Research Article Misplaced Inventory and Lead-Time …downloads.hindawi.com/journals/jam/2014/135284.pdf · Research Article Misplaced Inventory and Lead-Time in the Supply Chain:

Research ArticleMisplaced Inventory and Lead-Time inthe Supply Chain Analysis of Decision-Making onRFID Investment with Service Level

Li-Hao Zhang12 Ti-Jun Fan1 Wen-Chyuan Chiang3 and Feng Tao1

1 School of Business East China University of Science and Technology Shanghai 200237 China2 Krannert School of Management Purdue University West Lafayette IN 47907 USA3 College of Business University of Tulsa Tulsa 74104 USA

Correspondence should be addressed to Li-Hao Zhang leehowzhanghotmailcom

Received 13 February 2014 Accepted 25 June 2014 Published 23 July 2014

Academic Editor Zhi-Hong Guan

Copyright copy 2014 Li-Hao Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Radio-frequency identification (RFID) as the key technology of Internet of Things (IoT) has been hailed as a major innovationto solve misplaced inventory and reduce lead-time Many retailers have been pushing their suppliers to invest this technologyHowever its associated costs seem to prohibit its widespread application This paper analyzes the situation of service level in aretail supply chain which has resulted from misplaced inventory and lead-time By newsvendor model we analyze the differencebetween with- and without-RFID technologies in service level of centralized and decentralized supply chains respectively Thenwith different service levels we determine the tag cost thresholds at which RFID technology investment becomes profitable incentralized and decentralized supply chains respectively Furthermore we apply a linear transfer payment coefficient strategy tocoordinate with the decentralized supply chain It is found that whether the adoption of RFID technology improves the service leveldepends on the cost of RFID tag in the centralized system but it improves the service level in the decentralized system when onlythe supplier bears the cost of RFID tag Moreover the same cost thresholds of RFID tag with different service levels exist in boththe centralized and the decentralized cases

1 Introduction

11 Motivation Nowadays customer service level is one ofthe most important key performance indicators in the supplychain If it increases from 90 to 96 customers will surelybe much more satisfied and buy more from the supply chainin the USA Obviously customer service level will greatlyaffect the profit of supply chain To achieve higher customerservice level almost all the supply chains have placed a strongemphasis on the control of misplaced inventory and lead-time

Misplaced inventory one of the inventory inaccuracystyles in operations is widely discussed in the retailer supplychain [1ndash6] According to the surveys in DeHoratius andRaman [7] inventory records are inaccurate for 65 of thenearly 370000 inventory records observed across 37 retailstores 20 of the inventory records differ from the physical

stock by six or more items and approximately 12 ofthose items have no inventory on the shelf but therecorded inventory quantity is positive At the same timelead-time reduction can improve forecasting accuracy andthe speed of response to market changes get highersupply chain service levels and reduce excess inven-tory and stockouts [8] It has become one of the mostimportant sources of competitive advantage Granite Bay(httpwwwgranite-baycomreducing-lead-timehtml (Lastaccessed October 2008)) a US-based consulting firm hasreported that as a result of reducing many world-classmanufacturersrsquo lead-times by 50ndash80 they have increasedtheir market shares and improved their profitability GraniteBay regards lead-time reduction as the most important factorin achieving world-class operations

ldquoInternet of Things (IoT)rdquo is a landmark emergingstrategic industries in the 21st century and it was first coined

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 135284 17 pageshttpdxdoiorg1011552014135284

2 Journal of Applied Mathematics

by MIT in the late 1990s It refers to a ldquodevices or sensorsconnected worldrdquo where objects are connected monitoredand optimized through either wired wireless or hybridsystems The Cisco Internet Business Solutions Grouphas reported that in the next 10 years the IoT is allabout the potential economic value which will reach $144trillion all over the world while the share of the Chinesemarket will reach $18 trillion accounted for 12 of thetotal in the world (httpwwwnewskingusnews-4263717-Cisco-stressed-the-potential-value-of-the-interconnection-of-all-thingshtml (April 8 2013)) In 2012 IoT of theChinese industries has been reached yen 365 billion andit is expected to reach more than yen 500 billion in 2015(httpenwikipediaorgwikiInternet of Things January2014) Radio frequency identification (RFID) technologyis the key technical points of IoT and it is a way tosolve misplaced inventory and reduce lead-time RFIDis a contactless and low-power wireless communicationtechnology which has applications in many areas of asupply chain and it has been hailed as a major innovationto enhance the efficiency of a supply chain [9] The basicpremise of RFID is that a radio frequency (RF) reader canprovide real-time communication with numerous objectsat the same time reading as many as 200 tags in a secondwithout requiring contact or a direct line of sight [10] Theseadvanced identification and communication characteristicsof RFID can provide real-time information update andimprove productsrsquo traceability and visibility among supplychains as well as increasing the automation operationalprocess reducing the time of manual operations andincreasing efficiency of manufacture transportation storageand other links so it can eliminate the misplaced inventoryand reduce lead-time in the supply chains

However RFID technology is not yet cheap enough tobecome as common as barcoding technology In recent yearsseveral major retail chains have strongly promotedmdashor evenmandatedmdashthe adoption of RFID by their suppliers mostlyat the pallet level While RFID technology is suggested toenable substantial efficiency gains at the different stages of asupply chain the associated costs are by no means negligibleFor the retail supply chain due to the huge number ofcommodities involved the price of the RFID tag is the mostimportant factor For example Asif and Mandviwalla [11]estimated that a typical consumer packaged goods supplierwould spend between $13 million and $23 million for ship-ping 50 million cases per year just for the cost of RFID tagsAs a result the substantial cost of RFID seems to prohibit itswidespread use at the item level Therefore we must considerthe cost threshold for adoption of RFID tags in the supplychain and the change of the supply chain profit and servicelevel withwithout RFID technology In the recent yearsseveral major retail chains have strongly promotedmdashor evenmandatedmdashRFID adopting by their suppliers Generally theupstream enterprise adopts RFID technology and bears itscost but the downstream enterprise gains the profit It hasbeen pointed out by the practitioners that a compensationfor the upstream enterprise is necessary to ensure the supplychain operations with RFID technology

12 Research Questions andMain Results In order to addressthe above issues while the supply chain establishes a newbusiness model under IoT environment an RFID tag hasbeen labeled on each productWe focus on the improvementsthat result from increasing the accuracy of inventory infor-mation and speeding up the processes In order to ensurethe increase of the supply chainrsquos profit and performance itis important to balance the increase of service level with thecosts of RFID tagThis paper addresses the following researchquestions

(1) How do the misplaced inventory and lead-time affectservice level in a retail supply chain RFID technologycan eliminate the misplaced inventory and reducethe lead-time but it also increases the cost of supplychain Thus what is the difference between with andwithout RFID technology in service level of central-ized and decentralized supply chains respectively

(2) What is the decision-making on RFID technologyinvestment with different service levels of centralizedand decentralized supply chains respectively

(3) Howdoes RFID technology affect the playersrsquo profit indecentralized supply chain How can we coordinateit

In spite of the significance of IoT in retail supply chainsthe extant literature does not examine the above issues Theprimary motivation of this research is to address this gap Weuse theoretical and numerical example approaches to answerthe research questions We study two different settings (1) acentralized supply chain where the supplier owns the retailerand (2) a decentralized and uncoordinated supply chain

Firstly we define the service level according to Iyer andBergen [12] and it results from the misplaced inventoryand lead-time of a two-stage supply chain with a supplierand a retailer We assume that the market demand forecasterror varies with lead-time reduction Chen and Chuang [13]also assumed this in their simulation study We respectivelyanalyze the relationship not only between service level andmisplaced inventory but also between service level and lead-time And we find that the service level increases withthe decrease of the misplaced inventory rate but with theincrease of the lead-time reduction in both centralized anddecentralized supply chains When the supply chain adoptsRFID technology the service level increases with the decreaseof the unit cost of RFID tag in centralized supply chain butthe unit cost of RFID tag does not affect the service level indecentralized supply chain

Then we theoretically analyze the cost threshold of RFIDtag that becomes profitable in different service levels ofcentralized and decentralized supply chains We find that thetrends of service level and profit are different with differentRFID tag prices but the same cost-of-tag thresholds for RFIDtechnology with different service levels exist in both thecentralized and the decentralized cases

Finally when the variable tagging cost of RFID tech-nology under a threshold improves the profit of both thedecentralized supply chain and the retailer the manufac-turerrsquos profit will be reduced In this context we prove that

Journal of Applied Mathematics 3

the linear transfer payment coefficient strategy can be usedby the retailer to achieve Pareto improvement of the supplychain actorsrsquo profits We also provide a numerical example tovalidate our theoretical findings

The remainder of the paper is organized as follows Wediscuss related literature and our contributions in Section 2and present the base model with the assumptions andnotations in Section 3 We discuss the decision-making onRFID technology investment with the perspective of servicelevel in the centralizeddecentralized supply chain in Sections4 and 5 We provide numerical examples in Section 6 and inSection 7 we conclude our paper and point out the directionfor further researchThe proofs of key results are relegated toan appendix

2 Literature Review

Although the specific research questions of this paper havenot been addressed before there are certain streams ofexisting literature that are relevant to our research Our workis related to two streams of studies the first stream deals withthe service level of the supply chain and the second withthe adoption of RFID technology by the supply chain in IoTenvironment

The first stream of related literature consists of papersdealing with the service level of the supply chain Thiel et al[14] showed that the service-level quality is a nonmonotonefunction of the inaccuracy rate that is the service-levelquality increases up to an inventory record inaccuracy leveland subsequently decreases and they also found that thesame phenomenon between the function involving a level ofsafety stock defined by simulation and the function betweenthe service-level quality and inventory record inaccuracy Jhaand Shanker [15] studied an integrated production-inventorymodel and the buyer level demand is assumed to be inde-pendent and normally distributed They supposed a servicelevel constraint corresponding to each buyer is included intheir model which limits that the stock-out level per cycle foreach buyer is bounded Specifying a service level avoids thedifficult practical issue of explicitly estimating the shortagecost Chen and Chuang [13] found that the shortening oflead-time affects not only the retailerrsquos forecast accuracy butalso the service level of the supply chain Glasserman andWang [16] as well as Song and Yao [17] reached similarconclusions the reduction of lead-time can speed up theresponse and improve the service level to the customersraising the competitiveness of enterprises Sieke et al [8] usedthe service level references to the fraction of a manufacturerrsquosdemand filled by the supplier and analyzed two types ofservice level-based supply contracts that are designed by amanufacturer and offered to a supplierThey showed how thesupplier responds to the service level contracts and how thecontract parameters can be chosen such that the supply chainis coordinated

All of these studies supposed the service level as a con-straint in the supply chain designed a contract based on theservice level that coordinates the supply chain or found thatthe lead-time reductionwill affect service level in a qualitative

point of view In contrast we study the relationship not onlybetween the service level and the misplaced inventory ratebut also between the service level and the lead-time reductionin a quantitative point of view and we futurely explore thechanges of the service level impact on the revenue of thecentralized supply chain decentralized supply chain and itsplayers

More directly related to the present work is the literatureon RFID technology adoption in the IoT environment IoTsemantically means ldquoa worldwide network of interconnectedobjects uniquely addressable based on standard communi-cation protocolsrdquo For the management of supply chain underIoT the related literature about adoption of RFID technologyconsidered either the inventory inaccuracy elimination orlead-time reduction separately

In the Table 1 we can easily find the difference betweenthis paper and the most of existing literature The researchconcentrates on adoption of RFID technology to solve thesupply chainrsquos inventory inaccuracy problem as followsGaukler et al [18] assumed that the retailer acted as ifit were not aware of the inventory inaccuracy problemThis approach greatly simplifies the mathematical analysisreducing it to the level of a newsvendor problem In theirmodel the retailer is conscious of the uncertaintywith respectto inventory and adjusts its orders accordingly such that itsexecution problems also affect the manufacturer Rekik et al[3 4 19] performed several analyses in order to quantify theimpact of inventory inaccuracy on inventory managementThey defined separately the factors for inventory inaccuracysuch as misplaced items damage and spoilage and supplyerrors They dealt with each factor separately and analyzedthe effect of RFID technology on supply chain inventorythrough a newsvendor model They considered RFID to bea perfect technology that can eliminate all errors Camdereliand Swaminathan [20] considered the case where a fixed andknown proportion of the retailerrsquos order quantity becomesunavailable for sale due to misplacement at the retailer Allmisplaced products were recovered and salvaged at the endof the period They analyzed the impact of such proportionalloss on the retailerrsquos optimal order quantity They comparedthe performance of a decentralized supply chain with thatof a vertically integrated supply chain and investigated howcoordination can be achieved by means of buy-back andrevenue-sharing contracts Rekik and Sahin [6]modeled a setof scenarios depending on the technology available to trackshrinkage in the store withwithout RFIDThey assumed thatan inspection process is performed at a regular frequency of119873 selling periods The deployment of the RFID technologyproduces two benefits total visibility of the shrinkage rateand the elimination of shrinkage errors A comparison ofthe scenarios enables us to evaluate the economic impactof inventory record inaccuracies which can be significantparticularly in systems with a poor estimation of the errorparameter as well as with a high inspection cost

Only a few studies have been undertaken theoreticallyon reducing lead-time by adoption of RFID technology Forexample Sari [21] constructed a four-echelon supply chainsimulation model to verify the impact of RFID technologyon supply chain performance He concluded that integrating


Table1Difference

betweenthispapera

ndthee

xistingliterature

Service

level

Lead-time

Misp

laced

inventory

Sing

leperio

dCentralized

supp

lychain

Decentralized

supp

lychain

Coo

rdination

Mod

elQuantitativ

eresearch

Gaukler

etal[18]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Rekiketal[3419]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Sari[21]

lowastlowast

lowast(RS)

CamdereliandSw

aminathan[20]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

RekikandSahin[6]

lowastDyn

amicgame

lowast

ZhangandHu[22]

lowastlowast

lowastNew

svendo

rmod

ellowast

Fanetal[23]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Our

paper

lowastlowast

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast


RFID within the supply chain provides greater benefitsbased on the intensive collaboration between the partnersHe found RFID adoption to be the most beneficial inenvironments with longer lead-times and limited demanduncertainty The supply chain could use RFID systems toreduce lead-times thus increasing productsrsquo total usefulshelf life Zhang and Hu [22] studied the effect of lead-time compression on profit of supply chain players on theassumption that the market demand forecast error varieswith lead-time based on VMI Using a newsvendor modelthey discussed the constraint condition of RFID technologycost that can improve the profit of supply chain Fan et al[23] supposed the RFID can reduce the lead-time of thesupply chain and showed the revenue sharing contract cancoordinate the supply chain with RFID technology and theyfound that the interval of contract parameters would shrinkthe retailerrsquos optimal order quantity would reduce and thesupplierrsquos wholesale prices would rise with adoption of RFIDtechnology

The above authors have discussed the inventory inac-curacy or lead-time expected impact of RFID technologyon a supply chainrsquos profit however quantitative assessmentsof the economical results of RFID adoption not only ineliminating the misplacement but also in reducing lead-times for ordering products are limited in the existingliterature Our work contributes to the literature in that itjointly considers these two factors in the supply chain andit has presented the perspective of service level in bothcentralized and decentralized supply chains stemming frommisplaced inventory and lead-time withwithout consider-ation of deploying RFID technology Moreover as a newtechnology the cost of RFID tag limits its wide applicationand supply chains lack a model on which to base the decisionof whether or not to adopt RFID technology In contrast ourwork specifically focuses on the discussion of RFID tag costthresholds that can improve the profit of both the centralizedand decentralized supply chains with different service levelsand we find that a linear transfer payment coefficient strat-egy can make the players of decentralized supply chain toachieve a Pareto improvement

3 Model Assumptions and Notations

We consider a supply chain where a supplier produces asingle seasonal product and sells it to a retailer The retailersells the product in a store to end consumers The retailerrsquosordering decision is made within a one-period newsvendorframework Thus as the information flows from retailer tosupplier the product flows in the reverse direction

We assume thatmisplacement of inventory occurs imme-diately on delivery Hence as soon as the order is complete119897 (119897 = 1 minus 120572) proportion of the order gets misplaced in thestore and only 120572 (0 lt 120572 lt 1) proportion of the orderis available to the end customers during the selling seasonCustomers or employees can misplace items on the shopfloor or items can be misplaced in storage areas In mostreal life settings misplacement happens gradually over timeTherefore according to the retailerrsquos previous sales experience

and statistics a quantity of product 119902 ordered from the sup-plier can be divided into two portions (1) 120572119902 is the portionavailable for sale to consumers and (2) 119897119902 is the portionwhichis not available for sale due to misplacement In this paperwe assume that at the end of the period all misplaced itemswill be found and will have a residual value The adoption ofthe RFID technology has the potential to improve operationalprocesses and provide real-time information update due toIoTrsquos transparency and traceability we assume that eachSKU receives an RFID tag so the availability proportionbecomes 100 (120572 = 1) Since the implementation of RFIDtechnology in supply chains is relatively new the efficiencyrate of the readers can be below 100 in some instancesThiscan also be incorporated into our model easily by assumingthat availability goes to 120572 lt 1 instead of 1

We can easily understand the lead-time compressionwithRFID technology in the inset of Figure 1 Setting 119883 is thedemand within the period [0 119879] which is forecasted bythe retailer before purchase time We assume it follows anormal distribution whose normal density and distributionfunction are denoted by respectively 120601(119909) and Φ(119909) Thetime from the retailerrsquos order to receipt of the product is 119879hence 119879 is the lead-time of the supply chain If the supplychain does not adopt RFID technology the retailer needsto order product at time 0 The lead-time of the supplychain is 119879 at this moment and the demand forecast is 119883 sim

119873(120583 1205902

0) On the contrary if the supply chain establishes

a new business model under IoT environment an RFIDtag will be placed on each product The adoption of theRFID technology has the potential to increase the automationoperational processes reduce the time of manual operationsand improve the efficiency of manufacturing transportationstorage and other links the retailer can order at time 119905 whichmeans the lead-time is shortened to 119879 minus 119905 (0 le 119905 le 119879)where 119905 is the amount of lead-time compression with RFIDtechnology At this moment the demand forecast is 119883 sim

119873(120583 1205902

119905) With lead-time compression more information

about product sales and demands will be available to theretailer so themarket demand forecast will be more accurateWe set that once the order time 119905 has been decided thedemand within [0 119879] period forecasted is 120583 and the forecasterror standard deviation is 120590

119905= 1205900+ ((120590119879minus 1205900)119879)119905 In fact

Chen and Chuang [13] use a similar form to this demandvariance in their simulation study 1205902

0and 1205902119879are the variance

of market demand at time 119905 = 0 and at time 119905 = 119879 We knowthat the variance 120590 will decrease with the shortening of lead-time so the assumption of 120590

119879lt 1205900is established 119891(119909 0) and

119891(119909 119905) are the probability density function of119883 at time 0 andtime 119905 respectively and119865(119909 0) and119865(119909 119905) are the cumulativedensity function of119883 at time 0 and time 119905 respectively

The notations used throughout this paper are as follows119901 the unit product selling price 119904 the retailerrsquos shortage (lostsales) cost per unit 119903 the retailerrsquos inventory cost per unit119898the unit production cost 119908 the unit wholesale price 119888 theunit cost of RFID technology 119902 the market demand of theretailers predicted V the retailerrsquos salvage value per unit

In this paper we assume that the service level of thesupply chain (the available buying possibility of consumers)


Lead-time of the supply chain

Lead-time reductionwith RFID

Lead-time of the supplychain with RFID

0 t T

When the supply chain does notadopt RFID the retailer needs to

order product at time 0

When the supply chain adoptsRFID the retailer can order

product at time t

The retailer receives product at time T

Sales period

without RFID

Figure 1 The lead-time reduction with adoption of RFID technology

is greater than 05 in both the centralized and the decen-tralized cases withwithout consideration of deploying RFIDtechnology If the service level is less than 05 it means thatthe obtained rate of product is low and the supply chain needsto reorganize its practices

4 Centralized Supply Chain

This section studies the centralized supply chain with asupplier and a retailer It tackles the question of how tomaximize the overall profit of the supply chain throughdecision-making We study the profit of the supply chainwithwithout RFID technology and determine the cost-of-tag threshold at which RFID technology investment becomesprofitable with different service levels

41 Without Adoption of RFID Technology When the cen-tralized supply chain does not adopt RFID technology theretailer has to order 119902 at time 0 However portion 119897119902

is misplaced The centralized supply chainrsquos profit 120587119862(119902 0)

without adoption of RFID technology will be given by

120587119862(119902 0) = 119901 [int

120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909

minus 119904int+infin

120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119898119902

(1)

In (1) the first and second terms are the expected profit ofthe centralized case the third term is the price-cutting saleslosses the fourth term is the out-of-stock losses the fifth termis the residual value of misplacements and the sixth term isthe cost of manufacture

Performing first and second order differential on 119902 in (1)it can be derived that

120597120587119862(119902 0)

120597119902= 120572 [119901int

+infin

120572119902

119891 (119909 0) 119889119909 minus 119903int120572119902

0

119891 (119909 0) 119889119909

+ 119904int+infin

120572119902

119891 (119909 0) 119889119909] + 119897V minus 119898

= 120572 (119901 + 119904 minus V) + V minus 119898 minus 120572 (119901 + 119904 + 119903) 119865 (120572119902 0)

1205972120587119862(119902 0)

1205971199022= minus1205722(119901 + 119904 + 119903) 119891 (120572119902 0) lt 0

(2)

Through (2) it can be derived that profit function120587119862(119902 0)

is concave in 119902 Therefore the optimum 119876lowast1198620

of (1) is120597120587119862(119876lowast1198620 0)120597119902 = 0 The optimal order quantity can be

derived as follows

119865 (120572119876lowast

1198620 0) =

120572 (119901 + 119904 minus V) + V minus 119898

120572 (119901 + 119904 + 119903) (3)

Through (3) it can be derived that the misplacementcauses the retailer to increase its optimal order quantityin order that 120572119876lowast

1198620can satisfy the needs of the customers

because 119865(119909 0) ge 0 so 119865(120572119876lowast1198620 0) ge 0 We can get Lemma 1

Lemma 1 Given 119876lowast1198620

gt 0 there exists a threshold value120572 isin [119881

120572 1] such that the profits of the centralized supply chain

120587119862(120572119876lowast1198620 0) gt 0 Otherwise if and only if 120572 isin [0 119881

120572] will

make the profits of the centralized supply chain 120587119862(120572119876lowast1198620 0) lt

0 then the optimal strategy is not to place an order (119881120572

=

(119898 minus V)(119901 + 119904 minus V))

Because 119883 sim (120583 1205902) ((119883 minus 120583)120590) sim 119873(0 1) that is119865(120572119876lowast1198620 0) = Φ((120572119876lowast

1198620minus 120583)120590

0) we can derive the optimal

order quantity of the centralized supply chain 120572119876lowast1198620

= 120583 +

119885(1198781198620)1205900 According to Iyer and Bergen [12] the optimal

service level of the centralized supply chain without RFIDtechnology is defined as 119878

1198620= Φ((120572119876lowast

1198620minus 120583)120590

0) which

means the available buying possibility of consumers with theassumption in Section 2 obviously 05 le 119878

1198620le 1 119885(119878

1198620)


is the119885 value of a standard normal distribution that generatesa cumulative probability of 119878

1198620 Φ(119909) is the monotonically

increasing function We know that the more the numberof misplaced inventory the smaller the bounded randomvariable 120572 and the longer the lead-time the bigger thedemand forecast error 120590

0 From the service level function

1198781198620

= Φ((120572119876lowast

1198620minus 120583)120590

0) we can derive Lemma 2

Lemma 2 In the centralized supply chain the service levelincreases with the decrease of the misplaced inventory rate butwith the increase of the lead-time reduction

By substituting the optimal order quantity into (1) theconclusion can be drawn that the maximum expected profit120587lowast

119862(120572119876lowast1198620 0) of the centralized supply chain without adoption

of RFID technology is

120587lowast

119862(120572119876lowast

1198620 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(4)

42 With Adoption of RFID Technology When the supplieradopts RFID technology then the cost of each productchanges from 119898 to 119898 + 119888 The retailer can use the radiofrequency (RF) reader to get the accurate inventory and orderproduct at time 119905 We can derive the profits of the centralizedsupply chain 120587

119862(119902119905)with adoption of RFID technology as

follows

120587119862(119902 119905) = 119901 [int

119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus (119898 + 119888) 119902

(5)

In (5) the first and second terms are the expected profitsof the centralized case the third term is the price-cutting saleslosses the fourth term is out-of-stock losses and the fifthterm is the cost of production and RFID technology

Doing first and second order differential on 119902 in (5) it canbe derived that

120597120587119862(119902 119905)

120597119902= 119901int

+infin

119902

119891 (119909 119905) 119889119909 minus 119903int119902

0

119891 (119909 119905) 119889119909

+ 119904int+infin

119902

119891 (119909 119905) 119889119909 minus 119898 minus 119888

= (119901 + 119904 minus 119898 minus 119888) minus (119901 + 119904 + 119903) 119865 (119902 119905)

1205972120587119862(119902 119905)

1205971199022= minus (119901 + 119904 + 119903) 119891 (119902 119905) lt 0

(6)

Through (6) it can be derived that 120587119862(119902 119905) is the con-

cave function of 119902 therefore the optimum 119876lowast119862119905

of (5) is

120597120587119862(119876lowast119862119905 119905)120597119902 = 0 The optimal order quantity can be

derived as follows

119865 (119876lowast

119862119905 119905) =

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903 (7)

We can derive Lemma 3 from (7)

Lemma 3 Given 119876lowast

119862119905gt 0 there exists a threshold value

119888 isin [0 119880120574] such that the profits of the centralized supply chain

120587119862(119876lowast

119862119905 119905) ge 0 Otherwise if and only if 119888 isin [119880

120574 +infin] will

make the profits of the centralized supply chain 120587119862(119876lowast119862119905 119905) lt 0

then the optimal strategy is not adoption of RFID technology(119880120574= 119901 + 119904 minus 119898)

Because 119883 sim (120583 1205902) so ((119883 minus 120583)120590) sim 119873(0 1) that is119865(119876lowast

119862119905 119905) = Φ((119876

lowast

119862119905minus 120583)120590

119905) we can derive the optimal order

quantity 119876lowast119862119905

= 120583 + 119885(119878119862119905)120590119905 In this section the service level

is 119878119862119905

= Φ((119876lowast119862119905minus 120583)120590

119905) = (119901 + 119904 minus119898minus 119888)(119901 + 119904 + 119903) and we

can derive Lemma 4

Lemma 4 In centralized supply chain the service levelincreases with the decrease of the unit cost of RFID tag


119862(119876lowast119862119905 119905) of the centralized supply chain with adoption of

RFID technology is

120587lowast

119862(119876lowast

119862119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904(8)

43TheDifference betweenwith andwithout RFIDTechnologyin Service Level of the Centralized Supply Chain By compar-ing the service level values of 119878

1198620and 119878

119862119905 we can easily see

the difference between with and without RFID technology inservice level of the centralized supply chain inTheorem 5

Theorem5 The following findings hold for normal distributeddemand Only with the normal sales rate 120572 isin [119881

120572 1] will

the centralized supply chain operate without adoption of RFIDtechnology

(1) When the unit cost of RFID technology is 119888 lt 119880120572

adoption of the technology can enhance the service levelof the centralized supply chain namely 119878

119862119905gt 1198781198620(U120572=

(l120572)(m minus v))(2) When the unit cost of RFID technology is119880

120572le 119888 le 119880

120574

adoption of the technology can lower the service level ofthe centralized supply chain namely 119878

119862119905le 1198781198620

When the centralized supply chain adopts RFID tech-nology it can increase the service level through eliminatingthe misplaced inventory and reducing the lead-time but theextra cost of RFID tag will also decrease the service levelThe supply chain will choose the appropriate service level tobalance the relationship between the shortage cost and price-cutting sales cost If the RFID tag cost is in a range suchas 119888 isin [0 119880

120572) the service level of the supply chain can be


improved and the supply chain can maintain or lower theservice level according to its actual situation to get a largerprofit But when the RFID tag cost is in the other range suchas 119888 isin [119880

120572 119880120574] the positive effect on service level caused by

misplaced inventory elimination and the lead-time reductioncannot balance the negative effect of the over high RFIDtag cost in this situation the service level will decrease Theservice level can be maintained or enhanced according to theactual operations of the centralized supply chain

44 Decision-Making on RFID Technology Investment of theCentralized Supply Chain For the centralized supply chainwith a supplier and a retailer there are two choices

(1) Without adoption of RFID technology the supplychain can avoid its cost However it will suffer theloss caused bymisplaced inventory and too long lead-times of supply chain

(2) With adoption of RFID technology lead-times canbe shortened and misplaced inventory can be elim-inated However the supply chain must be able toafford the cost of RFID technology

We can get the decision-making on RFID investment inthe centralized supply chain fromTheorem 6

Theorem6 The following findings hold for normal distributeddemand Only with the threshold value 120572 isin [119881

120572 1] will the

centralized supply chain operate without adoption of RFIDtechnology

(1) When the service level of the centralized supply chainis enhanced with RFID technology the profit of thecentralized case will increase if and only if c le

min(U120572U120573) Otherwise it decreases (U

120573= (l120572)(mminus

v) + ((p + s + r)[1205900120601(Z(SC0)) minus 120590t120601(Z(SCt))]120583))

(2) When the service level of the centralized supply chainis lowered with RFID technology the profit of thecentralized case will increase if and only if U

120572le c le

min(U120573U120574) Otherwise it decreases

The findings of Theorems 5 and 6 indicate that thecentralized supply chain has more incentives to adopt thenew technology as the mean of service level or profit or bothincreasesHence Figure 2 shows if theRFID tag price in someregions the greatest benefits from RFID technology accrueto centralized supply chain under low inventory availabilityunder long lead-time and in expectation of higher servicelevel and profit And we also find the trends of service leveland profits are different with different RFID tag prices

5 Decentralized Supply Chain

This section studies the decentralized supply chain witha supplier and a retailer We assume that the supplier asa Stackelberg leader announces the wholesale price 119908 tothe retailer and the retailer as the follower orders productquantity taking into consideration the wholesale price andthe forecast demand of the market before season We study

the profit of each supply chain player withwithout deployingRFID technology and determine the cost-of-tag threshold atwhich RFID technology investment becomes profitable withdifferent service levels

51 Without Adoption of RFID Technology When the decen-tralized supply chain does not adopt RFID technology theretailer has to order 119902based on thewholesale price119908 at time 0However the item product 119897119902 is misplaced Taking 120587

119863119903(119902 0)

120587119863119904(119902 0) and 120587

119863(119902 0) as the expected profit of the retailer

supplier and decentralized supply chain without adoption ofRFID technologywill be given by

120587119863119903

(119902 0) = 119901 [int120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119908119902

(9)

120587119863119904

(119902 0) = (119908 minus 119898) 119902 (10)

120587119863(119902 0) = 120587

119863119903(119902 0) + 120587

119863119904(119902 0) (11)

Based on the same idea as shown in Section 41 theoptimal order quantity 119876lowast

1198630of (9) can be derived as follows

119865(120572119876lowast

11986300)

=120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903) (12)

Through (12) it can be derived that the misplaced inven-tory causes the retailer to increase its optimal order quantityin order that 120572119876lowast

1198630can satisfy the needs of the consumers

because 119865(1199090)

ge 0 so 119865(120572119876lowast

11986300)

ge 0 We can get the retailerrsquosorder strategy from Lemma 7



120572 1] such that the retailerrsquos profit 120587

119863119903(120572119876lowast1198630 0) ge 0

Otherwise if and only if 120572 isin [0 1198811015840120572] will make the retailerrsquos

profit 120587119863119903(120572119876lowast1198630 0) lt 0 then the optimal strategy is not to

place an order (1198811015840120572= (119908 minus V)(119901 + 119904 minus V))

We can derive the optimal order quantity of the decen-tralized supply chain 120572119876lowast

1198630= 120583 + 119885(119878

1198630)1205900 Obviously the

optimal service level of the decentralized supply chainwithoutRFID technology is 119878

1198630= Φ((120572119876lowast

1198630minus120583)120590

0) (05 le 119878

1198630le 1)

119885(1198781198630) is the 119885 value of a standard normal distribution that

generates a cumulative probability of 1198781198630 From the service

level function 1198781198630

= Φ((120572119876lowast1198630

minus 120583)1205900) we can derive

Lemma 8

Lemma 8 In the decentralized supply chain the service levelincreases with the decrease of the misplaced inventory rate butwith the increase of the lead-time reduction

As the retailerrsquos optimal order quantity determines theorder of the decentralized supply chain substituting theoptimal order quantity into (9) (10) and (11) we have the


IncreaseDecrease

Service level

Supply chainrsquos profit

Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c

Figure 2 Service level and profit changes of the centralized supply chain with different costs of RFID tag

maximum expected profit of the retailer supplier and decen-tralized supply chain without adoption of RFID technologyrespectively recorded as 120587

lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)

52 With Adoption of RFID Technology When the supplieradopts RFID technology then the cost of each productchanges from 119898 to 119898 + 119888 The retailers can order at time 119905taking 120587

119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587

119863(119902 119905) as the expected profit

of the retailer supplier and decentralized supply chain wehave

120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)


Lemma 9 The optimal order quantity of the decentralizedsupply chain with RFID technology is always greater than 0that is 119876lowast

119863119905gt 0 such that the retailerrsquos profit 120587

119863119903(119876lowast119863119905 119905) ge 0

then the optimal strategy is pushing its supplier to adopt RFIDtechnology

The optimal order quantity of the decentralized supplychain with RFID technology is 119876lowast

119863119905= 120583 + 119885(119878

119863119905)120590119905 In this

section the service level is 119878119863119905

= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus

119908)(119901 + 119904 + 119903) we can derive Lemma 10

Lemma 10 In the decentralized supply chain the unit cost ofRFID tag does not affect the service level

As the retailerrsquos optimal order quantity determines theorder of the decentralized supply chain substituting the opti-mal order quantity into (16) (17) and (18) we have the max-imum expected profit of the retailer supplier and decentral-ized supply chain with adoption of RFID technology respec-tively recorded as 120587lowast

119863119903(119876lowast119863119905 119905) 120587lowast119863119904(119876lowast119863119905 119905) and120587lowast

119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904 + (119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)

(22)

53 The Difference between with and without RFID Technol-ogy in Service Level of the Decentralized Supply Chain Bycomparing the service level values of 119878

1198630and 119878

119863119905 we can

easily find the difference between with and without RFIDtechnology in service level of the decentralized supply chaininTheorem 11

Theorem 11 The following findings hold for normal dis-tributed demand Only with the normal sales rate 120572 isin

[1198811015840

120572 1] will the decentralized supply chain operate without

adoption of RFID technology When the supplier adopts theRFID technology in the decentralized supply chain the servicelevel will be enhanced

RFID technology will increase the retailerrsquos service levelby eliminating the misplaced inventory and reducing thelead-time but the supplier undertakes the extra cost of RFIDtag thus the service level will increase Then the retailercan choose to maintain or probably reduce its service levelaccording to the actual situation to get more benefits

54 Decision-Making on RFID Technology Investment of theDecentralized Supply Chain

541 The Impact of Adopting RFID Technology on the Profit ofthe Decentralized Supply Chain

Proposition 12 When the normal sales rate 120572 isin [1198811015840

120572 1]

and the unit cost of RFID tag 119888 isin (0 1198801015840120573) the profit of

the decentralized supply chain will increase with adoptionof RFID technology Otherwise it decreases (1198801015840

120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])

542 The Impact of Adopting RFID Technology on theRetailerrsquos Profit

Proposition 13 When the profit of the decentralized supplychain increases with adoption of RFID technology the retailerrsquosprofit will always increase Otherwise it decreases

543 The Impact of Adopting RFID Technology on the Sup-plierrsquos Profit

Proposition 14 When the normal sales rate 120572 isin [119881120583 1] and

the unit cost of RFID tag 119888 isin (0 1198801015840

120572) the supplierrsquos profit will

increase Otherwise it decreases (1198801015840120572= ((120583119897120572)(119898minus119908) + (119908minus

119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))

From the above three propositions we find that theretailer is better off if and only if the decentralized supplychain is better off with adoption of RFID technology bysupplier As a consequence when the supplier bears the costof RFID tag the retailer gets the benefits from ldquospillovereffectrdquo of misplacement elimination and lead-time compres-sion There are three cases (1) both the supplier and theretailer benefit from an investment in RFID (2) only theretailer benefits from an investment in RFID but the profitof decentralized supply chain also increases and (3) no partybenefits from an investment in RFID

We can get the decision-making on RFID investment inthe decentralized supply chain fromTheorem 15

Theorem 15 The following findings hold for normal dis-tributed demand

(1) When the normal sales rate 120572 isin [119881120583 1] and the

unit cost of RFID tag 119888 isin (0 1198801015840120572) each playerrsquos profit

will increase with adoption of RFID technology in thedecentralized supply chain

(2) When the normal sales rate 120572 isin [119881120583 1] and the unit

cost of RFID tag 119888 isin (1198801015840120572 1198801015840120573) or when the normal sales

rate 120572 isin [1198811015840120572 119881120583) and the unit cost of RFID tag 119888 isin

(0 1198801015840120573) the profit of the decentralized supply chain and

the retailer increases while the supplierrsquos profit reduceswith adoption of RFID technology The linear transferpayment coefficient strategy can coordinate and achievea Pareto improvement in this case

In Figure 3 we can find that the trends of service leveland profit are different with different RFID tag prices andunderstand the above analyses more clearly

55 Coordination According to the Theorem 15 we find aninteresting phenomenon that the retailerrsquos profit increaseswhile the supplierrsquos profit decreases with adoption of RFIDtechnology but the profit of the decentralized supply chainalso increases The reasons of the supplierrsquos profit decreasewith RFID technology as follows firstly the supplier paysthe cost of RFID tag secondly with the misplaced inventoryelimination and the lead-time reduction the retailerrsquos optimalorder quantity 119876lowast

119863119905= 120583 + 119885(119878

119863119905)120590119905will decrease In order to

prevent the risk of supply-strand breaking and promote thesupplier adopting RFID technology to increase the servicelevel and profit of the decentralized supply chain we shouldcoordinate this phenomenon with a contract In this paperwe find that the linear transfer payment coefficient strategycan be used by the retailer to coordinate each player in thedecentralized system with RFID

Assume that the retailer transfers a part of the increasingprofit 120579Δ

119863119903(0 le 120579 le 1) to the supplier compensating its

losses caused by adoption of RFID technology If not onlythe retailerrsquos profit is incremented by (1 minus 120579)Δ

119863119903ge 0 but

also the supplierrsquos profit is incremented by Δ119863119904

+ 120579Δ119863119903

ge 0


Retailerrsquos profitSupplierrsquos profit

0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c

Normal sales rate 120572 isin [V120583 1]

Normal sales rate 120572 isin [V998400120572 V120583]

IncreaseDecrease

Figure 3 Service level and profit changes of the decentralized supply chain with different cost of RFID tag

the players can achieve a Pareto improvement We come tothe conclusion as follows

Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)

From (23) we can derive the coordination rules inTheorem 16


(1) When the normal sales rate 120572 isin [0 119881120583) thus 119876lowast

119863119905lt

119876lowast1198630 the coefficient of linear transfer payment should

satisfy 120579 isin [119880120579 1] then each playerrsquos profit will increase

with adoption of RFID technology (119880120579

= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))

(2) When the normal sales rate 120572 isin [119881120583 1] thus 119876lowast

119863119905ge



with adopting RFID technology

Thus both partied achieve a Pareto improvement andthe linear transfer payment coefficient strategy will ensurea coordination of the decentralized system with adoption ofRFID technology

6 Numerical Examples

To illustrate the proposed models we give numerical exam-ples as follows According to Camdereli and Swaminathan[20] as well as Chen and Chuang [13] we suppose 120583 = 5000

units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6

dollarsper unit 119904 = 4 dollarsper unit 119903 = 1 dollarperunit 119898 = 2 dollarsper unit 119908 = 3 dollarsper unit andV = 05 dollarper unitThe simulation data used for carryingout these numerical computations are assumed to representreal-world conditions as closely as possible

61TheDifference between with and without RFID Technologyin Service Level of the Supply Chain FromFigures 4 and 5 weknow that the unit price of RFID tag will influence the servicelevel in the centralized supply chain but it does not have effecton the service level in the decentralized supply chain And wecan directly demonstrate the conclusions of the Theorems 5and 11

62 Impact of Misplaced Inventory and Lead-Time on theMaximum Costs of RFID Tag Consider the supply chaincan bear maximum costs of RFID tag under the lead-timereduction 119905 isin [0 10] with three different normal sales ratessuch as 120572 = 08 120572 = 09 and 120572 = 1 According to theTheorems 6 and 15 we get Figures 6 and 7 with Matlab70We find that for both of the centralized supply chain andthe decentralized supply chain the supply chain can bear themaximum costs of RFID tag increase with the decrease ofthe normal sales rate but with the increase of the lead-timereduction

Then we compare Figure 6 with Figure 7 and find thatthe centralized case is more efficient than the decentralizedcase We find that the maximum costs of RFID tag which thecentralized case bearing is muchmore than the decentralizedcase bearing when the normal sales rate and the lead-timeare both same Now the RFID tag price is still expensivethe centralized case is more suitable for adopting RFIDtechnology If the supply chain has acapital constraint in


0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572

Figure 4 Service level change of the centralized supply chainwithwithout RFID technology

0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w

Figure 5 Service level change of the decentralized supply chainwithwithout RFID technology

adoption of RFID technology there is a solution that turnsthe supply chain into the centralized case

63 Impact of Adopting RFID Technology on the Playersrsquo Profitof the Decentralized Supply Chain Suppose the lead-timereduction is 119905 (from 0 to 10) and the unit cost of RFID tag is119888 = 02 we investigate the each playerrsquos profit variation withadoption of RFID technology by the decentralized supplychain with three different normal sales rates such as 120572 = 08120572 = 09 and 120572 = 1 which are shown in Figure 8 Wefind that each playerrsquos profit increment increases with thedecrease of the normal sales rate but with the increase ofthe lead-time reduction And we can directly demonstrate

0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10

Figure 6Themaximum costs of RFID tag in the centralized supplychain

0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10

Figure 7 The maximum costs of RFID tag in the decentralizedsupply chain

the conclusions of Proposition 13 if the profit increment ofthe decentralized supply chain is greater than zero thenthe retailerrsquos profit increment is always greater than zeroThen we have found that when the profit increment of thedecentralized supply chain is greater than zero while thesupplierrsquos profit increment is less than zero the linear transferpayment coefficient strategy can coordinate the supply chain

64 Coordination We assume the unit cost of RFID tag is119888 = 02 the normal sales rate is 120572 = 08 and the lead-timereduction is 119905 = 5 Thus 119881

120583= 0991 gt 120572 = 08 so we

take the coefficient of linear transfer payment 120579 isin (119880120579 1) into

consideration (the case that the services level 120572 isin [119881120583 1] and

coefficient of linear transfer payment 120579 isin (0 119880120579) can be solved

in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)

Figure 8 Impact of adopting RFID technology on the profit of decentralized supply chain and its players

With the coefficient of linear transfer payment 120579 isin [0 1]the profit variation of the decentralized supply chain andits members are shown in Table 2 We find that when thecoefficient of linear transfer payment 120579 ge 03403 eachplayerrsquos profit will increase at the same time with adoptionof RFID technology by the decentralized supply chain Thenthe decentralized case achieves Pareto improvement

7 Conclusions

As a new technology RFID can solve many problems insupply chain management However its cost limits its wideapplication and supply chains lack a model on which to basethe decision of whether or not to adopt RFID technologyThis paper considers a two-echelon supply with a supplierand a retailer which suffers from the misplaced inventoryand the lead-time and assumes that the market demandforecast error varies with lead-time reduction Consideringthe fact that RFID technology can eliminate the misplacedinventory and compress the lead-time of a supply chain wepresent a perspective of service level and use a newsvendormodel to study the difference of service level and profit ofthe centralizeddecentralized supply chains with adoption ofRFID technology

Our results first showed that in both centralized anddecentralized supply chain the service level not onlyincreases with the decrease of the number of misplacedproduct but also increases with the increase of the lead-timereduction

Secondly there is one distinguishing feature of our studythat we validated the adoption of RFID technology affectingthe service level of the supply chain We find that whetheradoption of RFID improves the service level depends on the

cost of the RFID tag in the centralized supply chain but itimproves the service level in the decentralized case regardlessof the cost

Thirdly there exist the same cost-of-tag thresholds withdifferent service levels in both the centralized and the decen-tralized cases Furthermore we prove that the linear transferpayment coefficient strategy can coordinate the decentralizedsupply chain with adoption of RFID technology while theprofit of supply chain and retailer increase but supplierrsquos profitreduces

It would be interesting to study the problem of technol-ogy adoption in multiple sales periods incorporating pricemarkdowns between periodsMoreover adding the fixed costof the RFID technology in our paper is also a meaningfulquestion But the cost sharing of the RFID tag is not usefulwhen there is a game between the players in the decentralizedsupply chain the supplier will transfer the addition RFID costthrough the wholesale price to the retailer (like [24])

Appendices

A Derivation of (4)By substituting the optimal order quantity into (1) theconclusion can be drawn that the maximum expected profit120587lowast119862(120572119876lowast1198620 0) of the centralized supply chain without adoption

of RFID technology is as follows

120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


Table 2 The profit variation of the decentralized supply chain and its members with coefficient of linear transfer payment 120579 (from 0 to 1)

120579The retailer The supplier The supply chain

120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394

minus 119898119876lowast

1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)

times (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)

B Derivation of (8)By substituting the optimal order quantity into (5) theconclusion can be drawn that the maximum expected profit

120587lowast


RFID technology is as follows

120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904

= (119901 + 119904 minus 119898 minus 119888) (120583 + 119885 (119878119862119905) 120590119905) minus 120590119905(119901 + 119904 + 119903)

times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904

= 120583 (119901 + 119904 minus 119898 minus 119888) + 119885 (119878119862119905) 120590119905(119901 + 119904 minus 119898 minus 119888)

minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)

Proof of Theorem 6 In order to study the influence of theadoption of RFID technology on the profit of the central-ized supply chain we determine the decision strategy foradoption of RFID technology by comparing profit modelsof the supply chain withwithout RFID technology If Δ

119862

stands for the profit variation of the centralized supply chain


withwithout consideration of deploying RFID technology itcan be derived that

Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)

We make a function 119910119862

= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905))] then we discuss it in two

different conditions

(A) When the service level is 05 le 1198781198620

lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When

the cost of RFID tag is 119888 le min119880120572 119880120573 the overall

profit of the supply chain can be increased

(B) When the service level is 05 le 119878119862119905

le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590

0gt 120590119905gt 0 we cannot

determine whether 1205900120601(119885(119878

1198620)) or 120590

119905120601(119885(119878

119862119905)) is

largerWhen 120601(119885(119878119862119905)) le ((1minus120572)120583120572(119901+119904+119903)120590

119905)(119898minus

V) + (1205900120590119905)120601(119885(119878

1198620)) it can be ensured that 119910

119862ge 0

when the cost of adopting RFID technology is 119880120572le

119888 le min(119880120573 119880120574) thus the overall profits of the supply

chain will increase

C Derivation of (13)

By substituting the optimal order quantity into (9) we havethemaximumexpected profit of the retailer without adoptionof RFID technology

120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)

D Derivation of (20)By substituting the optimal order quantity into (16) we havethe maximum expected profit of the retailer with adoption ofRFID technology

120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)

Proof of Theorem 11 By comparing the values of 1198781198630

and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908

119901 + 119904 + 119903minus120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)

According to (a) we can find that the service levelwill be improved with adoption of RFID technology by thedecentralized supply chain that is 119878

119863119905ge 1198781198630


Proof of Proposition 12 IfΔ119863stands for the profit variation of

the decentralized supply chain withwithout RFID technol-ogy it can be derived that

Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus120583119904 + (119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)

Here we define a function 119910119863

= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus

119885(1198781198630)1205900120572) we can discuss the service level in three cases

as belowWhen the service level is 05 le 119878

1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 also because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be

certain In this case the profit of supply chain willincrease when the cost of adopting RFID technology119888 le 119880

1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0

can be certain In this case the profit of supplychain will increase when the cost of adopting RFIDtechnology 119888 le 119880

1015840

120573

Proof of Proposition 13 In order to study the influence ofadopting RFID technology on retailerrsquos profit we comparethe retailerrsquos maximum excepted profit withwithout RFIDtechnology Suppose Δ

119863119903is retailerrsquos profit variation we get

Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)

When the service level is 05 le 1198781198630

lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905)) In this case the

retailerrsquos profit will increase with decentralized supply chainadopting RFID technology

Proof of Proposition 14 In order to study the influence ofadopting RFID technology on supplierrsquos profit we comparethe supplierrsquos maximum excepted profit withwithout RFIDtechnology Suppose Δ

119863119904is supplierrsquos profit variation we get

Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572) we can discuss it as below

When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)

Because 1205900ge 120590119905 we cannot determine whether 119885(119878

119863119905)120590119905or

119885(1198781198630)1205900120572 is larger in this case

(1) When 119885(119878119863119905)120590119905ge 119885(119878

1198630)1205900120572 there are two situa-

tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881

120583 1] are essential to make

the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we

can find that 120572 isin [119881120583 1] and the cost of RFID

technology 119888 isin (0 1198801015840120572) the supplierrsquos profit

will increase Otherwise it decreases (119881119911

=

119885(1198781198630)1205900119885(119878119863119905)120590119905)

(ii) 120572 isin [119881119911119881120583) then 119876lowast

119863119905lt 119876lowast1198630 so the function

119910119863119904

lt 0 In this case adopting RFID technologywill decrease the supplierrsquos profit

(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905

lt 119876lowast1198630 so the function 119910

119863119904lt 0 In this case

the adoption of RFID technology will decrease thesupplierrsquos profit

Conflict of Interests

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

Acknowledgments

For helpful feedback and comments the authors thank thefollowing for their helpful discussions Qi (Annabelle) FengGeorge J Shanthikumar Hui-xiao Yang Yan-yi Xu andLi Zhou This work was supported by China Scholarship


Council Foundation (201306740017) the National NaturalScience Foundation of China (71171082) and the New Cen-tury Excellent Talents in University of China (NCET-11-0637) The authors also wish to acknowledge the helpfulcomments provided by anonymous referees

References

[1] E Fleisch and C Tellkamp ldquoInventory inaccuracy and supplychain performance a simulation study of a retail supply chainrdquoInternational Journal of Production Economics vol 95 no 3 pp373ndash385 2005

[2] E Sahin J Buzacott and Y Dallery ldquoAnalysis of a newsvendorwhich has errors in inventory data recordsrdquo European Journalof Operational Research vol 188 no 2 pp 370ndash389 2008

[3] 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

[4] Y Rekik E Sahin Z Jemai and Y Dallery ldquoExecution errors inretail supply chains analysis of the case of misplaced productsrdquoInternational Journal of Systems Science vol 39 no 7 pp 727ndash740 2008

[5] E Sahin and Y Dallery ldquoAssessing the impact of inventory inac-curacies within a Newsvendor frameworkrdquo European Journal ofOperational Research vol 197 no 3 pp 1108ndash1118 2009

[6] Y Rekik and E Sahin ldquoExploring inventory systems sensitiveto shrinkage-analysis of a periodic review inventory under aservice level constraintrdquo International Journal of ProductionResearch vol 50 no 13 pp 3529ndash3546 2012

[7] N DeHoratius andA Raman ldquoInventory record inaccuracy anempirical analysisrdquoManagement Science vol 54 no 4 pp 627ndash641 2008

[8] M A Sieke R W Seifert and U W Thonemann ldquoDesigningservice level contracts for supply chain coordinationrdquo Produc-tion and Operations Management vol 21 no 4 pp 698ndash7142012

[9] G M Gaukler ldquoItem-level RFID in a retail supply chain withstock-out-based substitutionrdquo IEEE Transactions on IndustrialInformatics vol 7 no 2 pp 362ndash370 2011

[10] S Whang ldquoTiming of RFID adoption in a supply chainrdquoManagement Science vol 56 no 2 pp 343ndash355 2010

[11] Z Asif andM Mandviwalla ldquoIntegrating the supply chain withRFID a technical and business analysisrdquoCommunications of theAssociation for Information Systems vol 15 no 24 pp 393ndash4262005

[12] A V Iyer and M E Bergen ldquoQuick response in manufacturer-retailer channelsrdquo Management Science vol 43 no 4 pp 559ndash570 1997

[13] M S Chen and C C Chuang ldquoExtended newsboy problemwith shortage-level constraintsrdquo International Journal of Pro-duction Economics vol 67 no 3 pp 269ndash277 2000

[14] DThiel V Hovelaque and VThi Le Hoa ldquoImpact of inventoryinaccuracy on service-level quality in (Q R) continuous-reviewlost-sales inventorymodelsrdquo International Journal of ProductionEconomics vol 123 no 2 pp 301ndash311 2010

[15] J K Jha and K Shanker ldquoSingle-vendor multi-buyer integratedproduction-inventory model with controllable lead time andservice level constraintsrdquoAppliedMathematical Modelling Sim-ulation and Computation for Engineering and EnvironmentalSystems vol 37 no 4 pp 1753ndash1767 2013

[16] P Glasserman and Y Wang ldquoLeadtime-inventory trade-offs inassemble-to-order systemsrdquo Operations Research vol 46 no 6pp 858ndash871 1998

[17] J Song and D D Yao ldquoPerformance analysis and optimizationof assemble-to-order systems with random lead timesrdquo Opera-tions Research vol 50 no 5 pp 889ndash903 2002

[18] 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

[19] Y Rekik E Sahin and Y Dallery ldquoA comprehensive analysis ofthe Newsvendor model with unreliable supplyrdquo OR Spectrumvol 29 no 2 pp 207ndash233 2007

[20] A Z Camdereli and J M Swaminathan ldquoMisplaced inven-tory and radio-frequency identification (RFID) technologyinformation and coordinationrdquo Production and OperationsManagement vol 19 no 1 pp 1ndash18 2010

[21] K Sari ldquoExploring the impacts of radio frequency identification(RFID) technology on supply chain performancerdquo EuropeanJournal of Operational Research vol 207 no 1 pp 174ndash183 2010

[22] L Zhang and J Hu ldquoEffect of using RFID technology on supplychain profit and coordination based on VMIrdquo Chinese Journalof Management vol 10 no 4 pp 590ndash596 2013

[23] T Fan L Zhang F Wu and H X Yang ldquoEffectof RFIDtechnology on lead-time compression and coordination ofsupply chains revenuerdquoChinese Journal ofManagement Sciencevol 21 no 2 pp 114ndash122 2013

[24] L Zhang and T Fan ldquoResearch on the cost-sharing of adoptingRFID technology among supply chain enterpriserdquo ChineseJournal of Management Science vol 22 no 4 pp 25ndash35 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


by MIT in the late 1990s It refers to a ldquodevices or sensorsconnected worldrdquo where objects are connected monitoredand optimized through either wired wireless or hybridsystems The Cisco Internet Business Solutions Grouphas reported that in the next 10 years the IoT is allabout the potential economic value which will reach $144trillion all over the world while the share of the Chinesemarket will reach $18 trillion accounted for 12 of thetotal in the world (httpwwwnewskingusnews-4263717-Cisco-stressed-the-potential-value-of-the-interconnection-of-all-thingshtml (April 8 2013)) In 2012 IoT of theChinese industries has been reached yen 365 billion andit is expected to reach more than yen 500 billion in 2015(httpenwikipediaorgwikiInternet of Things January2014) Radio frequency identification (RFID) technologyis the key technical points of IoT and it is a way tosolve misplaced inventory and reduce lead-time RFIDis a contactless and low-power wireless communicationtechnology which has applications in many areas of asupply chain and it has been hailed as a major innovationto enhance the efficiency of a supply chain [9] The basicpremise of RFID is that a radio frequency (RF) reader canprovide real-time communication with numerous objectsat the same time reading as many as 200 tags in a secondwithout requiring contact or a direct line of sight [10] Theseadvanced identification and communication characteristicsof RFID can provide real-time information update andimprove productsrsquo traceability and visibility among supplychains as well as increasing the automation operationalprocess reducing the time of manual operations andincreasing efficiency of manufacture transportation storageand other links so it can eliminate the misplaced inventoryand reduce lead-time in the supply chains

However RFID technology is not yet cheap enough tobecome as common as barcoding technology In recent yearsseveral major retail chains have strongly promotedmdashor evenmandatedmdashthe adoption of RFID by their suppliers mostlyat the pallet level While RFID technology is suggested toenable substantial efficiency gains at the different stages of asupply chain the associated costs are by no means negligibleFor the retail supply chain due to the huge number ofcommodities involved the price of the RFID tag is the mostimportant factor For example Asif and Mandviwalla [11]estimated that a typical consumer packaged goods supplierwould spend between $13 million and $23 million for ship-ping 50 million cases per year just for the cost of RFID tagsAs a result the substantial cost of RFID seems to prohibit itswidespread use at the item level Therefore we must considerthe cost threshold for adoption of RFID tags in the supplychain and the change of the supply chain profit and servicelevel withwithout RFID technology In the recent yearsseveral major retail chains have strongly promotedmdashor evenmandatedmdashRFID adopting by their suppliers Generally theupstream enterprise adopts RFID technology and bears itscost but the downstream enterprise gains the profit It hasbeen pointed out by the practitioners that a compensationfor the upstream enterprise is necessary to ensure the supplychain operations with RFID technology

12 Research Questions andMain Results In order to addressthe above issues while the supply chain establishes a newbusiness model under IoT environment an RFID tag hasbeen labeled on each productWe focus on the improvementsthat result from increasing the accuracy of inventory infor-mation and speeding up the processes In order to ensurethe increase of the supply chainrsquos profit and performance itis important to balance the increase of service level with thecosts of RFID tagThis paper addresses the following researchquestions

(1) How do the misplaced inventory and lead-time affectservice level in a retail supply chain RFID technologycan eliminate the misplaced inventory and reducethe lead-time but it also increases the cost of supplychain Thus what is the difference between with andwithout RFID technology in service level of central-ized and decentralized supply chains respectively

(2) What is the decision-making on RFID technologyinvestment with different service levels of centralizedand decentralized supply chains respectively

(3) Howdoes RFID technology affect the playersrsquo profit indecentralized supply chain How can we coordinateit

In spite of the significance of IoT in retail supply chainsthe extant literature does not examine the above issues Theprimary motivation of this research is to address this gap Weuse theoretical and numerical example approaches to answerthe research questions We study two different settings (1) acentralized supply chain where the supplier owns the retailerand (2) a decentralized and uncoordinated supply chain

Firstly we define the service level according to Iyer andBergen [12] and it results from the misplaced inventoryand lead-time of a two-stage supply chain with a supplierand a retailer We assume that the market demand forecasterror varies with lead-time reduction Chen and Chuang [13]also assumed this in their simulation study We respectivelyanalyze the relationship not only between service level andmisplaced inventory but also between service level and lead-time And we find that the service level increases withthe decrease of the misplaced inventory rate but with theincrease of the lead-time reduction in both centralized anddecentralized supply chains When the supply chain adoptsRFID technology the service level increases with the decreaseof the unit cost of RFID tag in centralized supply chain butthe unit cost of RFID tag does not affect the service level indecentralized supply chain

Then we theoretically analyze the cost threshold of RFIDtag that becomes profitable in different service levels ofcentralized and decentralized supply chains We find that thetrends of service level and profit are different with differentRFID tag prices but the same cost-of-tag thresholds for RFIDtechnology with different service levels exist in both thecentralized and the decentralized cases

Finally when the variable tagging cost of RFID tech-nology under a threshold improves the profit of both thedecentralized supply chain and the retailer the manufac-turerrsquos profit will be reduced In this context we prove that




2 Literature Review









Table1Difference

betweenthispapera

ndthee

xistingliterature

Service

level

Lead-time

Misp

laced

inventory

Sing

leperio

dCentralized

supp

lychain

Decentralized

supp

lychain

Coo

rdination

Mod

elQuantitativ

eresearch

Gaukler

etal[18]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Rekiketal[3419]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Sari[21]

lowastlowast

lowast(RS)

CamdereliandSw

aminathan[20]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

RekikandSahin[6]

lowastDyn

amicgame

lowast

ZhangandHu[22]

lowastlowast

lowastNew

svendo

rmod

ellowast

Fanetal[23]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Our

paper

lowastlowast

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast









119873(120583 1205902



119873(120583 1205902



119905= 1205900+ ((120590119879minus 1205900)119879)119905 In fact












0 t T




product at time t


Sales period

without RFID








120587119862(119902 0) = 119901 [int

120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119898119902

(1)



120597120587119862(119902 0)

120597119902= 120572 [119901int

+infin

120572119902

119891 (119909 0) 119889119909 minus 119903int120572119902

0

119891 (119909 0) 119889119909

+ 119904int+infin

120572119902

119891 (119909 0) 119889119909] + 119897V minus 119898


1205972120587119862(119902 0)

1205971199022= minus1205722(119901 + 119904 + 119903) 119891 (120572119902 0) lt 0

(2)




derived as follows

119865 (120572119876lowast

1198620 0) =

120572 (119901 + 119904 minus V) + V minus 119898

120572 (119901 + 119904 + 119903) (3)








120572] will



=

(119898 minus V)(119901 + 119904 minus V))


1198620minus 120583)120590



= 120583 +



1198620= Φ((120572119876lowast

1198620minus 120583)120590

0) which


1198620le 1 119885(119878

1198620)






1198781198620

= Φ((120572119876lowast

1198620minus 120583)120590






120587lowast

119862(120572119876lowast

1198620 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(4)



follows

120587119862(119902 119905) = 119901 [int

119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus (119898 + 119888) 119902

(5)



120597120587119862(119902 119905)

120597119902= 119901int

+infin

119902

119891 (119909 119905) 119889119909 minus 119903int119902

0

119891 (119909 119905) 119889119909

+ 119904int+infin

119902

119891 (119909 119905) 119889119909 minus 119898 minus 119888


1205972120587119862(119902 119905)

1205971199022= minus (119901 + 119904 + 119903) 119891 (119902 119905) lt 0

(6)



of (5) is


derived as follows

119865 (119876lowast

119862119905 119905) =

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903 (7)





120587119862(119876lowast


120574 +infin] will




119862119905 119905) = Φ((119876

lowast

119862119905minus 120583)120590




is 119878119862119905

= Φ((119876lowast119862119905minus 120583)120590


can derive Lemma 4




RFID technology is

120587lowast

119862(119876lowast

119862119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904(8)


1198620and 119878




120572 1] will




119862119905gt 1198781198620(U120572=


120572le 119888 le 119880

120574


119862119905le 1198781198620












120572 1] will the




120573= (l120572)(mminus



120572le c le







119863119903(119902 0)

120587119863119904(119902 0) and 120587



120587119863119903

(119902 0) = 119901 [int120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119908119902

(9)

120587119863119904

(119902 0) = (119908 minus 119898) 119902 (10)

120587119863(119902 0) = 120587

119863119903(119902 0) + 120587

119863119904(119902 0) (11)



119865(120572119876lowast

11986300)

=120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903) (12)



because 119865(1199090)

ge 0 so 119865(120572119876lowast

11986300)





119863119903(120572119876lowast1198630 0) ge 0





1198630= 120583 + 119885(119878



1198630= Φ((120572119876lowast

1198630minus120583)120590

0) (05 le 119878

1198630le 1)




= Φ((120572119876lowast1198630


Lemma 8




IncreaseDecrease

Service level


Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c



lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)


119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587



120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)




119863119903(119876lowast119863119905 119905) ge 0



119863119905= 120583 + 119885(119878

119863119905)120590119905 In this


= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus





119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2 Literature Review









Table1Difference

betweenthispapera

ndthee

xistingliterature

Service

level

Lead-time

Misp

laced

inventory

Sing

leperio

dCentralized

supp

lychain

Decentralized

supp

lychain

Coo

rdination

Mod

elQuantitativ

eresearch

Gaukler

etal[18]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Rekiketal[3419]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Sari[21]

lowastlowast

lowast(RS)

CamdereliandSw

aminathan[20]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

RekikandSahin[6]

lowastDyn

amicgame

lowast

ZhangandHu[22]

lowastlowast

lowastNew

svendo

rmod

ellowast

Fanetal[23]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Our

paper

lowastlowast

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast









119873(120583 1205902



119873(120583 1205902



119905= 1205900+ ((120590119879minus 1205900)119879)119905 In fact












0 t T




product at time t


Sales period

without RFID








120587119862(119902 0) = 119901 [int

120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119898119902

(1)



120597120587119862(119902 0)

120597119902= 120572 [119901int

+infin

120572119902

119891 (119909 0) 119889119909 minus 119903int120572119902

0

119891 (119909 0) 119889119909

+ 119904int+infin

120572119902

119891 (119909 0) 119889119909] + 119897V minus 119898


1205972120587119862(119902 0)

1205971199022= minus1205722(119901 + 119904 + 119903) 119891 (120572119902 0) lt 0

(2)




derived as follows

119865 (120572119876lowast

1198620 0) =

120572 (119901 + 119904 minus V) + V minus 119898

120572 (119901 + 119904 + 119903) (3)








120572] will



=

(119898 minus V)(119901 + 119904 minus V))


1198620minus 120583)120590



= 120583 +



1198620= Φ((120572119876lowast

1198620minus 120583)120590

0) which


1198620le 1 119885(119878

1198620)






1198781198620

= Φ((120572119876lowast

1198620minus 120583)120590






120587lowast

119862(120572119876lowast

1198620 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(4)



follows

120587119862(119902 119905) = 119901 [int

119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus (119898 + 119888) 119902

(5)



120597120587119862(119902 119905)

120597119902= 119901int

+infin

119902

119891 (119909 119905) 119889119909 minus 119903int119902

0

119891 (119909 119905) 119889119909

+ 119904int+infin

119902

119891 (119909 119905) 119889119909 minus 119898 minus 119888


1205972120587119862(119902 119905)

1205971199022= minus (119901 + 119904 + 119903) 119891 (119902 119905) lt 0

(6)



of (5) is


derived as follows

119865 (119876lowast

119862119905 119905) =

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903 (7)





120587119862(119876lowast


120574 +infin] will




119862119905 119905) = Φ((119876

lowast

119862119905minus 120583)120590




is 119878119862119905

= Φ((119876lowast119862119905minus 120583)120590


can derive Lemma 4




RFID technology is

120587lowast

119862(119876lowast

119862119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904(8)


1198620and 119878




120572 1] will




119862119905gt 1198781198620(U120572=


120572le 119888 le 119880

120574


119862119905le 1198781198620












120572 1] will the




120573= (l120572)(mminus



120572le c le







119863119903(119902 0)

120587119863119904(119902 0) and 120587



120587119863119903

(119902 0) = 119901 [int120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119908119902

(9)

120587119863119904

(119902 0) = (119908 minus 119898) 119902 (10)

120587119863(119902 0) = 120587

119863119903(119902 0) + 120587

119863119904(119902 0) (11)



119865(120572119876lowast

11986300)

=120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903) (12)



because 119865(1199090)

ge 0 so 119865(120572119876lowast

11986300)





119863119903(120572119876lowast1198630 0) ge 0





1198630= 120583 + 119885(119878



1198630= Φ((120572119876lowast

1198630minus120583)120590

0) (05 le 119878

1198630le 1)




= Φ((120572119876lowast1198630


Lemma 8




IncreaseDecrease

Service level


Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c



lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)


119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587



120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)




119863119903(119876lowast119863119905 119905) ge 0



119863119905= 120583 + 119885(119878

119863119905)120590119905 In this


= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus





119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table1Difference

betweenthispapera

ndthee

xistingliterature

Service

level

Lead-time

Misp

laced

inventory

Sing

leperio

dCentralized

supp

lychain

Decentralized

supp

lychain

Coo

rdination

Mod

elQuantitativ

eresearch

Gaukler

etal[18]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Rekiketal[3419]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Sari[21]

lowastlowast

lowast(RS)

CamdereliandSw

aminathan[20]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

RekikandSahin[6]

lowastDyn

amicgame

lowast

ZhangandHu[22]

lowastlowast

lowastNew

svendo

rmod

ellowast

Fanetal[23]

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast

Our

paper

lowastlowast

lowastlowast

lowastlowast

lowastNew

svendo

rmod

ellowast









119873(120583 1205902



119873(120583 1205902



119905= 1205900+ ((120590119879minus 1205900)119879)119905 In fact












0 t T




product at time t


Sales period

without RFID








120587119862(119902 0) = 119901 [int

120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119898119902

(1)



120597120587119862(119902 0)

120597119902= 120572 [119901int

+infin

120572119902

119891 (119909 0) 119889119909 minus 119903int120572119902

0

119891 (119909 0) 119889119909

+ 119904int+infin

120572119902

119891 (119909 0) 119889119909] + 119897V minus 119898


1205972120587119862(119902 0)

1205971199022= minus1205722(119901 + 119904 + 119903) 119891 (120572119902 0) lt 0

(2)




derived as follows

119865 (120572119876lowast

1198620 0) =

120572 (119901 + 119904 minus V) + V minus 119898

120572 (119901 + 119904 + 119903) (3)








120572] will



=

(119898 minus V)(119901 + 119904 minus V))


1198620minus 120583)120590



= 120583 +



1198620= Φ((120572119876lowast

1198620minus 120583)120590

0) which


1198620le 1 119885(119878

1198620)






1198781198620

= Φ((120572119876lowast

1198620minus 120583)120590






120587lowast

119862(120572119876lowast

1198620 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(4)



follows

120587119862(119902 119905) = 119901 [int

119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus (119898 + 119888) 119902

(5)



120597120587119862(119902 119905)

120597119902= 119901int

+infin

119902

119891 (119909 119905) 119889119909 minus 119903int119902

0

119891 (119909 119905) 119889119909

+ 119904int+infin

119902

119891 (119909 119905) 119889119909 minus 119898 minus 119888


1205972120587119862(119902 119905)

1205971199022= minus (119901 + 119904 + 119903) 119891 (119902 119905) lt 0

(6)



of (5) is


derived as follows

119865 (119876lowast

119862119905 119905) =

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903 (7)





120587119862(119876lowast


120574 +infin] will




119862119905 119905) = Φ((119876

lowast

119862119905minus 120583)120590




is 119878119862119905

= Φ((119876lowast119862119905minus 120583)120590


can derive Lemma 4




RFID technology is

120587lowast

119862(119876lowast

119862119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904(8)


1198620and 119878




120572 1] will




119862119905gt 1198781198620(U120572=


120572le 119888 le 119880

120574


119862119905le 1198781198620












120572 1] will the




120573= (l120572)(mminus



120572le c le







119863119903(119902 0)

120587119863119904(119902 0) and 120587



120587119863119903

(119902 0) = 119901 [int120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119908119902

(9)

120587119863119904

(119902 0) = (119908 minus 119898) 119902 (10)

120587119863(119902 0) = 120587

119863119903(119902 0) + 120587

119863119904(119902 0) (11)



119865(120572119876lowast

11986300)

=120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903) (12)



because 119865(1199090)

ge 0 so 119865(120572119876lowast

11986300)





119863119903(120572119876lowast1198630 0) ge 0





1198630= 120583 + 119885(119878



1198630= Φ((120572119876lowast

1198630minus120583)120590

0) (05 le 119878

1198630le 1)




= Φ((120572119876lowast1198630


Lemma 8




IncreaseDecrease

Service level


Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c



lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)


119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587



120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)




119863119903(119876lowast119863119905 119905) ge 0



119863119905= 120583 + 119885(119878

119863119905)120590119905 In this


= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus





119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119873(120583 1205902



119873(120583 1205902



119905= 1205900+ ((120590119879minus 1205900)119879)119905 In fact












0 t T




product at time t


Sales period

without RFID








120587119862(119902 0) = 119901 [int

120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119898119902

(1)



120597120587119862(119902 0)

120597119902= 120572 [119901int

+infin

120572119902

119891 (119909 0) 119889119909 minus 119903int120572119902

0

119891 (119909 0) 119889119909

+ 119904int+infin

120572119902

119891 (119909 0) 119889119909] + 119897V minus 119898


1205972120587119862(119902 0)

1205971199022= minus1205722(119901 + 119904 + 119903) 119891 (120572119902 0) lt 0

(2)




derived as follows

119865 (120572119876lowast

1198620 0) =

120572 (119901 + 119904 minus V) + V minus 119898

120572 (119901 + 119904 + 119903) (3)








120572] will



=

(119898 minus V)(119901 + 119904 minus V))


1198620minus 120583)120590



= 120583 +



1198620= Φ((120572119876lowast

1198620minus 120583)120590

0) which


1198620le 1 119885(119878

1198620)






1198781198620

= Φ((120572119876lowast

1198620minus 120583)120590






120587lowast

119862(120572119876lowast

1198620 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(4)



follows

120587119862(119902 119905) = 119901 [int

119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus (119898 + 119888) 119902

(5)



120597120587119862(119902 119905)

120597119902= 119901int

+infin

119902

119891 (119909 119905) 119889119909 minus 119903int119902

0

119891 (119909 119905) 119889119909

+ 119904int+infin

119902

119891 (119909 119905) 119889119909 minus 119898 minus 119888


1205972120587119862(119902 119905)

1205971199022= minus (119901 + 119904 + 119903) 119891 (119902 119905) lt 0

(6)



of (5) is


derived as follows

119865 (119876lowast

119862119905 119905) =

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903 (7)





120587119862(119876lowast


120574 +infin] will




119862119905 119905) = Φ((119876

lowast

119862119905minus 120583)120590




is 119878119862119905

= Φ((119876lowast119862119905minus 120583)120590


can derive Lemma 4




RFID technology is

120587lowast

119862(119876lowast

119862119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904(8)


1198620and 119878




120572 1] will




119862119905gt 1198781198620(U120572=


120572le 119888 le 119880

120574


119862119905le 1198781198620












120572 1] will the




120573= (l120572)(mminus



120572le c le







119863119903(119902 0)

120587119863119904(119902 0) and 120587



120587119863119903

(119902 0) = 119901 [int120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119908119902

(9)

120587119863119904

(119902 0) = (119908 minus 119898) 119902 (10)

120587119863(119902 0) = 120587

119863119903(119902 0) + 120587

119863119904(119902 0) (11)



119865(120572119876lowast

11986300)

=120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903) (12)



because 119865(1199090)

ge 0 so 119865(120572119876lowast

11986300)





119863119903(120572119876lowast1198630 0) ge 0





1198630= 120583 + 119885(119878



1198630= Φ((120572119876lowast

1198630minus120583)120590

0) (05 le 119878

1198630le 1)




= Φ((120572119876lowast1198630


Lemma 8




IncreaseDecrease

Service level


Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c



lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)


119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587



120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)




119863119903(119876lowast119863119905 119905) ge 0



119863119905= 120583 + 119885(119878

119863119905)120590119905 In this


= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus





119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0 t T




product at time t


Sales period

without RFID








120587119862(119902 0) = 119901 [int

120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119898119902

(1)



120597120587119862(119902 0)

120597119902= 120572 [119901int

+infin

120572119902

119891 (119909 0) 119889119909 minus 119903int120572119902

0

119891 (119909 0) 119889119909

+ 119904int+infin

120572119902

119891 (119909 0) 119889119909] + 119897V minus 119898


1205972120587119862(119902 0)

1205971199022= minus1205722(119901 + 119904 + 119903) 119891 (120572119902 0) lt 0

(2)




derived as follows

119865 (120572119876lowast

1198620 0) =

120572 (119901 + 119904 minus V) + V minus 119898

120572 (119901 + 119904 + 119903) (3)








120572] will



=

(119898 minus V)(119901 + 119904 minus V))


1198620minus 120583)120590



= 120583 +



1198620= Φ((120572119876lowast

1198620minus 120583)120590

0) which


1198620le 1 119885(119878

1198620)






1198781198620

= Φ((120572119876lowast

1198620minus 120583)120590






120587lowast

119862(120572119876lowast

1198620 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(4)



follows

120587119862(119902 119905) = 119901 [int

119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus (119898 + 119888) 119902

(5)



120597120587119862(119902 119905)

120597119902= 119901int

+infin

119902

119891 (119909 119905) 119889119909 minus 119903int119902

0

119891 (119909 119905) 119889119909

+ 119904int+infin

119902

119891 (119909 119905) 119889119909 minus 119898 minus 119888


1205972120587119862(119902 119905)

1205971199022= minus (119901 + 119904 + 119903) 119891 (119902 119905) lt 0

(6)



of (5) is


derived as follows

119865 (119876lowast

119862119905 119905) =

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903 (7)





120587119862(119876lowast


120574 +infin] will




119862119905 119905) = Φ((119876

lowast

119862119905minus 120583)120590




is 119878119862119905

= Φ((119876lowast119862119905minus 120583)120590


can derive Lemma 4




RFID technology is

120587lowast

119862(119876lowast

119862119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904(8)


1198620and 119878




120572 1] will




119862119905gt 1198781198620(U120572=


120572le 119888 le 119880

120574


119862119905le 1198781198620












120572 1] will the




120573= (l120572)(mminus



120572le c le







119863119903(119902 0)

120587119863119904(119902 0) and 120587



120587119863119903

(119902 0) = 119901 [int120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119908119902

(9)

120587119863119904

(119902 0) = (119908 minus 119898) 119902 (10)

120587119863(119902 0) = 120587

119863119903(119902 0) + 120587

119863119904(119902 0) (11)



119865(120572119876lowast

11986300)

=120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903) (12)



because 119865(1199090)

ge 0 so 119865(120572119876lowast

11986300)





119863119903(120572119876lowast1198630 0) ge 0





1198630= 120583 + 119885(119878



1198630= Φ((120572119876lowast

1198630minus120583)120590

0) (05 le 119878

1198630le 1)




= Φ((120572119876lowast1198630


Lemma 8




IncreaseDecrease

Service level


Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c



lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)


119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587



120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)




119863119903(119876lowast119863119905 119905) ge 0



119863119905= 120583 + 119885(119878

119863119905)120590119905 In this


= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus





119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1198781198620

= Φ((120572119876lowast

1198620minus 120583)120590






120587lowast

119862(120572119876lowast

1198620 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(4)



follows

120587119862(119902 119905) = 119901 [int

119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus (119898 + 119888) 119902

(5)



120597120587119862(119902 119905)

120597119902= 119901int

+infin

119902

119891 (119909 119905) 119889119909 minus 119903int119902

0

119891 (119909 119905) 119889119909

+ 119904int+infin

119902

119891 (119909 119905) 119889119909 minus 119898 minus 119888


1205972120587119862(119902 119905)

1205971199022= minus (119901 + 119904 + 119903) 119891 (119902 119905) lt 0

(6)



of (5) is


derived as follows

119865 (119876lowast

119862119905 119905) =

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903 (7)





120587119862(119876lowast


120574 +infin] will




119862119905 119905) = Φ((119876

lowast

119862119905minus 120583)120590




is 119878119862119905

= Φ((119876lowast119862119905minus 120583)120590


can derive Lemma 4




RFID technology is

120587lowast

119862(119876lowast

119862119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904(8)


1198620and 119878




120572 1] will




119862119905gt 1198781198620(U120572=


120572le 119888 le 119880

120574


119862119905le 1198781198620












120572 1] will the




120573= (l120572)(mminus



120572le c le







119863119903(119902 0)

120587119863119904(119902 0) and 120587



120587119863119903

(119902 0) = 119901 [int120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119908119902

(9)

120587119863119904

(119902 0) = (119908 minus 119898) 119902 (10)

120587119863(119902 0) = 120587

119863119903(119902 0) + 120587

119863119904(119902 0) (11)



119865(120572119876lowast

11986300)

=120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903) (12)



because 119865(1199090)

ge 0 so 119865(120572119876lowast

11986300)





119863119903(120572119876lowast1198630 0) ge 0





1198630= 120583 + 119885(119878



1198630= Φ((120572119876lowast

1198630minus120583)120590

0) (05 le 119878

1198630le 1)




= Φ((120572119876lowast1198630


Lemma 8




IncreaseDecrease

Service level


Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c



lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)


119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587



120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)




119863119903(119876lowast119863119905 119905) ge 0



119863119905= 120583 + 119885(119878

119863119905)120590119905 In this


= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus





119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















120572 1] will the




120573= (l120572)(mminus



120572le c le







119863119903(119902 0)

120587119863119904(119902 0) and 120587



120587119863119903

(119902 0) = 119901 [int120572119902

0

119909119891 (119909 0) 119889119909 + int+infin

120572119902

120572119902119891 (119909 0) 119889119909]

minus 119903int120572119902

0

(120572119902 minus 119909) 119891 (119909 0) 119889119909


120572119902

(119909 minus 120572119902) 119891 (119909 0) 119889119909 + V119897119902 minus 119908119902

(9)

120587119863119904

(119902 0) = (119908 minus 119898) 119902 (10)

120587119863(119902 0) = 120587

119863119903(119902 0) + 120587

119863119904(119902 0) (11)



119865(120572119876lowast

11986300)

=120572 (119901 + 119904 minus V) + V minus 119908

120572 (119901 + 119904 + 119903) (12)



because 119865(1199090)

ge 0 so 119865(120572119876lowast

11986300)





119863119903(120572119876lowast1198630 0) ge 0





1198630= 120583 + 119885(119878



1198630= Φ((120572119876lowast

1198630minus120583)120590

0) (05 le 119878

1198630le 1)




= Φ((120572119876lowast1198630


Lemma 8




IncreaseDecrease

Service level


Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c



lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)


119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587



120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)




119863119903(119876lowast119863119905 119905) ge 0



119863119905= 120583 + 119885(119878

119863119905)120590119905 In this


= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus





119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










IncreaseDecrease

Service level


Service level


Service level


0

0

0

U120572

U120572

U120572

U120573

U120573

U120573

U120574

U120574

U120574

c

c

c



lowast

119863119903(120572119876lowast1198630 0) 120587lowast

119863119904(120572119876lowast1198630 0) and

120587lowast119863(120572119876lowast1198630 0)

120587lowast

119863119903(120572119876lowast

1198630 0) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

1198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(13)

120587lowast

119863119904(120572119876lowast

1198630 0) =

(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900) (14)

120587lowast

119863(120572119876lowast

1198630 0) = 120587

lowast

119863119903(120572119876lowast

1198630 0) + 120587

lowast

119863119904(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

(15)


119863119903(119902 119905) 120587

119863119904(119902 119905) and 120587



120587119863119903

(119902 119905) = 119901 [int119902

0

119909119891 (119909 119905) 119889119909 + int+infin

119902

119902119891 (119909 119905) 119889119909]

minus 119903int119902

0

(119902 minus 119909) 119891 (119909 119905) 119889119909


119902

(119909 minus 119902) 119891 (119909 119905) 119889119909 minus 119908119902

(16)

120587119863119904

(119902 119905) = (119908 minus 119898 minus 119888) 119902 (17)

120587119863(119902 119905) = 120587

119863119903(119902 119905) + 120587

119863119904(119902 119905) (18)



119865(119876lowast

119863119905119905)=119901 + 119904 minus 119908

119901 + 119904 + 119903 (19)




119863119903(119876lowast119863119905 119905) ge 0



119863119905= 120583 + 119885(119878

119863119905)120590119905 In this


= Φ((119876lowast119863119905

minus 120583)120590119905) = (119901 + 119904 minus





119863(119876lowast119863119905 119905)

120587lowast

119863119903(119876lowast

119863119905 119905) = (119901 + 119904 + 119903) [120583Φ (119885 (119878

119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(20)

120587lowast

119863119904(119876lowast

119863119905 119905) = (119908 minus 119898 minus 119888) (120583 + 119885 (119878

119863119905) 120590119905) (21)


120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120587lowast

119863(119876lowast

119863119905 119905) = 120587

lowast

119863119903(119876lowast

119863119905 119905) + 120587

lowast

119863119904(119876lowast

119863119905 119905)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


(22)


1198630and 119878

119863119905 we can



[1198811015840







120572 1]



120573= (120573 +

(119908 minus 119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) and 120573 =

(120583119897120572)(119898 minus V) + (119901 + 119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))])








119898)(119885(119878119863119905)120590119905minus 119885(119878

1198630)1205900120572))(120583 + 119885(119878

119863119905)120590119905) lt 1198801015840

120573and 119881

120583=

(120583 + 119885(1198781198630)1205900)(120583 + 119885(119878

119863119905)120590119905))














119863119905= 120583 + 119885(119878






119863119903ge 0 but


+ 120579Δ119863119903

ge 0



0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

0

Service level



Service level


U998400120573

U998400120573

U998400120572

c

c



IncreaseDecrease



Δ119863119904

+ 120579Δ119863119903

= 120583119897

120572(119898 minus 119908) + (119908 minus 119898)

times (119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

+ 120579120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))] ge 0

(23)




119863119905lt




= (120572(119908 minus

119898)(119876lowast1198630

minus119876lowast119863119905)+120572119888(120583+119885(119878

119863119905)120590119905))(120572120573+120583119897(119908minus119898)))


119863119905ge







units 1205900= 2000 units 120590

119879= 200 units 119879 = 10 weeks 119901 = 6






0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0206

1

0

1

2

0

04

08

cSC0

SCt

S = 05S

120572


0205081

2

4

6

02

05

08

SD0

SDt

S = 05

S

120572

w




0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10


0

2

4

0 5 10t

c max

120572 = 08

120572 = 09

120572 = 10




120583= 0991 gt 120572 = 08 so we




in the same way)


0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10minus05

0

05

1

15

2

25 times104

Δ120587

120572 = 08

t

Δ120587DR

Δ120587D

Δ120587DS

(a)

0 5 10minus1

0

1

2 times104

Δ120587

120572 = 09

t

Δ120587DR

Δ120587D

Δ120587DS

(b)

0 2 4 6 8 10minus1

0

1 times104

Δ120587

120572 = 10

t

Δ120587DR

Δ120587D

Δ120587DS

(c)



7 Conclusions







Appendices



120587lowast

119862(120572119876lowast

1198620 0) = 119901120572119876

lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620




120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120587lowast119863119903(1199025)

Δ119903

120587lowast119863119904(1199025)

Δ119904

120587lowast119863(1199025)

Δ

0 104576 71843 43069 minus24449 147645 4739402 90207 57474 57438 minus10080 147645 4739404 75839 43106 71806 4288 147645 4739406 61470 28737 86175 18657 147645 4739408 47102 14369 100543 33025 147645 473941 32733 00 114912 47394 147645 47394


1198620+ (119901 + 119903) int

120572119876lowast

1198620

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198620

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] minus 119903120572119876

lowast

1198620119865 (120572119876

lowast

1198620 0)

+ 119904120572119876lowast

1198620[1 minus 119865 (120572119876

lowast

1198620 0)] + V (1 minus 120572)119876

lowast

1198620


1198620minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198620))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198620)) minus 120583119904

= [119901 + 119904 +V119897 minus 119898

120572] (120583 + 119885 (119878

1198620) 1205900)

minus 1205900(119901 + 119904 + 119903)

times [119885 (1198781198620)Φ (119885 (119878

1198620)) + 120601 (119885 (119878

1198620))] minus 120583119904

= 120583 [119901 + 119904 +V119897 minus 119898

120572]

+ 119885 (1198781198620) 1205900[119901 + 119904 +

(1 minus 120572) V minus 119898

120572]

minus 119885 (1198781198620) 1205900(119901 + 119904 + 119903)


120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (120572119901 + 120572119904 minus 120572V + V minus 119898

120572119901 + 120572119904 + 120572119903)

minus 1205900(119901 + 119904 + 119903) 120601 (119885 (119878

1198620)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))]

minus 120583119904

(A1)


120587lowast



120587lowast

119862(119876lowast

119862119905 119905) = 119901119876

lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

+ (119901 + 119903) int119876lowast

119862119905

0

119909119891 (119909 119905) 119889119909 minus 119904int+infin

119876lowast

119862119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus 119903119876

lowast

119862119905119865 (119876lowast

119862119905 119905)

+ 119904119876lowast

119862119905[1 minus 119865 (119876

lowast

119862119905 119905)] minus (119898 + 119888)119876

lowast

119862119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119862119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119862119905)) minus 120583119904


times [119885 (119878119862119905)Φ (119885 (119878

119862119905)) + 120601 (119885 (119878

119862119905))] minus 120583119904


minus 119885 (119878119862119905) 120590119905(119901 + 119904 + 119903) (

119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= 120583 (119901 + 119904 + 119903) (119901 + 119904 minus 119898 minus 119888

119901 + 119904 + 119903)

minus 120590119905(119901 + 119904 + 119903) 120601 (119885 (119878

119862119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))]

minus 120583119904

(B1)


119862




Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Δ119862= 120587lowast

119862(119876lowast

119862119905 119905) minus 120587

lowast

119862(120572119876lowast

1198620 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119862119905)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198620)) minus 120590

0120601 (119885 (119878

1198620))] minus 120583119904

=(1 minus 120572) 120583

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198620)) minus 120590

119905120601 (119885 (119878

119862119905))] minus 120583119888

(B2)


= (120583119897120572)(119898 minus V) + (119901 + 119904 +

119903)[1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878




lt 119878119862119905 which

means 0 le 119885(1198781198620) lt 119885(119878

119862119905) then 120601(119885(119878

1198620)) gt

120601(119885(119878119862119905)) gt 0 Also because 120590

0gt 120590119905

gt 0 and1205900120601(119885(119878

1198620)) minus 120590

119905120601(119885(119878

119862119905)) gt 0 then 119910

119862gt 0 When




le 1198781198620 which

means 0 lt 119885(119878119862119905) le 119885(119878

1198620) then 0 lt 120601(119885(119878

1198620)) le

120601(119885(119878119862119905)) Also because 120590



1198620)) or 120590

119905120601(119885(119878

119862119905)) is


119905)(119898minus

V) + (1205900120590119905)120601(119885(119878


119862ge 0



chain will increase



120587lowast

119863119903(120572119876lowast

1198630 0) = 119901120572119876

lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630+ (119901 + 119903) int

120572119876lowast

1198630

0

119909119891 (119909 0) 119889119909


120572119876lowast

1198630

119909119891 (119909 0) 119889119909

= 119901120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)]

minus 119903120572119876lowast

1198630119865 (120572119876

lowast

1198630 0)

+ 119904120572119876lowast

1198630[1 minus 119865 (120572119876

lowast

1198630 0)] + V (1 minus 120572)119876

lowast

1198630


1198630minus (119901 + 119904 + 119903) 120590

0120601 (119885 (119878

1198630))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (1198781198630)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904

(C1)


120587lowast

119863119903(119876lowast

119863119905 119905) = 119901119876

lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

+ (119901 + 119903) int119876lowast

119863119905

0

119909119891 (119909 119905) 119889119909


119876lowast

119863119905

119909119891 (119909 119905) 119889119909

= 119901119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119903119876

lowast

119863119905119865 (119876lowast

119863119905 119905)

+ 119904119876lowast

119863119905[1 minus 119865 (119876

lowast

119863119905 119905)] minus 119908119876

lowast

119863119905

minus (119901 + 119904 + 119903) 120590119905120601 (119885 (119878

119863119905))

+ 120583 (119901 + 119904 + 119903)Φ (119885 (119878119863119905)) minus 120583119904

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]

minus 120583119904

(D1)


and 119878119863119905

we can find that

119878119863119905

minus 1198781198630

=119901 + 119904 minus 119908


120572 (119901 + 119904 + 119903)

=(119897120572) (119908 minus V)119901 + 119904 + 119903

ge 0

(a)


119863119905ge 1198781198630




Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Δ119863= 120587lowast

119863(119876lowast

119863119905 119905) minus 120587

lowast

119863(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))]


minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))]

minus 120583119904 +(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)

=120583119897

120572(119898 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

+ (119908 minus 119898)(119885 (119878119863119905) 120590119905minus119885 (1198781198630) 1205900

120572)

minus 119888 (120583 + 119885 (119878119863119905) 120590119905)

(D2)


= (120583119897120572)(119898 minus V) + (119901 +

119904 + 119903)[1205900120601(119885(119878

1198630)) minus 120590

119905120601(119885(119878

119863119905))] + (119908 minus 119898)(119885(119878

119863119905)120590119905minus



1198630lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878


1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878

119863119905))

(1) When 119885(119878119863119905)120590119905gt 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(2) When 119885(119878119863119905)120590119905= 119885(119878

1198630)1205900120572 then 119910

119863gt 0 can be


1015840

120573

(3) When 119885(119878119863119905)120590119905

lt 119885(1198781198630)1205900120572 if 119908 lt

120573(119885(1198781198630)1205900120572 minus 119885(119878

119863119905)120590119905) + 119898 then 119910

119863gt 0


1015840

120573



Δ119863119903

= 120587lowast

119863119903(119876lowast

119863119905 119905) minus 120587

lowast

119863119903(120572119876lowast

1198630 0)

= (119901 + 119904 + 119903) [120583Φ (119885 (119878119863119905)) minus 120590

119905120601 (119885 (119878

119863119905))] minus 120583119904

minus (119901 + 119904 + 119903) [120583Φ (119885 (1198781198630)) minus 120590

0120601 (119885 (119878

1198630))] minus 120583119904

=120583119897

120572(119908 minus V) + (119901 + 119904 + 119903)

times [1205900120601 (119885 (119878

1198630)) minus 120590

119905120601 (119885 (119878

119863119905))]

(D3)


lt 119878119863119905 which means

0 le 119885(1198781198630) lt 119885(119878

119863119905) thus 120601(119878

1198630) gt 120601(119878

119863119905) gt 0 because

1205900gt 120590119905gt 0 120590

0120601(119885(119878

1198630)) ge 120590

119905120601(119885(119878





Δ119863119904

= 120587lowast

119863119904(119876lowast

119863119905 119905) minus 120587

lowast

119863119904(120572119876lowast

1198630 0)

= [(119908 minus 119898 minus 119888) (120583 + 119885 (119878119863119905) 120590119905)]

minus [(119908 minus 119898)

120572(120583 + 119885 (119878

1198630) 1205900)]

(D4)


= (120583119897120572)(119898 minus 119908) + (119908 minus

119898)(119885(119878119863119905)120590119905minus 119885(119878


When 05 le 1198781198630

lt 119878119863119905 that means 0 le 119885(119878

1198630) lt 119885(119878

119863119905)


119863119905)120590119905or


(1) When 119885(119878119863119905)120590119905ge 119885(119878


tions

(i) 120572 isin [119881119911 1] and 120572 isin [119881


the function 119910119863119904

ge 0 Because 119881120583

gt 119881119911 we




=

119885(1198781198630)1205900119885(119878119863119905)120590119905)



119910119863119904


(2) When 119885(119878119863119905)120590119905lt 119885(119878

1198630)1205900120572 then 120572 isin (0 119881

119911] and

119876lowast119863119905






Acknowledgments




References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Misplaced Inventory and Lead-Time …downloads.hindawi.com/journals/jam/2014/135284.pdf · Research Article Misplaced Inventory and Lead-Time in the Supply Chain: