) 119876 (9)

where 119864min(119909 119876) = 119876 minus int1198760(119876 minus 119909)119891(119909)119889119909 and [119909 minus 119876]+ =

119876 minus 119864min(119909 119876) = int1198760(119876 minus 119909)119891(119909)119889119909 Since 1205972Π119888sc3120597119876

2=

minus119901119898minus 1199041198982120583119909lt 0 the preceding equation is a concave

function about 119876 that is it has the only optimal solution119876lowast which achieves the maximum profit of the whole supply

chain

DifferentiatingΠ119888sc3 with respect to119876 the optimal order-ing quantity and profit of the whole supply chain are givenby

119876119888lowast

sc3 = 2120583119909119901119898minus 119888 minus 119888

119905

119901119898minus 119904119898

Π119888lowast

sc3 =(119901119898minus 119888 minus 119905)

2

119901119898minus 119904119898

120583119909

(10)

53 The Advantages of Deployment with Technology Invest-ment In this section we focus on seeking answers fromthe following questions how to solve such issues as theassembly manufacture errors and shelf shrinkage by meansof multi-agent and RFID system more effectively What arethe levels of optimal investment that enjoy the technologyeconomically feasible in centralized supply chain

Since all members take the overall interests of the wholesupply chain into account they pursue the global profitmaximization and carry on the technology innovation withinformation sharing in the centralized scenario As we allknow if they do not introduce technologies they will sufferthe impacts of assembly errors and inaccurate inventoryConversely they implement it but undertake the technicalcost

To analyze the profit and loss with technology invest-ments the benefit function ΔΠ119888 is introduced Clearly whenΔΠ119888= 0 there exists a break-even point 1198881015840

119905which makes

technical investment always profitable In other words theincreased profits generated by innovative technology arealways larger than the investment cost ofmulti-agent softwareand RFID tags In detail one essential condition is to satisfythe inequality ΔΠ119888 gt 0

Proposition 1 The threshold value of technology investmentcost is 1198881015840

119905= 119901119898minus 119888 minus ((119901

119898120583120573120583120572minus 119888)radic(120590

2

120573+ 1205832

120573)(1205902120572+ 1205832120572))

When 119888119905lt 1198881015840

119905 the optimal solution is to make investment for

situation in the centralized scenario when 119888119905gt 1198881015840

119905 without

any investment on innovative technology being beneficial forthe whole supply chain

As can be seen from the above expression we find that119888119905has relationship not only with 120583

120572and 120583

120573 but also with the

selling price of manufacturer as well It further proves thatthe higher the selling price 119901

119898is the higher the technology


0 02 04 06 08 10

20

40

60

80

100

120

120572

120590120572 = 015

120590120572 = 025120590120572 = 035

c t

Figure 4 Variation of 119888119905with 120572 for different values of 120590

120572

cost that can be accepted by the supply chain system will beMeanwhile for a given 120583

120572 ΔΠ can be greater or less than

0 depending on the value of 120583119909 It appears that the benefit

achieved by the deployment of the multi-agent and RFIDsystem is comparable to the profit without any technologyinvestment Figure 4 from the analysis above represents thevariation of 119888

119905with 120572 for different values of 120590

120572 The more the

customer demand and the less effective the assembly systemis (that is the error rate 120572 is higher) the more importantthe technology investment will be for the decision whetherto adopt the information technology and the members inassembly supply chain can be benefited from the orderingquantity if we implement it Additionally the comparison ofthe Figure 5 reveals the variation of Δ119876 and ΔΠ with 120572 fordifferent demand

6 Optimal Strategy in Decentralized Scenario

In this section we mainly evaluate and analyze the impactof technology investment among three supply chain partnersOwning to independence relation they pursue themaximumprofits respectively Consequently the main issue will befocused on which partner takes the priority to employmulti-agent and RFID system and bear the higher costsThen two approaches are presented Similar to the previoussection the optimal benefits with or without adoption oftechnology by numerical solutions under the decentraliza-tion are also proposed In addition the retailer places anorder decision in terms of the wholesale price charged bythe manufacturer which book from the supplier and theoptimal order quantity has been the focus of description andexplanation

61 Optimal Decision in Decentralized Supply Chain withoutMulti-Agent and RFID Technology

611 The Basic Model of Two Suppliers According to theabove analysis the profit function of supplier 119894 that normallydepended on 119876

119894in decentralized scenario is given by

Π119863

119904119894

= 119908119894sdotmin (120572119876

119894 119902) minus 119904

119894[119902 minus 120572119876

119894]+

minus

2

sum

119894=1

119888119876119894 (11)

To make analysis and computation more convenientthe holding cost incurred by the supplier 119894 with less actualquantity delivered can be omitted Then we assume 119896

119894(119909) equiv

int119909

0120572119894119889119867119894(120572119894) according to the following formulas deduced

119864 [min (120572119894119876119894 119902)] = int

119902119876119894

0

120572119894119876119894ℎ119894(120572119894) 119889120572119894

+ int

1

119902119876119894

119902ℎ119894(120572119894) 119889120572119894

= 119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)

119864 [(119902 minus 120572119894119876119894)+

] = int

119902119876119894

0

(119902 minus 120572119894119876119894) ℎ119894(120572119894) 119889120572119894

= 119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)

(12)

We can easily get the expected profit of supplier

119864 [Π119863

119904119894

] = 119908119894[119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)]

minus 119904119894[119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)]

minus

2

sum

119894=1

119888119876119894

(13)

The first partial derivative ofΠ119863119904119894

with respect to119876119894can be

written as

120597Π119863

119904119894

120597119876119894

= 119908119894[119896119894(119902

119876119894

) + 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)]

minus 119904119894[119902ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

) minus 119896119894(119902

119876119894

)] minus 119888

= (119908119894+ 119904119894) 119896119894(119902

119876119894

) minus 119888

(14)


0

5000

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔΠ

=Π

c sc3minusΠ

c sc1

minus15000

minus10000

minus5000

(a)

minus500

0

500

1000

1500

2000

2500

3000

3500

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔQ

=Qc sc

3minusQc sc

1

(b)

Figure 5 Variation of Δ119876 and ΔΠ with 120572 for different values of 120583119909

Developing 120597Π119863119904119894

120597119876119894= 0 leads to the following expres-

sion meanwhile 1205972Π119863119904119894

1205971198762

119894= minus(119908

119894+ 119904119894)(11990221198763

119894) lt 0

1198911(1198761 1198762) = 119888 minus (119908

1+ 1199041) 1198961(119902

1198761

)

1198912(1198761 1198762) = 119888 minus (119908

2+ 1199042) 1198962(119902

1198762

)

(15)

For such setting we always have (1205971198911(1198761 1198762)1205971198762) minus

(1205971198912(1198761 1198762)1205971198762) = minus(119908

2+ 1199042)(11990221198763

2ℎ2)(119902119876

2) lt 0 for any

1198761and 119876

2 Thus as can be seen from the analysis above we

have the following important observation

Proposition 2 For any given order quantity from the man-ufacturer the sufficient conditions guaranteeing the existenceand uniqueness of the Nash equilibrium solution for thesuppliers under decentralized scenario are [1198761015840

1

lowast

1198761015840

2

lowast

]

612 The Best Response of The Manufacturer In the nextstage for Stackelberg leader-follower game we analyze themanufacturerrsquos best response

Π119863

1198981= int

119863119902

120572

[120572119902119908 minus 119904119898(119909 minus 120572119902)] ℎ (120572) 119889120572

+ int

120572

119863119902

119902119908ℎ (120572) 119889120572 minus 119888119902

(16)

where the first item represents that the manufacturer cannotsatisfy the retailerrsquos orders that is the manufacturerrsquos salesrevenue when 119863 gt 120572119902 The second item stands for the man-ufacturerrsquos sales revenue when 119863 lt 120572119902 From the equationabove we can get 1205972Π119863

11989811205971199022= minus(119908 + 119904

119898)(1199091199023ℎ)(120572) lt 0

which found that the manufacturerrsquos profit function Π1198631198981

isa concave function about the order quantity 119902 that is thereis only one optimal order quantity 119902lowast which makes themanufacturerrsquos profit be the maximum

int

119909119902lowast

120572

120572ℎ (120572) 119889120572 =119888

119908 + 119904119898

(17)

The optimal profit for manufacturer is given by

Π119863lowast

1198981= 119908119909 minus (119908 + 119904

119898) 119902119867(

119909

119902lowast) (18)

613 The Best Policy of the Retailer The retailerrsquos expectedprofit can be written as follow

Π119863

1199031= 119901119898int

120573

120573

119864min (119909 120573119902) 119892 (120573) 119889120573

+ 119904int

120573

120573

119864(120573119902 minus 119909)+

119892 (120573) 119889119909 119889120573 minus 119908119902

(19)

where 119864min(119909 120573119902) = 120573119902 minus int1205731199020(120573119902 minus 119909)119891(119909)119889119909 and 119864(120573119902 minus

119909)+= 120573119902 minus 119864min(119909 120573119902) = int120573119902

0(120573119902 minus 119909)119891(119909)119889119909 then we

simplify above equation

Π119863

1199031= minus119901119898minus 119904119898

4120583119909

(1205902

120573+ 1205832

120573) 1199022+ 119901119898120583120573119902 minus 119908119902 (20)


The optimal ordering quantity and profit of the retailerare respectively

119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)

62 Optimal Decision in Decentralized Supply Chain withMulti-Agent and RFID Technology In order to achieve thehighest performance of the supply chain it is necessaryto analyze the optimal decision in decentralized supplychain with multi-agent and RFID technology As we allknown if the retailer invests in information technologythe upstream ones cannot benefit from this otherwise theopposite Obviously from the perspective of the supply chainit will be better for the whole supply chain with the upstreamtomake an RampD expenditureTherefore here we assume thatthe technology investment can be actively supported by thesuppliers and manufacturer as a whole

The retailer can benefit from the free-ride RFID adoptionand also 120573 = 1 Thus the retailerrsquos expected profit function isas follow

Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)

Since 1205972Π11987721198631205971199022 = minus(119901119898minus 119904119898)119891(119902) lt 0 the profit of

retailer is a concave function about 119902 Then we solve the firstderivative and get the corresponding optimal order quantityand total profit of the retailer

119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583

120573lt 1 it is easy to prove that

Π119863lowast

1199031lt Π119863lowast

1199032is tenable In another word after implementing

the information technology the retailer can remotelymonitorthe inventory and track its position in real time directlywhich avoids the impacts of the worse service inefficienttransportation out of stock or overstock and other inac-curate inventory problems so that the retailer needs notundertake cost for this Thus the retailerrsquos profit increasesafter the upstream has deployed the information technologyThe Figure 6 respectively shows the impact of changes with120583119909and 119888119905on the variation of Π119863

119903and ΔΠ We can treat the

suppliers and the manufacturer as the same decision-makingsubject in this section

Proposition 3 If 119888119905lt 119888dagger

119905 the manufacturerrsquos profit increases

after investing information technology so that he is willing toimplement the technology If 119888

119905gt 119888dagger

119905 the increased cost resulted

from the suppliersrsquo application of the multi-agent and RFIDsystem is larger than the obtained income so the suppliersand manufacturer have no motivation to invest technologyand is unwilling to do where 119888dagger

119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)

It is concluded that 119888dagger119905is the highest technology cost

which can be accepted by the upstream firm in decentralizedassembly supply chain At this time the income that thetechnology brings to the suppliers or the manufacturer justequals the newly increased cost resulted from the applicationof the technology When the investment cost is higher than119888dagger

119905 the suppliers or manufacturer will not choose to employ

the technology actively On the contrary if this cost is lowerthan 119888dagger

119905 the income that the supplier or manufacturer obtains

by information technology is higher than the technologyinvestment so that they will all have enormous enthusiasmin investing it

According to the above propositions in the situationof 1198881015840119905gt 119888119905gt 119888dagger

119905 although the suppliersrsquo application of

information technology can increase the profit of the supplychain his profit will reduce Furthermore the upstream firmhas nomotivation to usemulti-agent andRFID system so thatthewhole supply chain cannot profit from the technology andthe maximum profit of the supply chain cannot be realizedThus it is necessary to study the supply chain coordinationin the situation of 1198881015840

119905lt 119888119905lt 119888dagger

119905to make the supplier

have enthusiasm to adopt this technology and reduce theinaccuracy of the supply chain inventory as well as increasethe performance of the supply chain and its members

7 Numerical Analysis

Finally a numerical example is taken to validate the ratio-nality and feasibility of the theoretics and method Whenthe customer demand follows uniform distribution pattern119909 sim 119880(0 200) we insert the following constant values intothe corresponding optimal expressions 119901

119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78

the upstream ones are unwilling to use RFID andmulti-agenttechnology because the profit will be reduced after adoptingit

Furthermore according to Figure 7 we can find thatthe increased profit of the manufacturer and the retaileris a linear function about 120573 The increased profit of themanufacturer monotonously reduces as the shelf inventoryerror rate increases while the increased profit of the retailermonotonously increases as the coefficient 120573 increases Whenthe factor of the inventory error 0 lt 120573 lt 04 themanufacturerrsquos profit is beneficial from reasonable utilizationof information technology at this time the upstream firmis willing to make a investment If 120573 gt 02 the retailerrsquosprofit increases after he has deployed RFID and multi-agent technology Therefore choosing the factor of the shelfinventory error rate at the interval of 02 lt 120573 lt 04 can


0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)

Figure 6 Variation of Π119863119903with 120583

119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1

Figure 7 Variation of ΔΠ119903and ΔΠ

119898with 120573 for different values of

119888119905

ensure that the profit increases to make both the upstreamand downstream firm achieve a win-win situation

Moreover the suppliers aim at encouraging the retailer toorder more goods but the retailer directs at freely riding theinnovative technology from the upstream ones Thus theycan consider the coordination model based on the revenuesharing That is the suppliers and manufacturer undertakethe technology investment cost alone and give price subsidyto the retailer on the condition that the wholesale price isnot changed Or the retailer gives parts of sales revenue tothe upstream ones at the end of a sales season in order to

stimulate and encourage the suppliers to increase the RampDinvestment

Being restrained by personal reason the members of thesupply chain will first consider their own benefit and will takethe maximization of the benefit of the whole supply chaininto consideration on the premise that their own benefit issatisfied Thus to ensure that the members are willing toaccept the revenue sharing contract it is necessary to guaran-tee that the profit obtained by both parties under this contractwith the supply chain that has adopted this technology isnot less than the one that has been obtained in decentralizedscenario before the contract is made In other words it isworth considering to ensure that the members of the supplychain can achieve win-win situation with the restraint of thecontract In other words the coordination mechanism is toensure the following two inequalities tenable ΔΠ

119903gt 0 and

ΔΠ119898gt 0

8 Concluding Remarks

The information technology has been heralded as a majorbreakthrough to enhance the efficiency of assembly sup-ply chain This paper mainly aims to explore the impactof information technology on supply chain managementespecially the prospects of multi-agent and RFID invest-ment value Meanwhile the possibility and reliability of themodel are examined through the qualitative and quantitativeanalyse of the multi-agent and RFID system investmentAdditionally supply chain partnership is an important factorof performance in supply chain operations It is significantto strengthen the supply chain cooperation managementFinally this paper mainly aims at a perfectly competitivemarket and carries out the research in the situation that themarket demands are uniformly distributed Future researches


should focus on whether the revenue sharing contract cancoordinate the supply chain when the market demands arenormal distribution As is apparent from the above descrip-tion the research of this paper has important theoreticalvalue and great practical significance

Notations

119909 The random variable representing thecustomerrsquos demand 119863

119901119898 The manufacturerrsquos selling price per item

119902119898 The ordering quantity of manufacturer

119904119898 The unit production shortage cost of

manufacturerV The salvage cost per item119876119894 The supplierrsquos production quantity 119894 = 1 2

119888 The unit production cost of supplier 119894119908119894 The wholesale price of supplier 119894

120572 The errors in assembly system (process)ℎ(119867) Pdf(cdf) characterizing 120572120573 The shelf inventory error rate (caused by

misplaced theft miscategory etc)119892(119866) Pdf(cdf) characterizing 120573119888119905 The technology investment cost (involving

multi-agent software and RFID tag cost)Π119888

sc The expected profit of whole supply chainin centralized scenario

Π119863

119904119898119903 The expected profit of suppliermanufacturer and retailer indecentralizedscenario

Acknowledgments

This research was supported in part by the National NaturalScience Foundation of China under Grant no 51175187 theNational High-Tech RampD Program of China under Grantno 2007AA04Z111 and the Grant of the University of MacauMYRG163 (Y1-L3) FBA11-LZT

References

[1] S Tzeng W Chen and F Pai ldquoEvaluating the business value ofRFID evidence from five case studiesrdquo International Journal ofProduction Economics vol 112 no 2 pp 601ndash613 2008

[2] A Ustundag and M Tanyas ldquoThe impacts of radio frequencyidentification (RFID) technology on supply chain costsrdquo Trans-portation Research E vol 45 no 1 pp 29ndash38 2009

[3] U Bagchi A Guiffrida L OrsquoNeill A Zeng and J Hayya ldquoTheeffect of RFIDon inventorymanagement and controlrdquo inTrendsin Supply Chain Design and Management pp 71ndash92 2007

[4] H Lee and O Ozer ldquoUnlocking the value of RFIDrdquo Productionand Operations Management vol 16 no 1 pp 40ndash64 2007

[5] A Atali H Lee and O Ozer ldquoIf the inventory managerknew value of visibility and RFID under imperfect inventoryinformationrdquo Social Science Research Network 2009

[6] M Tajima ldquoStrategic value of RFID in supply chain manage-mentrdquo Journal of Purchasing and Supply Management vol 13no 4 pp 261ndash273 2007

[7] E W T Ngai K K L Moon F J Riggins and C Y YildquoRFID research an academic literature review (1995ndash2005) andfuture research directionsrdquo International Journal of ProductionEconomics vol 112 no 2 pp 510ndash520 2008

[8] M M Hossain and V R Prybutok ldquoConsumer acceptance ofRFID technology an exploratory studyrdquo IEEE Transactions onEngineering Management vol 55 no 2 pp 316ndash328 2008

[9] E W T Ngai T C E Cheng S Au and K Lai ldquoMobilecommerce integrated with RFID technology in a containerdepotrdquo Decision Support Systems vol 43 no 1 pp 62ndash76 2007

[10] G M Gaukler R W Seifert and W H Hausman ldquoItem-levelRFID in the retail supply chainrdquo Production and OperationsManagement vol 16 no 1 pp 65ndash76 2007

[11] H S Heese ldquoInventory record inaccuracy double marginaliza-tion and RFID adoptionrdquo Production and Operations Manage-ment vol 16 no 5 pp 542ndash553 2007

[12] A Dutta H L Lee and S Whang ldquoRFID and operationsmanagement technology value and incentivesrdquo Productionand Operations Management vol 16 no 5 pp 646ndash655 2007

[13] Y Rekik E Sahin and Y Dallery ldquoAnalysis of the impact of theRFID technology on reducing product misplacement errors atretail storesrdquo International Journal of Production Economics vol112 no 1 pp 264ndash278 2008

[14] J G Szmerekovsky and J Zhang ldquoCoordination and adoptionof item-level RFID with vendor managed inventoryrdquo Interna-tional Journal of Production Economics vol 114 no 1 pp 388ndash398 2008

[15] E Bottani and A Rizzi ldquoEconomical assessment of the impactof RFID technology and EPC system on the fast-moving con-sumer goods supply chainrdquo International Journal of ProductionEconomics vol 112 no 2 pp 548ndash569 2008

[16] Y TuWZhou and S Piramuthu ldquoIdentifyingRFID-embeddedobjects in pervasive healthcare applicationsrdquo Decision SupportSystems vol 46 no 2 pp 586ndash593 2009

[17] S Wang S Liu andWWang ldquoThe simulated impact of RFID-enabled supply chain on pull-based inventory replenishmentin TFT-LCD industryrdquo International Journal of ProductionEconomics vol 112 no 2 pp 570ndash586 2008

[18] J Whitaker S Mithas and M S Krishnan ldquoA field study ofRFID deployment and return expectationsrdquo Production andOperations Management vol 16 no 5 pp 599ndash612 2007

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

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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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


Obviously the wide application of information technol-ogy will free people from the heavy labour Some large-scale retail enterprises such as Wal-Mart and METRO haveobtained obvious effects by trying out RFID technology inthe aspects of clothes and daily necessities However thehigh cost of RFID equipment limits its wider popularizationand application to more supply chains and enterprisesTherefore how to coordinate the upstream and downstreamfirms in the assembly system to use the RFID technologyto improve the performance of the supply chain is of greatsignificance

According to the report of American Shipper the averageoccurrence rate of demand exceeding supply and out-of-stock situations is 8 in global retail business Surprisinglyfor a 8000m2 retail store with the annual sales volume ofRMB 150 million it will lose RMB 84 million sales volumeonly due to the stockout per year The replenishment speedis 3 times faster than that of goods without the tags AfterWal-Mart adopts electronic product code (EPC) supportedby RFID technology to monitor goods the stockout ratereduces 16 For the goods tagged with RFID labels manualordering reduces by 10 RFID can decrease the quantity ofsafety inventory effectively and make quick response to thestockout [1] After BestBuy adopts RFID goods availabilityhas been added to 93 from 80 At present a famousBritish retailer TESCO introduces a device called smart shelfwhich can record each move of all products on the shelvesThis is also one of applications of RFID technology At thesame time RFID antennae are installed at the entrance andexit of the warehouse which are connected with warehousemanagement system Thus when the goods enter and leavethe warehouse electronic label information can be readpromptly Meanwhile the most accurate goods data can begained easily Additionally the accurate information aboutthe inventory quantity can be obtained as well The studyof American organs concerned shows that the accuracyrate of the inventory can reach 995 for the retail enter-prises applying information technology to inventory man-agement while other enterprises can only reach 80ndash85[2 3]

With the help of information technology enterpriseinformatization can be improved meanwhile industrial in-formation sharing can be promoted Real-time dynamic indi-cators in the supply chain such as inventory analysis marketand customer analysis and other data need monitoring byRFID sensor and are transmitted to the server throughinternet network Thus data information from differentchannels is gathered through cloud calculating data center ofthe internet of things so as to draw information strategiesigniting the profit point

Although complex product can be traced more accu-rately in real time with RFID technology one of the majorobstacles to adoption is the cost of RFID implementationRFID technology incurs huge cost including not only theinstallation investment but the costs of tags and readers aswell Motivated by the RFID investment issue we aim toexplore the prospects by mainly focusing on the optimalinvestment strategies in order to maximize the profits under

conditions with or without the perishable products valuetracing when the demand is stochastic in a two-level supplychain involving a manufacturer and a retailer We considerboth the centralized and decentralized scenario

Particularly the main research questions that this paperaddresses are as follows

(i) What are the levels of the optimal investment thatmake RFID adoption economically feasible in three-level centralized and decentralized assembly supplychains

(ii) How to combine the actual supply chain problemsuch as the shelf shrinkage errors in assemblyprocess and misplaced products to implement thetechnical innovation investment to be able to realizegreater investment returns

The remainder of this paper is organized as follows In thefollowing section we give a quick idea of the topics coveredin the literature review In addition literatures related tothis study will be reviewed In Section 3 we introduce thesystem architecture and benefit evaluation mainly involvingassembly manufacturing system based on multi-agent andRFIDThemodel analysis and themain results of this researchare presented in detail in Sections 4 to 6 The numericalanalysis is examined in Section 7 and we offer a summarytogether with a critique of the model and a direction forfuture research in the final section

2 Literature Review

RFID is a term that has been around for several years Inrecent years the analysis of RFID technology investment onvarious industries has attracted the attention of more andmore scholars As an exploratory research in essence morespecially the investment analysis based on mathematicalmodels is extremely scarce This section mainly builds onthree streams of literature the potential benefits and mostcommon application of RFID technology RFID investmenton supply chain with contracts and two- or three-levelsupply chain coordination as well as the development ofan RFID and multi-agent system based on manufacturingcontrol for industrial applications and dynamic logisticsprocess

The transaction and transportation errors misplacementdelay delivery and shrinkage errors are the main causes ofinventory inaccuracy all of which have been discussed asareas where RFID can increase accuracyThere are numerouspapers in the literature that have considered the advantagesof RFID technology The benefits under different categoriesare presented as follows Lee and Ozer [4] study a single-item periodic-review inventory system with replenishmentpolicy(sS) They integrate RFID technology with uncertaindemands and random distribution of transaction errors andobserve the inventory cost related to transaction errorsAtali et al [5] extend the abovementioned models theydevelop two models based on shrinkage such as theftsand damages misplacement and transaction or scanningerrors They consider that RFID can effectively reduce




checking

Interactivesystem


RFID tagsreader

antenna







Marketinganalysis

Customerbehaviour

Phys

ical

laye

rA

gent

laye

rAp

plic

atio

nla

yer


Subprocess (1)

Subprocess (2)

Process (2)

Process (1)

Agent node





RFID

+ ag

ent








Supplier 1Q1

120572

Supplier 2Q2

120572Qi

qm

qm



D(x)

120572120573120573

Customer
















0 2 4 6 8 1070

80

90

100

110

120

130

140

150

160

170

ct

Qclowast sc

3

(a)

0 2 4 6 8 10200

400

600

800

1000

1200

1400

1600

ct

Πclowast sc

3

(b)



sc3 with 119888119905


Π119888



119864min (119863 120572120573119876) = 120572120573119876 minus int120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

119864(120572120573119876 minus 119863)+

= 120572120573119876 minus 119864min (119909 120572120573119876)

= int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

(2)


Π119888

sc1 = 119901119898 int120572

120572

int

120573

120573

[120572120573119876 minus int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572

+ 119904int

120572

120572

int

120573

120573

[int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572 minus 2119888119876

(3)


Π119888

sc1 = 119901119898120583120572120583120573119876 minus119901119898minus 119904119898

4120583119909

sdot (1205902

120572+ 1205832

120572)

times (1205902

120573+ 1205832

120573)1198762minus 2119888119876

(4)




119876119888lowast

sc1 =2120583119909(119901119898120583120572120583120573minus 119888)

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

Π119888lowast

sc1 =(119901119898120583120572120583120573minus 119888)2

120583119909

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

(5)


119905respectively


Π119888

sc2 = 119901119898 sdotmin (1205721205731198761 120572120573119876

2 119863)

minus 119904119898[119863 minusmin (120572120573119876

1 120572120573119876

2)]+

+ V[min (1205721205731198761 120572120573119876

2) minus 119863]

+

minus 119888 (1198761+ 1198762)

(6)





Π119888

sc2=

Π119888


2

Π119888


1

Π119888


1

Π119888


2

(7)


Π119888lowast


2 119863)

minus 119904119898119863 + V sdotmin (120572120573119876

1 120572120573119876

2) minus 119888 (119876

1+ 1198762)

(8)




119905 The


Π119888

sc3 = 119901119898min (119863119876) + 119904119898[119876 minus 119863]

+minus 2 (119888 + 119888

119905) 119876 (9)



2=



chain


119876119888lowast

sc3 = 2120583119909119901119898minus 119888 minus 119888

119905

119901119898minus 119904119898

Π119888lowast

sc3 =(119901119898minus 119888 minus 119905)

2

119901119898minus 119904119898

120583119909

(10)




119905which makes



119905= 119901119898minus 119888 minus ((119901

119898120583120573120583120572minus 119888)radic(120590

2

120573+ 1205832

120573)(1205902120572+ 1205832120572))

When 119888119905lt 1198881015840



119905 without



120572and 120583





0 02 04 06 08 10

20

40

60

80

100

120

120572

120590120572 = 015

120590120572 = 025120590120572 = 035

c t


120572






120572 The more the







Π119863

119904119894

= 119908119894sdotmin (120572119876

119894 119902) minus 119904

119894[119902 minus 120572119876

119894]+

minus

2

sum

119894=1

119888119876119894 (11)


119894(119909) equiv

int119909


119864 [min (120572119894119876119894 119902)] = int

119902119876119894

0

120572119894119876119894ℎ119894(120572119894) 119889120572119894

+ int

1

119902119876119894

119902ℎ119894(120572119894) 119889120572119894

= 119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)

119864 [(119902 minus 120572119894119876119894)+

] = int

119902119876119894

0

(119902 minus 120572119894119876119894) ℎ119894(120572119894) 119889120572119894

= 119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)

(12)


119864 [Π119863

119904119894

] = 119908119894[119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)]

minus 119904119894[119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)]

minus

2

sum

119894=1

119888119876119894

(13)



written as

120597Π119863

119904119894

120597119876119894

= 119908119894[119896119894(119902

119876119894

) + 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)]

minus 119904119894[119902ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

) minus 119896119894(119902

119876119894

)] minus 119888

= (119908119894+ 119904119894) 119896119894(119902

119876119894

) minus 119888

(14)


0

5000

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔΠ

=Π

c sc3minusΠ

c sc1

minus15000

minus10000

minus5000

(a)

minus500

0

500

1000

1500

2000

2500

3000

3500

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔQ

=Qc sc

3minusQc sc

1

(b)


Developing 120597Π119863119904119894


sion meanwhile 1205972Π119863119904119894

1205971198762

119894= minus(119908

119894+ 119904119894)(11990221198763

119894) lt 0

1198911(1198761 1198762) = 119888 minus (119908

1+ 1199041) 1198961(119902

1198761

)

1198912(1198761 1198762) = 119888 minus (119908

2+ 1199042) 1198962(119902

1198762

)

(15)


(1205971198912(1198761 1198762)1205971198762) = minus(119908

2+ 1199042)(11990221198763

2ℎ2)(119902119876

2) lt 0 for any

1198761and 119876




1

lowast

1198761015840

2

lowast

]


Π119863

1198981= int

119863119902

120572

[120572119902119908 minus 119904119898(119909 minus 120572119902)] ℎ (120572) 119889120572

+ int

120572

119863119902

119902119908ℎ (120572) 119889120572 minus 119888119902

(16)


11989811205971199022= minus(119908 + 119904

119898)(1199091199023ℎ)(120572) lt 0



int

119909119902lowast

120572

120572ℎ (120572) 119889120572 =119888

119908 + 119904119898

(17)


Π119863lowast

1198981= 119908119909 minus (119908 + 119904

119898) 119902119867(

119909

119902lowast) (18)


Π119863

1199031= 119901119898int

120573

120573

119864min (119909 120573119902) 119892 (120573) 119889120573

+ 119904int

120573

120573

119864(120573119902 minus 119909)+

119892 (120573) 119889119909 119889120573 minus 119908119902

(19)


119909)+= 120573119902 minus 119864min(119909 120573119902) = int120573119902

0(120573119902 minus 119909)119891(119909)119889119909 then we


Π119863

1199031= minus119901119898minus 119904119898

4120583119909

(1205902

120573+ 1205832

120573) 1199022+ 119901119898120583120573119902 minus 119908119902 (20)



119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)



Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)



119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583


Π119863lowast

1199031lt Π119863lowast








119905gt 119888dagger



119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)










119905lt 119888119905lt 119888dagger





119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78




0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












checking

Interactivesystem


RFID tagsreader

antenna







Marketinganalysis

Customerbehaviour

Phys

ical

laye

rA

gent

laye

rAp

plic

atio

nla

yer


Subprocess (1)

Subprocess (2)

Process (2)

Process (1)

Agent node





RFID

+ ag

ent








Supplier 1Q1

120572

Supplier 2Q2

120572Qi

qm

qm



D(x)

120572120573120573

Customer
















0 2 4 6 8 1070

80

90

100

110

120

130

140

150

160

170

ct

Qclowast sc

3

(a)

0 2 4 6 8 10200

400

600

800

1000

1200

1400

1600

ct

Πclowast sc

3

(b)



sc3 with 119888119905


Π119888



119864min (119863 120572120573119876) = 120572120573119876 minus int120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

119864(120572120573119876 minus 119863)+

= 120572120573119876 minus 119864min (119909 120572120573119876)

= int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

(2)


Π119888

sc1 = 119901119898 int120572

120572

int

120573

120573

[120572120573119876 minus int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572

+ 119904int

120572

120572

int

120573

120573

[int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572 minus 2119888119876

(3)


Π119888

sc1 = 119901119898120583120572120583120573119876 minus119901119898minus 119904119898

4120583119909

sdot (1205902

120572+ 1205832

120572)

times (1205902

120573+ 1205832

120573)1198762minus 2119888119876

(4)




119876119888lowast

sc1 =2120583119909(119901119898120583120572120583120573minus 119888)

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

Π119888lowast

sc1 =(119901119898120583120572120583120573minus 119888)2

120583119909

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

(5)


119905respectively


Π119888

sc2 = 119901119898 sdotmin (1205721205731198761 120572120573119876

2 119863)

minus 119904119898[119863 minusmin (120572120573119876

1 120572120573119876

2)]+

+ V[min (1205721205731198761 120572120573119876

2) minus 119863]

+

minus 119888 (1198761+ 1198762)

(6)





Π119888

sc2=

Π119888


2

Π119888


1

Π119888


1

Π119888


2

(7)


Π119888lowast


2 119863)

minus 119904119898119863 + V sdotmin (120572120573119876

1 120572120573119876

2) minus 119888 (119876

1+ 1198762)

(8)




119905 The


Π119888

sc3 = 119901119898min (119863119876) + 119904119898[119876 minus 119863]

+minus 2 (119888 + 119888

119905) 119876 (9)



2=



chain


119876119888lowast

sc3 = 2120583119909119901119898minus 119888 minus 119888

119905

119901119898minus 119904119898

Π119888lowast

sc3 =(119901119898minus 119888 minus 119905)

2

119901119898minus 119904119898

120583119909

(10)




119905which makes



119905= 119901119898minus 119888 minus ((119901

119898120583120573120583120572minus 119888)radic(120590

2

120573+ 1205832

120573)(1205902120572+ 1205832120572))

When 119888119905lt 1198881015840



119905 without



120572and 120583





0 02 04 06 08 10

20

40

60

80

100

120

120572

120590120572 = 015

120590120572 = 025120590120572 = 035

c t


120572






120572 The more the







Π119863

119904119894

= 119908119894sdotmin (120572119876

119894 119902) minus 119904

119894[119902 minus 120572119876

119894]+

minus

2

sum

119894=1

119888119876119894 (11)


119894(119909) equiv

int119909


119864 [min (120572119894119876119894 119902)] = int

119902119876119894

0

120572119894119876119894ℎ119894(120572119894) 119889120572119894

+ int

1

119902119876119894

119902ℎ119894(120572119894) 119889120572119894

= 119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)

119864 [(119902 minus 120572119894119876119894)+

] = int

119902119876119894

0

(119902 minus 120572119894119876119894) ℎ119894(120572119894) 119889120572119894

= 119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)

(12)


119864 [Π119863

119904119894

] = 119908119894[119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)]

minus 119904119894[119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)]

minus

2

sum

119894=1

119888119876119894

(13)



written as

120597Π119863

119904119894

120597119876119894

= 119908119894[119896119894(119902

119876119894

) + 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)]

minus 119904119894[119902ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

) minus 119896119894(119902

119876119894

)] minus 119888

= (119908119894+ 119904119894) 119896119894(119902

119876119894

) minus 119888

(14)


0

5000

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔΠ

=Π

c sc3minusΠ

c sc1

minus15000

minus10000

minus5000

(a)

minus500

0

500

1000

1500

2000

2500

3000

3500

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔQ

=Qc sc

3minusQc sc

1

(b)


Developing 120597Π119863119904119894


sion meanwhile 1205972Π119863119904119894

1205971198762

119894= minus(119908

119894+ 119904119894)(11990221198763

119894) lt 0

1198911(1198761 1198762) = 119888 minus (119908

1+ 1199041) 1198961(119902

1198761

)

1198912(1198761 1198762) = 119888 minus (119908

2+ 1199042) 1198962(119902

1198762

)

(15)


(1205971198912(1198761 1198762)1205971198762) = minus(119908

2+ 1199042)(11990221198763

2ℎ2)(119902119876

2) lt 0 for any

1198761and 119876




1

lowast

1198761015840

2

lowast

]


Π119863

1198981= int

119863119902

120572

[120572119902119908 minus 119904119898(119909 minus 120572119902)] ℎ (120572) 119889120572

+ int

120572

119863119902

119902119908ℎ (120572) 119889120572 minus 119888119902

(16)


11989811205971199022= minus(119908 + 119904

119898)(1199091199023ℎ)(120572) lt 0



int

119909119902lowast

120572

120572ℎ (120572) 119889120572 =119888

119908 + 119904119898

(17)


Π119863lowast

1198981= 119908119909 minus (119908 + 119904

119898) 119902119867(

119909

119902lowast) (18)


Π119863

1199031= 119901119898int

120573

120573

119864min (119909 120573119902) 119892 (120573) 119889120573

+ 119904int

120573

120573

119864(120573119902 minus 119909)+

119892 (120573) 119889119909 119889120573 minus 119908119902

(19)


119909)+= 120573119902 minus 119864min(119909 120573119902) = int120573119902

0(120573119902 minus 119909)119891(119909)119889119909 then we


Π119863

1199031= minus119901119898minus 119904119898

4120583119909

(1205902

120573+ 1205832

120573) 1199022+ 119901119898120583120573119902 minus 119908119902 (20)



119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)



Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)



119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583


Π119863lowast

1199031lt Π119863lowast








119905gt 119888dagger



119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)










119905lt 119888119905lt 119888dagger





119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78




0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Supplier 1Q1

120572

Supplier 2Q2

120572Qi

qm

qm



D(x)

120572120573120573

Customer
















0 2 4 6 8 1070

80

90

100

110

120

130

140

150

160

170

ct

Qclowast sc

3

(a)

0 2 4 6 8 10200

400

600

800

1000

1200

1400

1600

ct

Πclowast sc

3

(b)



sc3 with 119888119905


Π119888



119864min (119863 120572120573119876) = 120572120573119876 minus int120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

119864(120572120573119876 minus 119863)+

= 120572120573119876 minus 119864min (119909 120572120573119876)

= int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

(2)


Π119888

sc1 = 119901119898 int120572

120572

int

120573

120573

[120572120573119876 minus int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572

+ 119904int

120572

120572

int

120573

120573

[int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572 minus 2119888119876

(3)


Π119888

sc1 = 119901119898120583120572120583120573119876 minus119901119898minus 119904119898

4120583119909

sdot (1205902

120572+ 1205832

120572)

times (1205902

120573+ 1205832

120573)1198762minus 2119888119876

(4)




119876119888lowast

sc1 =2120583119909(119901119898120583120572120583120573minus 119888)

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

Π119888lowast

sc1 =(119901119898120583120572120583120573minus 119888)2

120583119909

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

(5)


119905respectively


Π119888

sc2 = 119901119898 sdotmin (1205721205731198761 120572120573119876

2 119863)

minus 119904119898[119863 minusmin (120572120573119876

1 120572120573119876

2)]+

+ V[min (1205721205731198761 120572120573119876

2) minus 119863]

+

minus 119888 (1198761+ 1198762)

(6)





Π119888

sc2=

Π119888


2

Π119888


1

Π119888


1

Π119888


2

(7)


Π119888lowast


2 119863)

minus 119904119898119863 + V sdotmin (120572120573119876

1 120572120573119876

2) minus 119888 (119876

1+ 1198762)

(8)




119905 The


Π119888

sc3 = 119901119898min (119863119876) + 119904119898[119876 minus 119863]

+minus 2 (119888 + 119888

119905) 119876 (9)



2=



chain


119876119888lowast

sc3 = 2120583119909119901119898minus 119888 minus 119888

119905

119901119898minus 119904119898

Π119888lowast

sc3 =(119901119898minus 119888 minus 119905)

2

119901119898minus 119904119898

120583119909

(10)




119905which makes



119905= 119901119898minus 119888 minus ((119901

119898120583120573120583120572minus 119888)radic(120590

2

120573+ 1205832

120573)(1205902120572+ 1205832120572))

When 119888119905lt 1198881015840



119905 without



120572and 120583





0 02 04 06 08 10

20

40

60

80

100

120

120572

120590120572 = 015

120590120572 = 025120590120572 = 035

c t


120572






120572 The more the







Π119863

119904119894

= 119908119894sdotmin (120572119876

119894 119902) minus 119904

119894[119902 minus 120572119876

119894]+

minus

2

sum

119894=1

119888119876119894 (11)


119894(119909) equiv

int119909


119864 [min (120572119894119876119894 119902)] = int

119902119876119894

0

120572119894119876119894ℎ119894(120572119894) 119889120572119894

+ int

1

119902119876119894

119902ℎ119894(120572119894) 119889120572119894

= 119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)

119864 [(119902 minus 120572119894119876119894)+

] = int

119902119876119894

0

(119902 minus 120572119894119876119894) ℎ119894(120572119894) 119889120572119894

= 119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)

(12)


119864 [Π119863

119904119894

] = 119908119894[119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)]

minus 119904119894[119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)]

minus

2

sum

119894=1

119888119876119894

(13)



written as

120597Π119863

119904119894

120597119876119894

= 119908119894[119896119894(119902

119876119894

) + 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)]

minus 119904119894[119902ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

) minus 119896119894(119902

119876119894

)] minus 119888

= (119908119894+ 119904119894) 119896119894(119902

119876119894

) minus 119888

(14)


0

5000

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔΠ

=Π

c sc3minusΠ

c sc1

minus15000

minus10000

minus5000

(a)

minus500

0

500

1000

1500

2000

2500

3000

3500

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔQ

=Qc sc

3minusQc sc

1

(b)


Developing 120597Π119863119904119894


sion meanwhile 1205972Π119863119904119894

1205971198762

119894= minus(119908

119894+ 119904119894)(11990221198763

119894) lt 0

1198911(1198761 1198762) = 119888 minus (119908

1+ 1199041) 1198961(119902

1198761

)

1198912(1198761 1198762) = 119888 minus (119908

2+ 1199042) 1198962(119902

1198762

)

(15)


(1205971198912(1198761 1198762)1205971198762) = minus(119908

2+ 1199042)(11990221198763

2ℎ2)(119902119876

2) lt 0 for any

1198761and 119876




1

lowast

1198761015840

2

lowast

]


Π119863

1198981= int

119863119902

120572

[120572119902119908 minus 119904119898(119909 minus 120572119902)] ℎ (120572) 119889120572

+ int

120572

119863119902

119902119908ℎ (120572) 119889120572 minus 119888119902

(16)


11989811205971199022= minus(119908 + 119904

119898)(1199091199023ℎ)(120572) lt 0



int

119909119902lowast

120572

120572ℎ (120572) 119889120572 =119888

119908 + 119904119898

(17)


Π119863lowast

1198981= 119908119909 minus (119908 + 119904

119898) 119902119867(

119909

119902lowast) (18)


Π119863

1199031= 119901119898int

120573

120573

119864min (119909 120573119902) 119892 (120573) 119889120573

+ 119904int

120573

120573

119864(120573119902 minus 119909)+

119892 (120573) 119889119909 119889120573 minus 119908119902

(19)


119909)+= 120573119902 minus 119864min(119909 120573119902) = int120573119902

0(120573119902 minus 119909)119891(119909)119889119909 then we


Π119863

1199031= minus119901119898minus 119904119898

4120583119909

(1205902

120573+ 1205832

120573) 1199022+ 119901119898120583120573119902 minus 119908119902 (20)



119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)



Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)



119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583


Π119863lowast

1199031lt Π119863lowast








119905gt 119888dagger



119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)










119905lt 119888119905lt 119888dagger





119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78




0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 1070

80

90

100

110

120

130

140

150

160

170

ct

Qclowast sc

3

(a)

0 2 4 6 8 10200

400

600

800

1000

1200

1400

1600

ct

Πclowast sc

3

(b)



sc3 with 119888119905


Π119888



119864min (119863 120572120573119876) = 120572120573119876 minus int120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

119864(120572120573119876 minus 119863)+

= 120572120573119876 minus 119864min (119909 120572120573119876)

= int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909

(2)


Π119888

sc1 = 119901119898 int120572

120572

int

120573

120573

[120572120573119876 minus int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572

+ 119904int

120572

120572

int

120573

120573

[int

120572120573119876

0

(120572120573119876 minus 119909)119891 (119909) 119889119909]

times 119892 (120573) ℎ (120572) 119889120573 119889120572 minus 2119888119876

(3)


Π119888

sc1 = 119901119898120583120572120583120573119876 minus119901119898minus 119904119898

4120583119909

sdot (1205902

120572+ 1205832

120572)

times (1205902

120573+ 1205832

120573)1198762minus 2119888119876

(4)




119876119888lowast

sc1 =2120583119909(119901119898120583120572120583120573minus 119888)

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

Π119888lowast

sc1 =(119901119898120583120572120583120573minus 119888)2

120583119909

(119901119898minus 119904119898) (1205902120572+ 1205832120572) (1205902

120573+ 1205832

120573)

(5)


119905respectively


Π119888

sc2 = 119901119898 sdotmin (1205721205731198761 120572120573119876

2 119863)

minus 119904119898[119863 minusmin (120572120573119876

1 120572120573119876

2)]+

+ V[min (1205721205731198761 120572120573119876

2) minus 119863]

+

minus 119888 (1198761+ 1198762)

(6)





Π119888

sc2=

Π119888


2

Π119888


1

Π119888


1

Π119888


2

(7)


Π119888lowast


2 119863)

minus 119904119898119863 + V sdotmin (120572120573119876

1 120572120573119876

2) minus 119888 (119876

1+ 1198762)

(8)




119905 The


Π119888

sc3 = 119901119898min (119863119876) + 119904119898[119876 minus 119863]

+minus 2 (119888 + 119888

119905) 119876 (9)



2=



chain


119876119888lowast

sc3 = 2120583119909119901119898minus 119888 minus 119888

119905

119901119898minus 119904119898

Π119888lowast

sc3 =(119901119898minus 119888 minus 119905)

2

119901119898minus 119904119898

120583119909

(10)




119905which makes



119905= 119901119898minus 119888 minus ((119901

119898120583120573120583120572minus 119888)radic(120590

2

120573+ 1205832

120573)(1205902120572+ 1205832120572))

When 119888119905lt 1198881015840



119905 without



120572and 120583





0 02 04 06 08 10

20

40

60

80

100

120

120572

120590120572 = 015

120590120572 = 025120590120572 = 035

c t


120572






120572 The more the







Π119863

119904119894

= 119908119894sdotmin (120572119876

119894 119902) minus 119904

119894[119902 minus 120572119876

119894]+

minus

2

sum

119894=1

119888119876119894 (11)


119894(119909) equiv

int119909


119864 [min (120572119894119876119894 119902)] = int

119902119876119894

0

120572119894119876119894ℎ119894(120572119894) 119889120572119894

+ int

1

119902119876119894

119902ℎ119894(120572119894) 119889120572119894

= 119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)

119864 [(119902 minus 120572119894119876119894)+

] = int

119902119876119894

0

(119902 minus 120572119894119876119894) ℎ119894(120572119894) 119889120572119894

= 119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)

(12)


119864 [Π119863

119904119894

] = 119908119894[119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)]

minus 119904119894[119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)]

minus

2

sum

119894=1

119888119876119894

(13)



written as

120597Π119863

119904119894

120597119876119894

= 119908119894[119896119894(119902

119876119894

) + 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)]

minus 119904119894[119902ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

) minus 119896119894(119902

119876119894

)] minus 119888

= (119908119894+ 119904119894) 119896119894(119902

119876119894

) minus 119888

(14)


0

5000

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔΠ

=Π

c sc3minusΠ

c sc1

minus15000

minus10000

minus5000

(a)

minus500

0

500

1000

1500

2000

2500

3000

3500

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔQ

=Qc sc

3minusQc sc

1

(b)


Developing 120597Π119863119904119894


sion meanwhile 1205972Π119863119904119894

1205971198762

119894= minus(119908

119894+ 119904119894)(11990221198763

119894) lt 0

1198911(1198761 1198762) = 119888 minus (119908

1+ 1199041) 1198961(119902

1198761

)

1198912(1198761 1198762) = 119888 minus (119908

2+ 1199042) 1198962(119902

1198762

)

(15)


(1205971198912(1198761 1198762)1205971198762) = minus(119908

2+ 1199042)(11990221198763

2ℎ2)(119902119876

2) lt 0 for any

1198761and 119876




1

lowast

1198761015840

2

lowast

]


Π119863

1198981= int

119863119902

120572

[120572119902119908 minus 119904119898(119909 minus 120572119902)] ℎ (120572) 119889120572

+ int

120572

119863119902

119902119908ℎ (120572) 119889120572 minus 119888119902

(16)


11989811205971199022= minus(119908 + 119904

119898)(1199091199023ℎ)(120572) lt 0



int

119909119902lowast

120572

120572ℎ (120572) 119889120572 =119888

119908 + 119904119898

(17)


Π119863lowast

1198981= 119908119909 minus (119908 + 119904

119898) 119902119867(

119909

119902lowast) (18)


Π119863

1199031= 119901119898int

120573

120573

119864min (119909 120573119902) 119892 (120573) 119889120573

+ 119904int

120573

120573

119864(120573119902 minus 119909)+

119892 (120573) 119889119909 119889120573 minus 119908119902

(19)


119909)+= 120573119902 minus 119864min(119909 120573119902) = int120573119902

0(120573119902 minus 119909)119891(119909)119889119909 then we


Π119863

1199031= minus119901119898minus 119904119898

4120583119909

(1205902

120573+ 1205832

120573) 1199022+ 119901119898120583120573119902 minus 119908119902 (20)



119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)



Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)



119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583


Π119863lowast

1199031lt Π119863lowast








119905gt 119888dagger



119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)










119905lt 119888119905lt 119888dagger





119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78




0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Π119888

sc2=

Π119888


2

Π119888


1

Π119888


1

Π119888


2

(7)


Π119888lowast


2 119863)

minus 119904119898119863 + V sdotmin (120572120573119876

1 120572120573119876

2) minus 119888 (119876

1+ 1198762)

(8)




119905 The


Π119888

sc3 = 119901119898min (119863119876) + 119904119898[119876 minus 119863]

+minus 2 (119888 + 119888

119905) 119876 (9)



2=



chain


119876119888lowast

sc3 = 2120583119909119901119898minus 119888 minus 119888

119905

119901119898minus 119904119898

Π119888lowast

sc3 =(119901119898minus 119888 minus 119905)

2

119901119898minus 119904119898

120583119909

(10)




119905which makes



119905= 119901119898minus 119888 minus ((119901

119898120583120573120583120572minus 119888)radic(120590

2

120573+ 1205832

120573)(1205902120572+ 1205832120572))

When 119888119905lt 1198881015840



119905 without



120572and 120583





0 02 04 06 08 10

20

40

60

80

100

120

120572

120590120572 = 015

120590120572 = 025120590120572 = 035

c t


120572






120572 The more the







Π119863

119904119894

= 119908119894sdotmin (120572119876

119894 119902) minus 119904

119894[119902 minus 120572119876

119894]+

minus

2

sum

119894=1

119888119876119894 (11)


119894(119909) equiv

int119909


119864 [min (120572119894119876119894 119902)] = int

119902119876119894

0

120572119894119876119894ℎ119894(120572119894) 119889120572119894

+ int

1

119902119876119894

119902ℎ119894(120572119894) 119889120572119894

= 119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)

119864 [(119902 minus 120572119894119876119894)+

] = int

119902119876119894

0

(119902 minus 120572119894119876119894) ℎ119894(120572119894) 119889120572119894

= 119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)

(12)


119864 [Π119863

119904119894

] = 119908119894[119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)]

minus 119904119894[119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)]

minus

2

sum

119894=1

119888119876119894

(13)



written as

120597Π119863

119904119894

120597119876119894

= 119908119894[119896119894(119902

119876119894

) + 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)]

minus 119904119894[119902ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

) minus 119896119894(119902

119876119894

)] minus 119888

= (119908119894+ 119904119894) 119896119894(119902

119876119894

) minus 119888

(14)


0

5000

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔΠ

=Π

c sc3minusΠ

c sc1

minus15000

minus10000

minus5000

(a)

minus500

0

500

1000

1500

2000

2500

3000

3500

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔQ

=Qc sc

3minusQc sc

1

(b)


Developing 120597Π119863119904119894


sion meanwhile 1205972Π119863119904119894

1205971198762

119894= minus(119908

119894+ 119904119894)(11990221198763

119894) lt 0

1198911(1198761 1198762) = 119888 minus (119908

1+ 1199041) 1198961(119902

1198761

)

1198912(1198761 1198762) = 119888 minus (119908

2+ 1199042) 1198962(119902

1198762

)

(15)


(1205971198912(1198761 1198762)1205971198762) = minus(119908

2+ 1199042)(11990221198763

2ℎ2)(119902119876

2) lt 0 for any

1198761and 119876




1

lowast

1198761015840

2

lowast

]


Π119863

1198981= int

119863119902

120572

[120572119902119908 minus 119904119898(119909 minus 120572119902)] ℎ (120572) 119889120572

+ int

120572

119863119902

119902119908ℎ (120572) 119889120572 minus 119888119902

(16)


11989811205971199022= minus(119908 + 119904

119898)(1199091199023ℎ)(120572) lt 0



int

119909119902lowast

120572

120572ℎ (120572) 119889120572 =119888

119908 + 119904119898

(17)


Π119863lowast

1198981= 119908119909 minus (119908 + 119904

119898) 119902119867(

119909

119902lowast) (18)


Π119863

1199031= 119901119898int

120573

120573

119864min (119909 120573119902) 119892 (120573) 119889120573

+ 119904int

120573

120573

119864(120573119902 minus 119909)+

119892 (120573) 119889119909 119889120573 minus 119908119902

(19)


119909)+= 120573119902 minus 119864min(119909 120573119902) = int120573119902

0(120573119902 minus 119909)119891(119909)119889119909 then we


Π119863

1199031= minus119901119898minus 119904119898

4120583119909

(1205902

120573+ 1205832

120573) 1199022+ 119901119898120583120573119902 minus 119908119902 (20)



119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)



Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)



119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583


Π119863lowast

1199031lt Π119863lowast








119905gt 119888dagger



119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)










119905lt 119888119905lt 119888dagger





119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78




0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 02 04 06 08 10

20

40

60

80

100

120

120572

120590120572 = 015

120590120572 = 025120590120572 = 035

c t


120572






120572 The more the







Π119863

119904119894

= 119908119894sdotmin (120572119876

119894 119902) minus 119904

119894[119902 minus 120572119876

119894]+

minus

2

sum

119894=1

119888119876119894 (11)


119894(119909) equiv

int119909


119864 [min (120572119894119876119894 119902)] = int

119902119876119894

0

120572119894119876119894ℎ119894(120572119894) 119889120572119894

+ int

1

119902119876119894

119902ℎ119894(120572119894) 119889120572119894

= 119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)

119864 [(119902 minus 120572119894119876119894)+

] = int

119902119876119894

0

(119902 minus 120572119894119876119894) ℎ119894(120572119894) 119889120572119894

= 119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)

(12)


119864 [Π119863

119904119894

] = 119908119894[119876119894119896119894(119902

119876119894

) + 119902119867119894(119902

119876119894

)]

minus 119904119894[119902119867119894(119902

119876119894

) minus 119876119894119896119894(119902

119876119894

)]

minus

2

sum

119894=1

119888119876119894

(13)



written as

120597Π119863

119904119894

120597119876119894

= 119908119894[119896119894(119902

119876119894

) + 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)]

minus 119904119894[119902ℎ119894(119902

119876119894

)(minus119902

1198762

119894

)

minus 119876119894

119902

119876119894

ℎ119894(119902

119876119894

)(minus119902

1198762

119894

) minus 119896119894(119902

119876119894

)] minus 119888

= (119908119894+ 119904119894) 119896119894(119902

119876119894

) minus 119888

(14)


0

5000

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔΠ

=Π

c sc3minusΠ

c sc1

minus15000

minus10000

minus5000

(a)

minus500

0

500

1000

1500

2000

2500

3000

3500

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔQ

=Qc sc

3minusQc sc

1

(b)


Developing 120597Π119863119904119894


sion meanwhile 1205972Π119863119904119894

1205971198762

119894= minus(119908

119894+ 119904119894)(11990221198763

119894) lt 0

1198911(1198761 1198762) = 119888 minus (119908

1+ 1199041) 1198961(119902

1198761

)

1198912(1198761 1198762) = 119888 minus (119908

2+ 1199042) 1198962(119902

1198762

)

(15)


(1205971198912(1198761 1198762)1205971198762) = minus(119908

2+ 1199042)(11990221198763

2ℎ2)(119902119876

2) lt 0 for any

1198761and 119876




1

lowast

1198761015840

2

lowast

]


Π119863

1198981= int

119863119902

120572

[120572119902119908 minus 119904119898(119909 minus 120572119902)] ℎ (120572) 119889120572

+ int

120572

119863119902

119902119908ℎ (120572) 119889120572 minus 119888119902

(16)


11989811205971199022= minus(119908 + 119904

119898)(1199091199023ℎ)(120572) lt 0



int

119909119902lowast

120572

120572ℎ (120572) 119889120572 =119888

119908 + 119904119898

(17)


Π119863lowast

1198981= 119908119909 minus (119908 + 119904

119898) 119902119867(

119909

119902lowast) (18)


Π119863

1199031= 119901119898int

120573

120573

119864min (119909 120573119902) 119892 (120573) 119889120573

+ 119904int

120573

120573

119864(120573119902 minus 119909)+

119892 (120573) 119889119909 119889120573 minus 119908119902

(19)


119909)+= 120573119902 minus 119864min(119909 120573119902) = int120573119902

0(120573119902 minus 119909)119891(119909)119889119909 then we


Π119863

1199031= minus119901119898minus 119904119898

4120583119909

(1205902

120573+ 1205832

120573) 1199022+ 119901119898120583120573119902 minus 119908119902 (20)



119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)



Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)



119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583


Π119863lowast

1199031lt Π119863lowast








119905gt 119888dagger



119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)










119905lt 119888119905lt 119888dagger





119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78




0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5000

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔΠ

=Π

c sc3minusΠ

c sc1

minus15000

minus10000

minus5000

(a)

minus500

0

500

1000

1500

2000

2500

3000

3500

0 02 04 06 08 1120572

120583x = 50

120583x = 100120583x = 150

ΔQ

=Qc sc

3minusQc sc

1

(b)


Developing 120597Π119863119904119894


sion meanwhile 1205972Π119863119904119894

1205971198762

119894= minus(119908

119894+ 119904119894)(11990221198763

119894) lt 0

1198911(1198761 1198762) = 119888 minus (119908

1+ 1199041) 1198961(119902

1198761

)

1198912(1198761 1198762) = 119888 minus (119908

2+ 1199042) 1198962(119902

1198762

)

(15)


(1205971198912(1198761 1198762)1205971198762) = minus(119908

2+ 1199042)(11990221198763

2ℎ2)(119902119876

2) lt 0 for any

1198761and 119876




1

lowast

1198761015840

2

lowast

]


Π119863

1198981= int

119863119902

120572

[120572119902119908 minus 119904119898(119909 minus 120572119902)] ℎ (120572) 119889120572

+ int

120572

119863119902

119902119908ℎ (120572) 119889120572 minus 119888119902

(16)


11989811205971199022= minus(119908 + 119904

119898)(1199091199023ℎ)(120572) lt 0



int

119909119902lowast

120572

120572ℎ (120572) 119889120572 =119888

119908 + 119904119898

(17)


Π119863lowast

1198981= 119908119909 minus (119908 + 119904

119898) 119902119867(

119909

119902lowast) (18)


Π119863

1199031= 119901119898int

120573

120573

119864min (119909 120573119902) 119892 (120573) 119889120573

+ 119904int

120573

120573

119864(120573119902 minus 119909)+

119892 (120573) 119889119909 119889120573 minus 119908119902

(19)


119909)+= 120573119902 minus 119864min(119909 120573119902) = int120573119902

0(120573119902 minus 119909)119891(119909)119889119909 then we


Π119863

1199031= minus119901119898minus 119904119898

4120583119909

(1205902

120573+ 1205832

120573) 1199022+ 119901119898120583120573119902 minus 119908119902 (20)



119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)



Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)



119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583


Π119863lowast

1199031lt Π119863lowast








119905gt 119888dagger



119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)










119905lt 119888119905lt 119888dagger





119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78




0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119902lowast

1= 2120583119909

119901119898120583120573minus 119908

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

Π119863lowast

1199031

=

(119901119898120583120573minus 119908)2

(119901119898minus 119904) (120590

2

120573+ 1205832

120573)

120583119909

(21)



Π119863

1199032= 119901119898119864min (119909 119902) + 119904

119898119864(119902 minus 119909)

+

= 119901119898[119902 minus int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909]

+ 119904119898int

119902

0

(119902 minus 119909) 119891 (119909) 119889119909 minus 119901119908119902

(22)



119902lowast

2= 2120583119909

119901119898minus 119901119908

119901119898minus 119904119898

Π119863lowast

1199032=(119901119898minus 119901119908)2

119901119898minus 119904119898

120583119909

(23)

Due to 120590120573gt 0 and 0 lt 120583


Π119863lowast

1199031lt Π119863lowast








119905gt 119888dagger



119905= 119901119908minus 119888 minus [119901

119908minus (119901119908+

119904119898)119866(119902119876

lowast

1)](119901119898120583120573minus 119901119908)(1205902

120573+ 1205832

120573)(119901119898minus 119901119908)










119905lt 119888119905lt 119888dagger





119908= 20 119888 = 12

119901119908= 20 119901

119898= 30 V = 8 and 119904

119898= 5 when 0 lt 119888

119905lt 78




0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 2000

200

400

600

800

1000

1200

120583x

ΠDlowast

r1

ΠDlowast

r2

ΠDlowast

r

(a)

0 2 4 6 8 10minus250

minus200

minus150

minus100

minus50

0

50

100

ΔΠ

=Π

Dlowast

m2minusΠ

Dlowast

m1

ct

(b)


119909and ΔΠ with 119888

119905

minus1000

minus500

0

500

1000

1500

2000

0 02 04 06 08 1120573

ΔΠ

120573 sim ΔΠr

120573 sim ΔΠm

120573 sim ΔΠr 25120573 sim ΔΠm 25

ct = 1 ct =

ct = ct =1



119888119905





119903gt 0 and

ΔΠ119898gt 0





Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Notations











Π119863


Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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