research article optimal decisions for adoption of item

Research ArticleOptimal Decisions for Adoption of Item-Level RFID in a RetailSupply Chain with Inventory Shrinkage under CVaR Criterion

Chunming Xu1 and Daozhi Zhao2

1Department of Mathematics College of Science Tianjin University of Technology Tianjin 300384 China2College of Management and Economics Tianjin University Tianjin 300072 China

Correspondence should be addressed to Chunming Xu chunmingxutjuteducn

Received 6 January 2016 Accepted 21 February 2016

Academic Editor Gabriella Bretti

Copyright copy 2016 C Xu and D Zhao 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

This paper investigates the effect of item-level RFID on inventory shrinkage in the retail supply chain which consists of a risk-neutralmanufacturer and a risk-averse retailer Under conditional value-at-risk (CVaR) criterion twodifferent supply chain settingsare discussed as follows In the centralized setting we develop the models in both RFID case and no RFID case respectivelyComparisons between the two cases are made In particular a sufficient condition is given to judge whether to adopt item-levelRFID In the decentralized setting we focus on discussing two different contract types including wholesale price contact andrevenue sharing contract Finally number examples and sensitivity analysis are given to illustrate the proposed models The resultsshow that for the centralized system the sales-available rate the recovery rate and the tag cost are mainly the driving factorsin evaluating the benefit of an item-level RFID In particular when the sales-available rate and the tag cost are quite small andthe recovery rate is higher the supply chain partnersrsquo profits obtained by investment for RFID are improved significantly For thedecentralized system under revenue sharing contract Pareto improving outcome and coaffording risk can be achieved if the retailersets an appropriate parameter for the manufacturer

1 Introduction

(1) Motivation In traditional inventory system it is oftenassumed that inventory record and physical inventory (actualon-hand inventory) are identical but in the real world theinventory record can hardly match the physical inventorythat is inventory inaccuracy is widespread and inevitable inthe store or warehousebackroom [1 2] Usually there arethree different kinds of causes of inventory inaccuracy

(i) inventory shrinkage theft from the shelf productspoilage and product damage and so forth

(ii) misplacement products that are placed on the wrongshelf are not accessible to customers due to improperusage of the storage area

(iii) transaction errors the wrong item codes are recordedat the cash register

In recent years inventory inaccuracy is common in theretail industries and it has gained more and more attention

of some researchers Kang and Gershwin [3] noted that in aglobal retailerrsquos stores investigated only 70ndash75 of inventoryrecord was accurate in the best performing retail storeduring its annual inventory audit DeHoratius and Raman [4]reported that 65 of the inventory records in retail stores didnot match the physical stock from about 370000 examinedSKUs (stock-keeping-units) Moreover 20 of the inventoryrecords differed from the physical stock by six or moreitems By many investigations on how to generate errors inthe retail inventory operation process Rekik [1] concludedthat inventory shrinkage is the main cause of inventoryinaccuracy Additionally related studies also showed thatthe nonsale inventory shrinkage usually leads to unavailabledemand for end consumers and themagnitude of such lossesstemming from inventory shrinkage is huge Bednarz et al [5]estimated that US retailers suffered a $313 billion loss due toshrinkage in 2002 According to ECR Europe [6] the value oflost inventory due to shrinkage in 2000 was 134 billion eurosfor retailers in Europe

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 7834751 17 pageshttpdxdoiorg10115520167834751

2 Discrete Dynamics in Nature and Society

Nowadays however radio frequency identification(RFID) technology can be used to track products andmanageinventory based on its ability to higher visibility and it hasbeen seen as a promising solution for inventory shrinkage inthe supply chain arena [7] Generally RFID has two principaladvantages high-frequency monitoring and nonline ofsight reading which yield information that is both moreaccurate and timely [8] Recently in order to characterizemore practical features Hervert-Escobar et al [9] utilizedthe information obtained in consecutive read attempts tohelp identify a tag in RFID implementation and developed aheuristic method of selection based on Hamming distancecomputer simulation is used to illustrate the validity of theproposed method Talavera et al [10] also studied RFIDimplementation in the steel industry where RFID technologyshows that the shared inventory improvement and the rate ofobsolescence reduction are related to inventorymanagementAs information technologies continue to improve and theircosts continue to decrease obtaining more accurate real-time inventory information is becoming increasinglycost-effective As a result more and more large companies(such as Wal-Mart Proctor amp Gamble and Gillette) areguided by high-quality information in their daily operationsand decision making [11]

Motivated by the issue of RFID technology adoption inthe industry (especially in supply chain management) westudy the effect of item-level RFID on inventory shrinkagein the retail supply chain frame To get more general resultswe extend previous knowledge of inventory inaccuracy anditem-level RFID by incorporating risk-averse considerationAs mentioned above in reality it is difficult for some com-panies to bear losses stemming from inventory shrinkageWhen facing a higher inventory shrinkage rate and a greaterdemanduncertainty the revenue obtained by selling productscan not balance the increased loss supply chain managersmust take more risks caused by inventory shrinkage anddemand uncertainty and the results in the risk-neutral casemay be considered as unrealistic Therefore the questionwe are concerned about is how the risk aversion levelaffects their optimal decisions in supply chain system withor without item-level RFID under conditional value-at-risk(CVaR) criterion (see Section 31) In addition many existingstudies incorporated price-dependent stochastic demand orRFID technology into supply chain models but few of themexplored RFID technology for retail inventory shrinkageprice-dependent stochastic demand and risk issue simulta-neously Our paper will cover these gaps

(2) Related Literature

(i) RFID and Inventory Inaccuracy Although recentresearchers have given some literature reviews on RFIDtechnology [12 13] the related research on RFIDimplementation in inventory management is relatively newWe here only focus on reviewing some recent studies onanalyzing the impact of item-level RFID on the reductionof inventory inaccuracies De Kok et al [14] consideredcost-benefit trade-off between inventory costs and the costsof RFID regarding shrinkage and proved that this break-even

price is highly related to the value of the items that arelost the shrinkage fraction and the remaining shrinkageafter employing RFID Heese [15] studied inventory recordinaccuracy in a supply chain model with a manufacturerand a retailer and analyzed the impact of inventory recordinaccuracy on optimal stocking decisions and profits Bycontrasting optimal decisions in a decentralized supply chainwith those in an integrated supply chain they concluded thatinventory record inaccuracy exacerbates the inefficienciesresulting from double marginalization in decentralizedsupply chains Rekik et al [16] developed a newsvendormodel and analyzed the RFID adoption strategy withcoordination to improve the supply chainrsquos performanceunder retail inventory inaccuracy that is subject to errorsstemming from execution problems Based on imperfectinventory records and unobserved lost sales Mersereau[17] also discussed a periodic review inventory systemthat explores one- and two-period versions of the problemand demonstrated several mechanisms by which the errorprocess and associated record inaccuracy can impact optimalreplenishment Considering identical error distributions andcounting costs Kok and Shang [2] investigated how todesign cycle-count policies from the perspective of the entiresupply chain for a two-stage system provided a simplerecursion to evaluate the system cost proposed a heuristicto obtain effective base-stock levels and proved that it ismore effective to conduct more frequent cycle counts at thedownstream stage Taking into account inventory inaccuracystemming from shrinkage and delivery errors Sarac et al[18] investigated a simulation model with RFID technologyin a three-level supply chain and evaluated the qualitativeand quantitative impacts of RFID technologies on supplychain system performances and profits The aforementionedarticles all assumed demand to be a constant or a stochasticvariable To our knowledge there has been no work oninvestigating the impact of inventory shrinkage on supplychain with price-dependent stochastic demand In thispaper bearing in mind the fact that the demand of product isaffected by price we discuss a supply chain model with retailinventory shrinkage and assess the benefit of the item-levelRFID implementation

(ii) Risk Aversion Risk aversion issues in supply inventorymanagement have been extensively studied in the pastdecades There are mean variance (MV) value-at-risk (VaR)and conditional value-at-risk (CVaR) in traditional riskanalysis approaches Gan et al [19] used MV approach tostudy the supply chain coordination in a two-echelon supplychain with risk-averse agents Under MV framework Choiet al [20] considered two-echelon supply chain coordinationproblemwhen the partners take differentsame risk attitudesthe result shows that the whole supply chain coordinationdepends on how different the risk related thresholds betweenthe two supply chain agents are Choi [21] developed thesupply chain models for a multiperiod retail replenishmentproblem with and without RFID under the MV frameworkand analytically discussed the use of RFID under vendor-managed inventory (VMI) scheme in a two-echelon single-manufacturer single-retailer supply chain Considering

Discrete Dynamics in Nature and Society 3

the application of RFID technology to eliminate themisplace-ment problems Chen et al [22] focused on analyzing howthe risk attitude affects the supply chain members incentivesto adopt RFID and the corresponding coordination contractwhere the central semideviation is adopted to measure theretailerrsquos risk attitude Ozler et al [23] utilized VaR as the riskmeasure in a newsvendor framework and investigated themultiproduct newsvendor problem under a VaR constraintBased on price-dependent demands Chen et al [24]explored the CVaR objective as the decision criterion in thenewsvendor problem They analyzed the optimal pricingand stocking decisions and derived sufficient conditions forthe existence of unique solution and further revealed theneat monotonicity properties associated with the optimalpricing and ordering decisions Chiu and Choi [25] studiedthe price-dependent newsvendor problem with a VaRobjective they discussed both the linear and multiplicativeprice-dependent demand distributions cases and analyticallyderived the optimal solutions for the problem under a VaRobjective Focusing on VaR constraint and CVaR as the riskmeasures of the downside risk Wu et al [26] investigatedprofit maximization versus risk approaches for the standardnewsvendor problem with uncertainty in demand as well asa generalized version with uncertainty in the shortage costDifferent from MV approaches the upside of variance isnot considered as the risk-averse decision-maker in realitythe upside of variance can be viewed as the surprising gainsfrom investment Most risk-averse decision-makers onlycare about the downside losses rather than the upside gainsThus VaR and CVaR approaches are more intuitive andcomprehensive to reflect decision-makerrsquos risk attitude butas Ahmed et al [27] pointed out compared to VaR approachCVaR can be consistent with second-order stochasticdominance rules Furthermore Section 31 also shows thatCVaR approach has attractive computational characteristicsHowever our work is not to argue how much better CVaRis than the other approaches Rather we only utilize CVaRas a risk measurement Contrary to the above papers wejust limit focusing on the effect of item-level RFID on theinventory shrinkage problem and take into account retailerrsquosrisk caused by demand uncertainty and nonsale shrinkage insupply chain frame

(iii) RFID and Inventory Shrinkage Inventory shrinkage is themain cause of inventory inaccuracy there are some relatedstudies to our paper in adopting the RFID technology foreliminating inventory shrinkage For example consideringthe holding cost Rekik et al [28] investigated the problem ofhaving theft in store under a service level constraint and theyanalyzed the impact of theft errors and the value of the RFIDon the inventory system under the assumption of perfectRFID technology which could eliminate theft errors com-pletely while we suppose that only a part of shrinkage errorscan be eliminated due to imperfect RFID which is closer toreality Although Fan et al [29 30] also studied the problemof RFID technology for reducing inventory shrinkage underimperfect RFID they assumed that the demand followed auniformdistributionwith known parameters when analyzing

the threshold value of tag cost while our work does not havethis assumption

Under CVaR criterion the contribution of this paperis threefold (1) we derive optimal supply chains decisionswith and without item-level RFID in centralized setting andprovide a sufficient condition to judge whether to adopt item-level RFID (2)we also derive optimal supply chains decisionswith item-level RFID in decentralized setting and discuss thenecessity of supply chain coordination in this case and (3)we design wholesale price contract with revenue sharing toachieve a win-win situation for supply chain partners

The reminder of the paper is organized as follows Inthe following section model descriptions notations andassumptions are presented Section 3 firstly gives definition ofCVaR and then focuses on analyzing supply chainrsquos optimaldecisions in centralized case with and without item-levelRFID under CVaR criterion and discusses how the optimaldecisions change with model parameters finally we judgewhether to adopt item-level RFID for assessing the benefitof the item-level RFID implementation In Section 4 weexplore the optimal policies for a decentralized supply chainwith two widely used contracts including wholesale pricecontract and revenue sharing contract and discuss supplychain coordination for achieving the best performance ofthe entire supply chain Section 5 includes number examplesand the sensitivity analysis of the parameters in the proposedmodels Finally we conclude with a summary of this paperand point out the direction for future research

2 Model Descriptions and Assumptions

We consider a two-echelon supply chain with one manufac-turer and one retailer A single short-life-cycle or seasonalproduct is provided by themanufacturer and then the retailersells it to end consumers Noting that there exist nonsaleinventory shrinkage phenomena and demand uncertaintyin many retail industries and after making the payment tothe manufacturer the retailer manages and maintains theretail inventory system alone themanufacturer however hasno money pumped into retail inventory that is the retailerbears all risk associated with nonsale inventory shrinkageand demand uncertainty and the manufacturer has no riskThus we here assume that the retailer is risk-averse and themanufacturer is risk-neutral

To model the impact of the nonsale retail inventoryshrinkage problems we define 120572 to be the ratio betweenthe sales-available on-hand inventory and the total physicalinventory in retail store Related researches show that item-level RFID can achieve a higher product availability but theeffect of RFID is imperfect and inventory shrinkage problemscan be only eliminated partly [31] Similar to the researchof Fan et al [29] on item-level RFID in retail inventoryconsidering the inventory shrinkage problems the followingassumption is made in our research when the retailer doesnot resort to a smart inventory system with item-level RFIDhe or she orders 119876 units from the manufacturer and only120572119876 units are sales-available the other (1 minus 120572)119876 units areunavailable to end customers due to inventory shrinkageerrors but when item-level RFID are used in inventory


No RFID case RFID case

units

Salable

Order quantity

Sales-available quantity

Shrinkage quantity

units

units

Recovery quantity

Shrinkage quantity


Salable

Shrinkage

Shrinkage

Q units

120572Qunits120572Q

(1 minus 120572)Q

units

+

120573(1 minus 120572)Q

(1 minus 120573)(1 minus 120572)Q

Figure 1 RFID implementation versus no RFID case in the retail inventory

system 120573(1 minus 120572)119876 units for inventory shrinkage can bepurchased by end customers and the other (1 minus 120573)(1 minus 120572)119876units may remain as nonsale inventory shrinkage problems(see Figure 1)

We adopt the following notation throughout this paper

(1) 119863(sdot) denotes end consumer demand during the sell-ing season

(2) 119901denotes the retailer sets retail price per unit product(3) 119908 denotes the manufacturer sets wholesale price per

unit product(4) 119888119872

denotes manufacturers marginal production costat the production stage that is 119908 gt 119888

119872

(5) 119905 denotes RFID tag price per unit product(6) 119888119877denotes operating cost per unit product at the retail

stage for inventory handling shelf-space usage andso forth 119901 gt 119888

119877

(7) 119891(sdot) denotes probability density function (PDF)(8) 119865(sdot) denotes cumulative distribution function (CDF)(9) 120576 denotes random variable with PDF 119891(sdot) and CDF

119865(sdot)(10) 119909ℎ(119909) denotes the general failure rate function of

demand distribution(11) 119902 denotes the retailers order quantity from the manu-

facturer during the single period(12) 119889(119901) denotes deterministic and decreasing function

of retail price 119901(13) 119897 denotes stocking factor of inventory(14) 120578

119877denotes the retailerrsquos risk aversion value that is

120578119877isin (0 1]

(15) 120578119862denotes the risk aversion value of the whole supply

chain that is 120578119862isin (0 1]

(16) 120587No119862(sdot) denotes total expected channel profit of cen-

tralized system without item-level RFID(17) Π1015840(sdot) denotes total expected channel profit of cen-

tralized system without item-level RFID under CVaRcriterion that is Π1015840(sdot) = CVaR(120587No

119862(sdot))

(18) 120587RF119862(sdot) denotes total expected channel profit of cen-

tralized system under item-level RFID(19) Π(sdot) denotes total expected channel profit of cen-

tralized system with item-level RFID under CVaRcriterion that is Π(sdot) = CVaR(120587RF

119862(sdot))

(20) 120587No119877(sdot) denotes the retailerrsquos expected profit without

item-level RFID(21) Π1015840

119877(sdot) denotes the retailerrsquos expected profit with-

out item-level RFID under CVaR criterion that isΠ1015840

119877(sdot) = CVaR(120587No

119877(sdot))

(22) 120587RF119877(sdot) denotes the retailerrsquos expected profit with item-

level RFID(23) Π

119877(sdot) denotes the retailerrsquos expected profit with item-

level RFID under CVaR criterion that is Π119877(sdot) =

CVaR(120587RF119877(sdot))

(24) 120587No119872(sdot) denotes the manufacturerrsquos expected profit

without item-level RFID(25) Π1015840

119872(sdot) denotes the manufacturerrsquos expected profit

without item-level RFID under CVaR criterion thatis Π119877(sdot) = CVaR(120587No

119872(sdot))

(26) 120587RF119872(sdot) denotes the manufacturerrsquos expected profit

with item-level RFID(27) Π


with item-level RFID under CVaR criterion that isΠ119872(sdot) = CVaR(120587RF

119872(sdot))


In addition we make the following assumptions

(1) To limit the number of parameters considered inmodel analysis we only consider RFID tag cost thefixed costs of RFID implementation include readersystem infrastructure maintenance and support andIT investments are not part of ourmodelThe detailedassessment of the above fixed costs is provided byseveral studies [32 33]

(2) For simplicity at the end of the selling season anyunsold retail product bears no salvage value or dis-posal cost in retail store at the same time we assumeunsatisfied demand incurs no loss of goodwill cost(ie shortage penalty) Related studies [34 35] showthat the assumptions of zero salvage value or holdingcost and zero loss of goodwill cost are appropriatereflections of reality for season or short life-cycleproducts

(3) We assume that the end customer demand 119863(119901) hasthe multiplicative functional form that is 119863(119901) =

119889(119901)120576 where 120576 is supported on [1198601015840 1198611015840] with 1198611015840 gt

1198601015840 ge 0 119865(sdot) is strictly increasing and differentiableon [1198601015840 1198611015840] and 119865(1198601015840) = 0 119865(1198611015840) = 1

(4) We consider the power form of price-dependentdemand factor throughout this paper that is 119889(119901) =119860119901minus119896 where119860 gt 0 119896 gt 1 see Petruzzi and Dada [36]for an excellent review and extensions

(5) In the power form of 119889(119901) = 119860119901minus119896 following Petruzziand Dada [36] we define 119909ℎ(119909) equiv 119909119891(119909)[120578 minus 119865(119909)]

that denotes the GFR (generalized failure rate) func-tion of demand distribution under CVaR criterionassume that it has the strictly increasing property(119909ℎ(119909))

1015840 gt 0 The IGFR (increasing generalized fail-ure rate) assumption is mild condition because itcaptures the most common distributions such as theuniform the normal and the exponential as well asthe gamma andWeibull families subject to parameterrestrictions

3 Centralized Policies under CVaR Criterion

31 Definition of CVaR CVaRmeasures a conditional expec-tation of the realized profit when the realized profit isnot more than a certain quantile of profit which is oftenconcerned with risk-averse decision-makers It is a coherentrisk measure with attractive computational characteristicsand consequently it is widely used in the financial fieldsFollowing Rockafellar andUryasev [37 38] andWu et al [26]CVaR maximizes the average profit of the profit falling belowa certain quantile level which is defined as the maximumprofit at a specified confidence level More formally for thegiven distribution of the profit function 120587(x y) CVaR can betreated as follows

CVaR120578(120587 (x y)) = 119864 [120587 (x y) | 120587 (x y) le Θ

120578(y)]

=1

120578int120587(xy)leΘ120578

120587 (x y) 119892 (y) 119889y(1)

where 119864[sdot] denotes expectation operator and 120578 isin (0 1]

reflects the degree of risk aversion that is a lower valueimplies a higher degree of risk aversion and 120578 = 1 implies riskneutrality x denotes decision vector y denotes randomvector 119892(y) denotes the probability density function of therandom vector y and Θ

120578denotes 120578-quantile of the random

vector y that is

Θ120578(y) = sup 120592 | Prob 120587 (x y) le 120592 le 120578 (2)

In addition a more generalized formula is introduced tocompute CVaR as follows

CVaR120578(120587 (x y))

= max120592isinR

120592 +1

120578119864 [min (120587 (x y) minus 120592 0)]

(3)

It is worth mentioning that Rockafellar and Uryasev[37 38] proved that (1) and (3) are equivalent under thegeneralized condition but as compared to (1) (3) is moreconvenient to be used inmathematical calculation and analy-sisTherefore wewill adopt (3) tomodel risk-averse problemswith retail inventory shrinkage errors in supply chain

32 CentralizedModels under CVaRCriterion In the central-ized supply chain setting we consider two different cases thatis one with item-level RFID and another without item-levelRFID We first give the general expected profit as functionsof 119901 and 119897 and characterize the optimal decisions to thecentralized systemwith item-level RFID and then we exploreoptimal decisions to the centralized system with no RFIDunder CVaR criterion Finally for assessing the benefit ofthe item-level RFID implementation we give a sufficientcondition to make supply chain manager judge whether toadopt item-level RFID

321 Model with Item-Level RFID Based on the above nota-tions and assumptions the expected profit function of thecentralized system with item-level RFID can be written as

120587RF119862(119901 119897) = 119901119864 [min (1199021015840 119863 (119901))] minus (119888

119872+ 119888119877+ 119905) 119902 (4)

where 1199021015840 is sales-available product quantity in the retail inven-tory that is 1199021015840 = 120572119902+120573(1minus120572)119902 For the end customer demand119863(119901) = 119889(119901)120576 following Petruzzi and Dada [36] we define119897 equiv 1199021015840119889(119901) as stocking factor By substituting 1199021015840 = 119889(119901)119897 into

(4) then (4) is equivalent to

120587RF119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877+ 119905)119898119897 (5)

where 119898 = 1[120572 + 120573(1 minus 120572)] and (119897 minus 119909)+= max0 (119897 minus 119909)

denotes the quantity of unsold retail product due to demanduncertainty and nonsale inventory shrinkage at the end of theselling season

In what follows consider the losses caused by demanduncertainty and nonsale inventory shrinkage may lead to themarket risk by the assumption presented in Section 2 Theretailer is risk-averse and the manufacturer has no any riskso the retailer risk attitude should be viewed as the whole


supply chain risk aversion level that is 120578119862= 120578119877The following

lemma is listed for obtaining the optimal decisions of the cen-tralized system with item-level RFID under CVaR criterion

Lemma 1 Under the CVaR constraint let Π(119901 119897 V) =

119862119881119886119877(120587119877119865119862(119901 119897)) for given 119901 119897 the unique optimal Vlowast(119901 119897) =

119889(119901)[119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] maximizes Π(119901 119897 V) where 1198601015840 le

119897 le 119865minus1(120578119877) lt 1198611015840

Proof ByΠ(119901 119897 V) = CVaR(120587RF119862(119901 119897)) from (3) the expected

profit function of the centralized system under CVaR crite-rion is shown by

Π(119901 119897 V)

= maxVisinR

V +1

120578119877

119864 [min (120587RF119862(119901 119897) minus V 0)]

(6)

Substituting (5) into (6) we have

Π(119901 119897 V) = V minus1

120578119877

int1198611015840

1198601015840

V minus 119889 (119901) 119901 [119897 minus (119897 minus 119909)+]

minus (119888119872+ 119888119877+ 119905)119898119897

+

119889119865 (119909)

(7)

Equation (7) can be rewritten by

Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

(8)

For any given 119901 and 119897 we easily get the following(1)When V le 119889(119901)[1199011198601015840 minus (119888

119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V

120597Π (119901 119897 V)120597V

= 1

(9)

(2)When 119889(119901)[1199011198601015840 minus (119888119872+ 119888119877+ 119905)119898119897] lt V le 119889(119901)[119901119897 minus

(119888119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V minus1

120578119877

int(V+119889(119901)(119888119872+119888119877+119905)119898119897)119901119889(119901)

1198601015840

V minus 119889 (119901)

sdot [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

119865(V + 119889 (119901) (119888

119872+ 119888119877+ 119905)119898119897

119901119889 (119901))

(10)

In particular

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[1199011198601015840minus(119888119872+119888119877+119905)119898119897]= 1

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[119901119897minus(119888119872+119888119877+119905)119898119897]= 1 minus

1

120578119877

119865 (119897)

(11)

(3)When V gt 119889(119901)[119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

lt 0

(12)

Based on the above cases (1)ndash(3) we can concludethat Π(119901 119897 V) is a concave function of V Let Vlowast(119901 119897) =

argmaxVisinRΠ(119901 119897 V) combining cases (1) (2) and (3) it canbe shown that

Vlowast (119901 119897) isin (119889 (119901) [1199011198601015840 minus (119888119872+ 119888119877+ 119905)119898119897] 119889 (119901)

sdot [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]]

(13)

Next in order to prove

Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (14)

where 1198601015840 le 119897 le 119865minus1(120578119877) lt 1198611015840 the following discussions are

listed(a) If 119897 lt 119865minus1(120578

119877) then

Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (15)

and we therefore have

Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) [[119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

minus119901

120578119877

int119897

1198601015840

(119897 minus 119909) 119889119865 (119909)]

(16)

(b) If 119897 ge 119865minus1(120578119877) then

Vlowast (119901 119897) = 119889 (119901) [119901119865minus1 (120578119877) minus (119888119872+ 119888119877+ 119905)119898119897] (17)

and we have

Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) 119901119865minus1 (120578119877)

minus (119888119872+ 119888119877+ 119905)119898119897 minus

119901

120578119877

int119865minus1(120578119877)

1198601015840

119865 (119909) 119889119909

(18)


It follows from (18) that

120597Π (119901 119897 Vlowast (119901 119897))120597119897

= minus119889 (119901) (119888119872+ 119888119877+ 119905)119898 lt 0 (19)

that is Π(119901 119897 Vlowast(119901 119897)) is decreasing in 119897 isin (119865minus1(120578119877) 1198611015840) so

119897 = 119865minus1(120578119877)maximizes Π(119901 119897 Vlowast(119901 119897))

In conclusion combining (a) and (b) it follows from thefacts that

Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (20)

where 1198601015840 le 119897 le 119865minus1(120578119877) lt 1198611015840 This completes the proof

By Lemma 1 we know that for any given 119901 119897 the uniqueoptimal Vlowast(119901 119897) = 119889(119901)[119901119897 minus (119888

119872+ 119888119877+ 119905)119898119897] maximizes

Π1(119901 119897 V) By substituting Vlowast(119901 119897) into (6) we get the general

expected profit function of the centralized system with item-level RFID under CVaR criterion

Π(119901 119897) = 119889 (119901) [119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(21)

where Λ(119897) = int1198971198601015840(119897 minus 119909)119889119865(119909)

Now the following theoremwill give the optimal decisionto the centralized system with item-level RFID under CVaRcriterion

Theorem 2 Under the CVaR constraint for any given 119897 isin

[1198601015840 1198611015840] and 119889(119901) = 119860119901minus119896 if 119909ℎ(119909) is IGFR that is (119909ℎ(119909))1015840 gt0 the optimal stocking factor (119897119877119865

119862)lowast is uniquely determined by

119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (22)

and the unique optimal order quantity (119902119877119865119862)lowast is listed by

(119902119877119865

119862)lowast

= 119860119898(119897119877119865

119862)lowast[

[

(120578119877minus 119865 ((119897119877119865

119862)lowast

))

119898120578119877(119888119872+ 119888119877+ 119905)

]

]

119896

(23)

Proof By 119902 = 119898119897119889(119901) and 119889(119901) = 119860119901minus119896 we have 119901 =

(119860119898119897119902)1119896 For any given 119897 119902 substituting 119901 = (119860119898119897119902)

1119896

into (21) (21) can be written by

Π(119897 119902) = 119860(119902

119860119898119897)minus(1minus119896)119896

[119897 minus1

120578119877

Λ (119897)]

minus (119888119872+ 119888119877+ 119905) 119902

(24)

and taking the first-order partial derivation of Π(119897 119902) withrespect to 119897 we obtain that the necessary condition formaximizing Π(119897 119902) is

120597Π (119897 119902)

120597119897=11986011198961199021minus1119896

1198961205781198771198972minus1119896

[120578119877119897 minus 119896119897119865 (119897) + (119896 minus 1) Λ (119897)]

= 0

(25)

Let 119885(119897) = 120578119877119897 minus 119896119897119865(119897) + (119896 minus 1)Λ(119897) and notice that the first

factor in (25) is always positive so first necessary conditiononly requires that the optimal stocking factor (119897RF

119862)lowast satisfies

119885(119897) = 0 solving 119885(119897) = 0 we get the optimal (119897RF119862)lowast

determined by 119865(119897) = (120578119877119897 + (119896 minus 1)Λ(119897))119896119897

Next we will prove the existence of the optimal (119897RF119862)lowast

It is obvious that 119885(119897) is continuous in the support set [1198601015840119865minus1(120578

119877)] After some manipulation we get119885(1198601015840) = 1198601015840120578

119877gt 0

and 119885(119865minus1(120578119877)) = (1 minus 119896)120578

119877119865minus1(120578

119877) + (119896 minus 1) int

119865minus1(120578119877)

1198601015840

119865(119909)119889119909

Since int119865minus1(120578119877)

1198601015840 119865(119909)119889119909 le 119865(119865minus1(120578

119877))(119865minus1(120578

119877) minus 1198601015840) =

120578119877(119865minus1(120578

119877) minus 1198601015840) we have 119885(119865minus1(120578

119877)) le (1 minus 119896)1198601015840 lt 0

hence there exists the optimal (119897RF119862)lowast that satisfies 119885(119897) = 0

in the support set (1198601015840 119865minus1(120578119877))

Furthermore to verify the uniqueness of the optimal(119897RF119862)lowast we have 1198851015840(119897) = (120578

119877minus 119865(119897))(1 minus 119896119897ℎ(119897)) and 11988510158401015840(119897) =

minus1198851015840(119897)ℎ(119897)minus119896(120578119877minus119865(119897))(119897ℎ(119897))1015840 Since (119897ℎ(119897))1015840 gt 0 by Lemma 1

119897 le 119865minus1(120578119877) 119897 isin [1198601015840 1198611015840] we easily gain 11988510158401015840(119897)|

1198851015840(119897)=0

lt

0 which implies that 119885(119897) is unimodal function Thus theoptimal (119897RF

119862)lowast is unique

From (22) we find that the optimal stocking factor (119897RF119862)lowast

does not depend on the order quantity 119902 Substituting (22)into (24) we get

Π((119897RF119862)lowast

119902)

=1198961198601119896119898minus1+1119896 ((119897RF

119862)lowast

)1+1119896

1199021minus1119896 (120578119877minus 119865 ((119897RF

119862)lowast

))

(119896 minus 1) 120578119877

minus (119888119872+ 119888119877+ 119905) 119902

(26)

In what follows we can show that

119889Π((119897RF119862)lowast

119902)

119889119902

=1198601119896119898minus1+1119896

((119897RF119862)lowast

)1+1119896

119902minus1119896

(120578119877minus 119865 ((119897

RF119862)lowast

))

120578119877

minus (119888119872+ 119888119877+ 119905)

(27)

and solving 119889Π((119897RF119862)lowast 119902)119889119902 = 0 we get (119902RF

119862)lowast

=

119860119898(119897RF119862)lowast[(120578119877minus 119865((119897RF

119862)lowast))119898120578

119877(119888119872+ 119888119877+ 119905)]119896 Meanwhile

we easily gain 1198892Π1((119897RF119862)lowast 119902)1198891199022|

119902=(119902RF119862)lowast lt 0 According to

the second-order sufficient condition there exists the uniqueoptimal (119902RF

119862)lowast that maximizes Π((119897RF

119862)lowast 119902) This completes

the proof

The above theorem shows that it does not have anyrequirement on problem parameters other than the demanddistribution itself to determine the optimal decisions ofthe centralized system under CVaR criterion It should bepointed out that in Theorem 2 when 120578

119877= 1 and 120572 = 1

the optimal inventory factor (119897RF119862)lowast is the same as Wang et al

[39] and Li and Hua [40] In addition for the optimal order


quantity (119902RF119862)lowast by 119901 = (119860119898119897119902)1119896 we can get the optimal

retail price

(119901RF119862)lowast

=119898120578119877(119888119872+ 119888119877+ 119905)

120578119877minus 119865 ((119897RF

119862)lowast) (28)

Substituting (119901RF119862)lowast and (119897RF

119862)lowast into (21) the maximum

expected profit of the centralized system with item-levelRFID under CVaR criterion is given by

Π((119897RF119862)lowast

)

=119860 (119897RF119862)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

((120578119877minus 119865 ((119897RF

119862)lowast

))

120578119877

)

119896

(29)

322 Model without Item-Level RFID Similarly by theassumption presented above for the case without item-levelRFID (where 119905 = 0 and 120573 = 0) let 1198981015840 = 1120572 the expectedprofit function of the centralized system under no RFID canbe written as

120587No119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877)1198981015840119897 (30)

Now let Π1015840(119901 119897) = maxVisinRCVaR(120587No119862(119901 119897)) the optimal

decision to the centralized system without item-level RFIDunder CVaR criterion is given by the following theorem

Theorem3 In the centralized systemwithout item-level RFIDif 119909ℎ(119909) is IGFR then the decision vector ((119901119873119900

119862)lowast (119897119873119900119862)lowast) is the

unique maximizer of Π1015840(119901 119897) where

(119901119873119900

119862)lowast

=1198981015840120578119877(119888119872+ 119888119877)

120578119877minus 119865 ((119897119873119900

119862)lowast) (31)

and (119897119873119900119862)lowast is described by

119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (32)

Proof This proof is similar to the proof procedures ofTheorem 2 thus we here omit this proof

Similarly according to Theorem 3 we can easily derivethe optimal order quantity as

(119902No119862)lowast

= 1198601198981015840(119897

No119862)lowast[

[

120578119877minus 119865 ((119897

No119862)lowast

)

1198981015840120578119877(119888119872+ 119888119877)]

]

119896

(33)

and the maximum expected profit of the centralized systemwithout item-level RFID under CVaR criterion is given by

Π1015840((119897

No119862)lowast

)

=119860 (119897No119862)lowast

(119896 minus 1) [1198981015840 (119888119872 + 119888119877)]119896minus1

(120578119877minus 119865 ((119897No

119862)lowast

)

120578119877

)

119896

(34)

The following proposition discusses how the optimaldecisions change with model parameters in the centralizedsystem under CVaR criterion

Proposition 4 If 119909ℎ(119909) is IGFR then the following hold

(1) (119897119877119865119862)lowast= (119897119873119900119862)lowast

(2) Both (119897119877119865119862)lowast and (119897119873119900

119862)lowast are not affected by the sales-

available proportion 120572 but they are increasing in 120578119877

(3) Both (119901119877119865119862)lowast and (119901119873119900

119862)lowast are decreasing in 120572 let119866(119909) =

(119909ℎ(119909)[119896119865(119909) minus 120578119877] minus 119865(119909))(1 minus 119896119909ℎ(119909)) and thus

(i) when 119866((119897119877119865119862)lowast) gt 0 (119901119877119865

119862)lowast is increasing in 120578

119877

when 119866((119897119877119865119862)lowast) lt 0 (119901119877119865

119862)lowast is decreasing in 120578

119877

when119866((119897119877119865119862)lowast) = 0 (119901119877119865

119862)lowast is not affected by 120578

119877

(ii) when119866((119897119873119900119862)lowast) gt 0 (119901119873119900-119877119865


119877

when 119866((119897119873119900119862)lowast) lt 0 (119901119873119900


119877

when119866((119897119873119900119862)lowast) = 0 (119901119873119900


119877

Proof Part (1) Comparing (22) with (32) we can easily get theresult that (119897No

119862)lowast= (119897RF119862)lowast

Part (2) Since (22) and (32) do not involve the sales-available proportion 120572 both (119897No

119862)lowast and (119897RF

119862)lowast are not affected

by 120572 From (22) the optimal (119897RF119862)lowast satisfies 119885((119897RF

119862)lowast) =

0 By the implicit function rule 119889(119897RF119862)lowast119889120578119877

= minus(119897RF119862)lowast

(120597119885((119897RF119862)lowast)120597(119897RF119862)lowast) according to proof ofTheorem 2119885(119860) =

119860120578119877gt 0 119885(119865minus1(120578

119877)) le (1 minus 119896)119860 lt 0 and (119897RF

119862)lowast solving

119885((119897RF119862)lowast) = 0 is unique It implies 120597119885((119897RF

119862)lowast)120597(119897RF119862)lowastlt 0

Thus we get 119889(119897RF119862)lowast119889120578119877gt 0 Similar to the proof procedures

of 119889(119897RF119862)lowast119889120578119877gt 0 we can gain 119889(119897No

119862)lowast119889120578119877gt 0

Part (3) From (28) and (31) we easily show that (119901RF119862)lowast is

increasing in119898 and (119901No119862)lowast is increasing in1198981015840 in conjunction

with 119898 = 1[120572 + 120573(1 minus 120572)] and 1198981015840 = 1120572 they imply that 119898and 1198981015840 are decreasing in 120572 we thus have the fact that both(119901No119862)lowast and (119901RF

119862)lowast are decreasing in 120572

(i) From (28) taking the first derivative of (119901RF119862)lowast

with respect to 120578119877 we have 119889(119901RF

119862)lowast119889120578119877

=

((120578119877119891((119897RF119862)lowast)(119889(119897RF119862)lowast119889120578119877) minus 119865(119897RF

119862)lowast)[120578119877

minus

119865(119897RF119862)lowast]2)119898(119888119872+ 119888119877+ 119905) furthermore by the proof

of Proposition 4(2) we get 119889(119897RF119862)lowast119889120578119877= minus(119897RF

119862)lowast

1198851015840(119897)|119897=(119897

RF119862)lowast By simply substituting 119889(119897RF

119862)lowast119889120578119877

into 119889(119901RF119862)lowast119889120578119877 so we derive that 119889(119901RF

119862)lowast119889120578119877=

(119898(119888119872

+ 119888119877+ 119905)[120578

119877minus 119865(119897RF

119862)lowast]2)119866((119897RF119862)lowast) which

implies that the monotone behavior of (119901RF119862)lowast can

be determined by the sign of 119866((119897RF119862)lowast) Thus when

119866((119897RF119862)lowast) gt 0 (119901RF


119877 when

119866((119897RF119862)lowast) lt 0 (119901RF


119877 when

119866((119897RF119862)lowast) = 0 (119901RF


119877

(ii) The proof is similar to the proof procedures of Part3(i) we can gain the monotone behavior of (119901No

119862)lowast

with respect to 120578119877 thus we here omit this proof

Proposition 4(1) implies that the optimal stocking factordoes not depend on whether the centralized system adoptsitem-level RFID or not and it seems to depend heavily ondemand distribution


Proposition 4(2) states that the optimal stocking factors(119897RF119862)lowast and (119897No

119862)lowast are independent of the sales-available pro-

portion parameter they only depend on the risk aversionvalue and increase with 120578

119877increases Because a higher value

of 120578119877implies a lower degree of risk aversion it implies that a

lower degree of risk aversion may lead to a higher stockingfactor that is in the centralized case if a supply chainmanager has less fear of risk he or she always tends to ordermore to meet market demand no matter whether to employitem-level RFID or not

Proposition 4(3) states that the optimal retail prices(119901

RF119862)lowast and (119901No

119862)lowast deceasewith the sales-available proportion

increases it means that a higher sales-available rate may beable to make supply chain manager set a lower retail pricefor attracting customers to buy more but the relationshipbetween the optimal retail price and the risk aversion leveldoes not absolutely increase or decrease it depends on thesign of 119866((119897RF

119862)lowast) or 119866((119897No

119862)lowast) that is for adopting item-level

RFID case when 119866((119897RF119862)lowast) gt 0 the optimal retail price

increases with 120578119877increases and it implies that if a supply

chain manager is risk-averse enough he or she is more likelyto set a lower retail price to avoid the risk caused by mar-ket uncertainty and nonsale inventory shrinkage when119866((119897

RF119862)lowast) lt 0 the optimal retail price decreases with 120578

119877

increases it means that if a supply chainmanager has less fearof risk he or she may raise hisher retail price and order lessto balance the relationship between the expected benefit andthe risk when 119866((119897RF

119862)lowast) = 0 the optimal retail price is not

affected on the risk aversion level it only depends on somespecial demand distributions

To assess the benefit of the item-level RFID imple-mentation in the centralized situation under CVaR cri-terion we introduce the auxiliary function as Δ =

ln[Π((119897RF119862)lowast)Π1015840((119897No

119862)lowast)] where Π((119897RF

119862)lowast) and Π1015840((119897No

119862)lowast) are

given by (29) and (34) respectively Note that the auxiliaryfunction Δ can be used to judge whether to adopt item-level RFID that is if Δ gt 0 it means an item-level RFIDimplementation can bringmore expected profit thannoRFIDcase in centralized system but if Δ le 0 it means that ascompared to item-level RFID system one case without RFIDis a better choice We will discuss how the model parametersaffect item-level RFID implementation in the following

Proposition 5 (1) Δ is independent of 120578119877and is decreasing in

119905(2) Δ is decreasing in 120572 but is increasing in 120573

Proof Part (1) By (29) and (34) after some single algebra wederive the function as

Δ = (119896 minus 1) ln [120572 + 120573 (1 minus 120572)

120572sdot

119888119872+ 119888119877

119888119872+ 119888119877+ 119905

] (35)

From (35) Δ does not have 120578119877 so Δ is independent of 120578

119877 and

the conclusion that Δ decreases in 119905 is obviousPart (2) Equation (35) can also be written as Δ = (119896 minus

1) ln[[(1 minus 120573) + 120573120572] sdot ((119888119872+ 119888119877)(119888119872+ 119888119877+ 119905))] so we easily

reach the conclusion thatΔ is decreasing in 120572 Using a similarargument we can also gain that Δ is increasing in 120573

In fact (35) can be viewed as a sufficient condition tojudge whether to adopt item-level RFID and Proposition 5states that although the retailer is risk-averse the judgmentfunction Δ is not affected by the risk-averse level 120578

119877 it

only depends on some parameters such as 120572 120573 and 119905 Inother words the risk-averse level is not an effective incentivefor supply chain manager to adopt item-level RFID systemhowever the sales-available rate and the tag cost are mainlydriving factors in evaluating the benefit of an item-level RFIDIn light of this we will give the threshold values of 120572 120573 and 119905in the following theorem

Theorem 6 Under the CVaR constraint for 0 lt 120572 120573 le 1 if119905 isin (0 119905) (119905 = 120573(119888

119872+ 119888119877)(1 minus 120572)120572) then Δ gt 0 if 119905 isin [119905 +infin)

then Δ le 0

Proof From (35) in order to show Δ gt 0 (le0) we only needto show [(1 minus 120573) + 120573120572] sdot ((119888

119872+ 119888119877)(119888119872+ 119888119877+ 119905)) gt 1 (le1)

which can be written by 119905 lt (ge) 119905 = 120573(119888119872+ 119888119877)(1 minus 120572)120572

Therefore we have the following if 119905 isin (0 119905) then Δ gt 0 if119905 isin [119905 +infin) then Δ(120572 120573 119905) le 0

Theorem 6 gives a threshold value of 119905 that is 119905 = 120573(1 minus

120572)(119888119872+ 119888119877)120572 and when the RFID tag cost 119905 is lower than

threshold value 119905 item-level RFID implementation can bringmore expected profit otherwise the supply chain will sufferlosses at 119905 ge 119905 Likewise the threshold values of 120572 120573 aresummarized in Table 1 for more details and furthermorethe impact of the key parameters on supply chains optimaldecisions will be discussed in Example 1

From the discussion above we know that the model withitem-level RFID is more generalized than no RFID case inthe centralized supply chain system that is when 119905 = 0 and119898 = 119898

1015840 themodelwith item-level RFID reduces to themodelwithout RFID system Therefore we only explore one casewith RFID technology in the following decentralized supplychain analysis the other scenarios are shown in Table 2

4 Decentralized Policies underCVaR Criterion

In this section we explore the optimal policies for a decen-tralized supply chain with a separate manufacturer and aseparate retailer and then we discuss a wholesale pricecontract Furthermore we study a revenue sharing contractfor coordinating the supply chain which concentrates onthe allocation of the expected sale revenue between themanufacturer and the retailer

41 Wholesale Price Contract We here consider that facingnonsale inventory shrinkage phenomena and demand uncer-tainty in the retail setting the retailer (like Wal-Mart Targetetc) takes the initiative in employing RFID for achievinga higher product availability and bears all of the RFIDtags cost The manufacturer needs to decide wholesale pricecontract parameters to achieve hisher performance Theorder quantity is delivered to the retailer before the sellingseason and transfer payments are made between supplychain players based on the agreed contract


Table 1 Item-level RFID implementation cases for the key parameters 120572 120573 and 119905

The parameter The parameterrsquos threshold value The parameterinterval The sign of Δ Use RFID

120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo

Table 2 Summary of optimal decisions and profits in different scenarios under CVaR criterion

Decentralized systems Centralized systemsRFID No RFID RFID No RFID

Optimal sale price 119896

119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840

Optimal order quantity (119896 minus 1

119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896

Optimal wholesale price119905 + 119888119877+ 119896119888119872

119896 minus 1

119888119877+ 119896119888119872

119896 minus 1mdash mdash

Retailerrsquos expected profit (119896 minus 1

119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1

Π1015840lowast mdash mdash

Manufacturerrsquos expected profit (119896 minus 1

119896)

119896

Πlowast

(119896 minus 1

119896)

119896


Supply chain expected profit ((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast

Note119872 = 1198721015840 = 120578119877(120578119877 minus 119865((119897RF119862 )lowast))119873 = 119888119872 + 119888119877 + 119905119873

1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)

119896minus1119872119896 and Π1015840lowast = 119860(119897RF

119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896

In decentralized supply chain system with item-levelRFID under CVaR criterion the retailerrsquos expected profitfunction is similar to the function of centralized system inSection 3 so we here directly give the following

120587RF119877(119901 119897)

= 119889 (119901) 119901 [119897 minus (119897 minus 119909)+] minus (119908 + 119888

119877+ 119905)119898119897

(36)

and substituting (35) into (3) the retailerrsquos expected profitfunctionwith item-level RFIDunderCVaR criterion is shownby

CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877

sdot 119864 [min (120587RF119877(119901 119897) minus V

119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF

119877(119901 119897)) similar to the proof

of Lemma 1 there also exists the unique optimal Vlowast(119901 119897) =119860119901minus119896119897[119901 minus (119908 + 119888

119877+ 119905)119898] which maximizes Π

119877(119901 119897 V(119901 119897))

and the retailerrsquos expected decision function becomes

Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)

In what follows we provide a theorem for getting theretailerrsquos optimal decision

Theorem 7 In decentralized setting for the retailer if 119909ℎ(119909)is IGFR then the decision vector ((119901119877119865

119877)lowast (119897119877119865119877)lowast) is the unique

maximizer of Π119877(119901 119897) where (119897119877119865

119877)lowast is determined by

119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)

Proof Similar to the proof of Theorem 2 thus we here omitthis proof

According to Theorem 7 by 119901 = (119860119898119897119902)1119896 we can

derive the retailerrsquos optimal order quantity as

(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)


and the maximum expected profit is given by

Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)

From Theorems 2 3 and 7 we easily find that theretailers optimal stocking factor is always equal to that of thecentralized system that is (119897RF

119877)lowast= (119897

RF119862)lowast= (119897

No119862)lowast it seems

to depend heavily on demand distribution and risk-averselevel and does not depend on some parameters such as 119905 120572and 120573

Knowing the retailerrsquos order quantity (119902RF119877)lowast the manu-

facturerrsquos expected profit function is easily written as

Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)

For obtaining the manufacturerrsquos optimal decision weshow the following theorem

Theorem 8 The optimal wholesale price for manufacturer isunique and is given by 119908lowast = (119905 + 119888

119877+ 119896119888119872)(119896 minus 1)

Proof Recall that (119897RF119877)lowast chosen by the retailer does not

depend on 119908 from (41) we can gain that the necessary con-dition for the maximum of Π

119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)

and notice that the first four terms in the left part of (42) areeach positive so it only requires the optimal wholesale price119908lowast which satisfies [1 minus 119896(119908 minus 119888

119872)(119908 + 119888

119877+ 119905)] = 0 After

simple manipulation we give 119908lowast = (119905 + 119888119877+ 119896119888119872)(119896 minus 1)

Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)

and we therefore conclude that Π119872(119908) is strictly concave in

119908 and the optimal wholesale price 119908 = 119908lowast is unique

Remark 9 In fact substituting 119908 = 119908lowast into (41) and (42)both the optimal retailerrsquos expected profit and the optimalmanufacturerrsquos expected profit are shown respectively thatis

Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

sdot (119896 minus 1

119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)

Under a wholesale price contract it is not difficult to find that

Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)

lt Π((119897RF119862)lowast

)

(47)

which shows that the total of decentralized supply chain profitis always lower than the centralized case More specificallyit means that more than 26 percent (((119890 minus 2)119890) lowast 100)of the whole supply chainrsquos profit is lost due to doublemarginalization meanwhile it also implies that there existsa potential incentive to coordinate between supply chainplayers

In what follows we discuss supply chain coordination forachieving the best performance of the entire supply chainFollowing Cachon [41] and He et al [42] a contract designedby themanufacturer is said to coordinate the supply chain if itsatisfies the first-order condition of centralized supply chainrsquosprofit function at (119897RF

119862) and (119902RF

119862) By the above discussion

we know (119897RF119877)lowast= (119897RF119862)lowast so there is only a need to satisfy

(119902RF119877)lowast= (119902RF119862)lowast After simple manipulation we can see that

there exists 119908 = 119888119872

which is required to coordinate theretailerrsquos order quantity However it will directly lead to zeroprofit for the manufacturer so the wholesale price contractcannot coordinate the supply chain


42 Wholesale Price Contract with Revenue Sharing In thissection we further discuss supply chain coordination underwholesale price contract with revenue sharingThemanufac-turer offers a lower wholesale price to the retailer and sharesa fraction of sale revenue earned by the retailer The retailerdecides sale price and order quantity No money changeshands unless an item is sold Let 120574 be a proportion of salerevenue kept by the retailer and the other 1 minus 120574 is shared bythe manufacturer

Under wholesale price contract with revenue sharingusing similar arguments the retailerrsquos general expected profitfunction under CVaR criterion is

Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)

and the manufacturerrsquos general expected profit function is

Π119872(119908 120574) = 119889 (119901)

sdot [(1 minus 120574) 119901 + (119908 minus 119888119872)119898] 119897 minus

(1 minus 120574) 119901Λ (119897)

120578119877

(49)

Theorem 10 Under the wholesale price contract with revenuesharing for given 120574 if contract parameters satisfy the condition119908 = 120574119888

119872+ (120574 minus 1)(119888

119877+ 119905) and 119889(119901) = 119860119901minus119896 then the vector

((119901119877119865119862)lowast (119897119877119865119862)lowast) is also optimal decision in decentralized case

Proof Substituting119908 = 120574119888119872+(120574minus1)(119888

119877+119905) and 119889(119901) = 119860119901minus119896

into (48) and (49) we have

Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)

For a given 120574 it means that when Π(119901 119897) reaches itsmaximum value in ((119901RF

119862)lowast (119897RF119862)lowast) Π119877(119901 119897 120574) and Π

119872(119908 120574)

can also achieve their optimal expected profits respectivelythat is the vector ((119901RF

119862)lowast (119897RF119862)lowast) is also optimal decision in

that case

From Theorem 10 we can see that under the wholesaleprice contract with revenue sharing an arbitrary allocationof the optimal centralized supply chain profit between themanufacturer and the retailer can be achieved by changingthe proportion parameter 120574

Further from (29) (45) and (46) combined withTheorem 10 we have the next corollary

Corollary 11 If the revenue sharing proportion parameter 120574satisfies the fact that 119908 = 120574119888

119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)

The above corollary shows that setting appropriate con-tract parameters the wholesale price contract with revenuesharing can coordinate the supply chain and both themanufacturer and the retailer get expected profits higher thantheir respective reservation expected profits It is implied thatif properly designed it is attractive for both parties of supplychain to accept the coordination contract and the contractparameter 120574 depends on the partnersrsquo bargaining power

In fact from the above discussion we can also see thatwhile gaining more profit from sale revenue the manufac-turer takes a part of risk caused by demand uncertainty andnonsale inventory shrinkage in return for the retailer heshebears less risk and earns more profit from the coordinationcontract which can achieve a win-win situation for supplychain partners

5 Number Examples and Sensitivity Analysis

In this section we implement sensitivity analysis of the keyparameters 120572 120573 and 119905 to illustrate the impact of changesof parameters on supply chainrsquos optimal decisions underCVaR criterion and then we further explore the impact ofthe retailerrsquos risk attitude inventory shrinkage rate demanduncertainty and price-elasticity index of the demand on thetotal supply chainrsquos profit with coordination for getting moreinsights In our number examples the base values of theparameters are listed as follows 120572 = 06 120573 = 08 119905 = 02119896 = 31198601015840 = 0119860 = 500 120578

119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06

the random component of the demand 120576 is assumed to followthe normal distribution with 120583 = 50 and 1205902 = 52

Example 1 (the impact of 120572 120573 and 119905) In this subsectionwe calculate supply chainrsquos optimal solutions with threepossible cases (1) different values of 120572 and different valuesof 120573 (2) different values of 120572 and different values of 119905 (3)different values of 120573 and different values of 119905 The sensitivityanalysis is performed by respectively changing the valueof two parameters but keeping other parameters constantFor notational convenience we do not distinguish betweenRFID case and no RFID case that is we generally use 119902lowast

119877

to represent the retailerrsquos optimal order quantity with RFIDor without RFID under CVaR criterion and the similarnotations will be used in 119901

lowast

119877 119908lowast Πlowast

119877 Πlowast119872 119901lowast119862 119902lowast119862 and Πlowast

The corresponding results are shown in Tables 3ndash5From Table 3 no matter which case happens the central-

ized system or the decentralized system we can observe thefollowing (1) the centralized supply chainrsquos optimal profitsΠlowast the retailerrsquos optimal profits Πlowast

119877 and the manufacturerrsquos

optimal profits Πlowast119872all increase as the sales-available propor-

tion 120572 and the recovery rate 120573 increase (2) the centralizedoptimal order quantity 119902lowast

119862and the retailerrsquos optimal order

quantity 119902lowast119877all increase as120572 and120573 increase (3) the centralized

optimal retail price 119901lowast119862and the retailerrsquos optimal retail price

119901lowast119877all decrease as 120572 and 120573 increase but the manufacturerrsquos

wholesale price 119908lowast is not influenced by the sales-availableproportion 120572 and the positive value of the recovery rate 120573Table 3 also illustrates that compared to no RFID case (suchas 120572 = 06 120573 = 0) if item-level RFID performs quite well


Table 3 The optimal decisions and profits with varying 120572 and 120573

120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877

119908lowast Πlowast119877

Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943


(such as 120573 = 07 08 09) the benefits from RFID is largerthan the costs incurred by adopting RFID and the higherthe recovery rate the more superior the item-level RFID inimproving the supply chain performance

From Table 4 when the recovery rate is constant anditem-level RFID is employed Πlowast Πlowast

119877 Πlowast119872 119902lowast119862 and 119902lowast

119877all

decrease as the tag cost 119905 increases but 119901lowast119862 119901lowast119877 and 119908lowast

all increase as the tag cost 119905 increases In other words forthe retailer the manufacturer and the centralized systemwhen the recovery rates remain unchanged their benefits willbe reduced due to higher tag price In fact by comparingdifferent cases (such as120572 = 09 119905 = 03 and120572 = 06 119905 = 01) itis not difficult to find that when sales-available proportion isfairly small and the tag price is lower the effect of adoptingitem-level RFID is much better but when the tag price ishigher the benefits from RFID could not compensate for thecosts incurred by using RFID and no RFID case is a betterchoice (such as 120572 = 06 119905 = 0 and 120572 = 06 119905 = 05)

From Table 5 when the tag prices remain unchanged asthe recovery rate increases Πlowast Πlowast


119877increase

and 119901lowast119862and 119901lowast

119877decrease but when the recovery rates remain

unchanged as the tag price increases the opposite of theaforementioned results is true In addition we notice thatalthough item-level RFID performs quite well (such as 119905 = 07and 120573 = 09) as compared to no RFID case (such as 119905 = 0 and120573 = 0) it does not offer a better decision due to higher tagcost and the higher the tag cost the greater the harm to thesupply chain performance

From the above analysis we find that the tag costthe recovery rate and the sales-available proportion caninfluence supply chain performance to be specific when thesales-available proportion is of lower level (higher shrinkagerate) tag cost is quite small and when the recovery rate isof higher level it is very important for adopting the item-level RFID to improve supply chainrsquos performanceThereforein practice there is a need to exactly assess the thresholdvalues of the tag cost the recovery rate and the sales-availableproportion to make better choices

Example 2 (the impact of 120590 1 minus 120572 and 119896) In this subsectionwe firstly calculate the total supply chainrsquos profit by varying 120590from 0 to 10 in different risk settings (see Figure 2) Next forthe given 120590 = 5 and 119896 = 2 we vary 1 minus 120572 from 005 to 04to study the impacts of the shrinkage rate on the total supplychainrsquos profit with coordination in different risk settings (seeFigure 3) Finally we change the price-elasticity index from21 to 29 in different RFID tag costs to explore the impact ofthe price-elasticity index on the total supply chainrsquos profitwithcoordination (see Figure 4)

In Figure 2 interestingly we find that for each risksetting as the standard deviation 120590 increases the total supplychainrsquos profit with coordination firstly increases and thendecreases but when 120590 is smaller the more the supply chainsare risk-averse the more all the supply chains get profit fromcoordination when 120590 is greater the opposite of the aboveresult is true An intuitive explanation for this fact is thata larger value of the standard deviation 120590 means a higher

1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion

Figure 2 Impact of changes of 120590 on the total supply chainrsquos profitwith coordination in different 120578

119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500

Figure 3 Impact of changes of 1minus120572 on the total supply chainrsquos profitwith coordination in different 120578

119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04



uncertainty in demand in early stage for the risk-aversesupply chain the superiority of improving performance isobvious but in later stage for the risk-averse supply chainwith more fear of risk the more dramatic the uncertaintyin demand the more dramatic the decrease in total supplychain sales revenue They care about their expected profits ina conservative attitude

Recall the above discussion 1 minus 120572 can be viewed asshrinkage rate which is often concerned with risk-aversesupply chain in the retail setting In Figure 3 we find thatregardless of risk neutrality (120578

119862= 1) or risk aversion (120578

119862lt 1)

the total supply chainrsquos profit decreases as the shrinkage rateincreases and meanwhile the more the supply chains arerisk-averse the more the total supply chainrsquos profit is forcoordination Moreover we also find that compared to riskneutrality case the risk-averse supply chain always gets moreprofits

For the power form of price-dependent demand factor119889(119901) = 119860119901

minus119896 the parameter 119896 is the price-elasticity indexand the larger the value of 119896 is the more sensitive thedemand is to a change in price In Figure 4 for the case withor without item-level RFID the total supply chainrsquos profitdecreases as the price-elasticity index increases and as thetag cost increases the total supply chainrsquos profit decreasesThat is because the larger the price-elasticity index the moredramatic the decrease in demand with an increase in retailprice The supply chain only gets less profits from the lessend customer demand and meanwhile with price-elasticityindex increasing the supply chain with RFID always getsmore profits than the one without RFID

In fact from Figures 2 3 and 4 the total supply chainperformance from coordination not merely depends heavilyon the isoprice-elastic demand function form we also findthat for different risk levels the effect of the demanduncertainty and the shrinkage rate on the total supply chainrsquosprofit is very significant therefore it is necessary for supplychainmanager to exactly evaluate these parameters in the realworld

6 Summary and Conclusions

Most of the literatures on the application of RFID inimproving product availability assumed that the reliabilityof RFID is very perfect and shrinkage errors can be elim-inated completely However in practice nonsale inventoryshrinkage problems can only be eliminated partly due toRFID misreading In this paper we further explore supplychain optimization and coordination from imperfect RFIDperspective We develop supply chain models with price-dependent stochastic demand in both centralized scenarioand decentralized scenario under a conditional value-at-risk(CVaR) criterion and we analyze the optimal supply chainrsquosdecisions in the two different scenarios

In centralized scenario we give the optimal supply chainrsquosdecisions under item-level RFID or no RFID case (seeTable 2) worthwhile to mention is that regardless of item-level RFID case or no RFID case the optimal stocking factorshave no any requirement on model parameters other than

the demand distribution itself and the risk aversion level andthe optimal stocking factor does not depend on whether toadopt item-level RFID or not in particular setting 120578

119877= 1

and 120572 = 1 the optimal inventory factor is the same as theone proposed by Wang et al [39] and Li and Hua [40] theincentives of the centralized system to employ RFID are notaffected by the risk aversion level they mainly depend on therelative values of the sales-available rate the recovery rate ofRFID and the tag cost respectively

In decentralized scenario we only explore one case withRFID technology and consider two widely used contractsincluding wholesale price contract and revenue sharingcontract Unfortunately we find that the total of decentralizedsupply chain profit is no more than 74 ((2119890) lowast 100) of thecentralized supply chain profit due to double marginalizationunder wholesale price contract Furthermore we discusssupply chain coordination for achieving the best performanceof the entire supply chain under revenue sharing contractwe find that Pareto improving outcome will be achieved ifthe retailer sets an appropriate revenue sharing proportionparameter for the manufacturer interestingly under thiscontract for the manufacturer and the retailer while gettingexpected profits higher than their respective reservationexpected profits they actually achieve coaffording risk whichcould have been taken by the retailer alone

Finally we implement sensitivity analysis of the keyparameters to illustrate the impact of changes of parameterson supply chain performance and the total supply chainrsquosprofit with coordination number results show that especiallyif the sales-available proportion is of lower level (highershrinkage rate) tag cost is quite small and when the recoveryrate is of higher level supply chain partnersrsquo profits obtainedby investment in RFID are improved significantly and fordifferent risk levels the effects of the demand uncertainty andthe shrinkage rate on supply chainrsquos profit with coordinationare very obvious

There are several possible extensions for further researchIn this paper we only consider and analyze the supplychain including a risk-averse retailer and a risk-neutralmanufacturer on this topic A natural extension is to assumethat supply chain composed of a manufacturer and a retaileris risk-averse In addition it is worth consideration that onemanufacturer sells to two or multiple competing retailersWhat strategy can be used to coordinate these agentsrsquoperformance when the retailers face inventory shrinkageand demand uncertainty It would also be interesting toinvestigate how different risk attitudes affect supply chainperformance

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China (nos 71472134 and 71072155)


References

[1] Y Rekik ldquoInventory inaccuracies in the whole sale supplychainrdquo International Journal of Production Economics vol 5 no2 pp 3ndash10 2010

[2] A G Kok and K H Shang ldquoEvaluation of cycle-count policiesfor supply chains with inventory inaccuracy and implica-tions on RFID investmentsrdquo European Journal of OperationalResearch vol 237 no 1 pp 91ndash105 2014

[3] Y Kang and S B Gershwin ldquoInformation inaccuracy in inven-tory systems stock loss and stockoutrdquo IIE Transactions vol 37no 9 pp 843ndash859 2005

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

[5] A Bednarz D Dubie and R Langford Playing Tag NewsFactor Network 2003

[6] ECR Europe Shrinkage A Collaborative Approach to ReducingStock Loss in the Supply Chain ECR Europe Brussels Belgium2003

[7] 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

[8] H Y Dai andM M Tseng ldquoThe impacts of RFID implementa-tion on reducing inventory inaccuracy in a multi-stage supplychainrdquo International Journal of Production Economics vol 139no 2 pp 634ndash641 2012

[9] L Hervert-Escobar N R Smith J R Rodrıguez-Cruz and LE Cardenas-Barron ldquoMethods of selection and identificationof RFID tagsrdquo International Journal of Machine Learning andCybernetics vol 6 no 5 pp 847ndash857 2015

[10] H E Talavera J Banks N R Smith and L E Cardenas-BarronldquoEnhancing the management of shared inventory in the steelindustry using RFID an alternative to bar codesrdquo InternationalJournal of Machine Learning and Cybernetics vol 6 no 5 pp733ndash745 2015

[11] J J Roh A Kunnathur and M Tarafdar ldquoClassification ofRFID adoption an expected benefits approachrdquo Information ampManagement vol 46 no 6 pp 357ndash363 2009

[12] E W T Ngai K K L Moon F J Riggins and C Y Yi ldquoRFIDresearch an academic literature review (1995ndash2005) and futureresearch directionsrdquo International Journal of Production Eco-nomics vol 112 no 2 pp 510ndash520 2008

[13] A Sarac N Absi and S Dauzere-Peres ldquoA literature review onthe impact of RFID technologies on supply chainmanagementrdquoInternational Journal of Production Economics vol 128 no 1 pp77ndash95 2010

[14] A G De Kok K H Van Donselaar and T van Woensel ldquoAbreak-even analysis of RFID technology for inventory sensitiveto shrinkagerdquo International Journal of Production Economicsvol 112 no 2 pp 521ndash531 2008

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

[16] Y Rekik Z Jemai E Sahin and Y Dallery ldquoImprovingthe performance of retail stores subject to execution errorscoordination versus RFID technologyrdquo OR Spectrum vol 29no 4 pp 597ndash626 2007

[17] A J Mersereau ldquoInformation-sensitive replenishment wheninventory records are inaccuraterdquo Production and OperationsManagement vol 22 no 4 pp 843ndash856 2013

[18] A Sarac N Absi and S Dauzere-Peres ldquoImpacts of RFIDtechnologies on supply chains a simulation study of a three-level supply chain subject to shrinkage and delivery errorsrdquoEuropean Journal of Industrial Engineering vol 9 no 1 pp 27ndash52 2015

[19] X Gan S P Sethi and H Yan ldquoCoordination of supply chainswith risk-averse agentsrdquo Production and Operations Manage-ment vol 13 pp 135ndash147 2004

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

[21] T-M Choi ldquoCoordination and risk analysis of VMI supplychains with RFID technologyrdquo IEEE Transactions on IndustrialInformatics vol 7 no 3 pp 497ndash504 2011

[22] S ChenHWang Y Xie andCQi ldquoMean-risk analysis of radiofrequency identification technology in supply chain with inven-torymisplacement risk-sharing and coordinationrdquoOmega vol46 pp 86ndash103 2014

[23] A Ozler B Tan and F Karaesmen ldquoMulti-product newsvendorproblem with value-at-risk considerationsrdquo International Jour-nal of Production Economics vol 117 no 2 pp 244ndash255 2009

[24] YChenMXu andZ Zhang ldquoA risk-averse newsvendormodelwith CVaR criterionrdquo Operations Research vol 57 pp 1040ndash1044 2009

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

[26] M Wu S X Zhu and R H Teunter ldquoNewsvendor problemwith random shortage cost under a risk criterionrdquo InternationalJournal of Production Economics vol 145 no 2 pp 790ndash7982013

[27] S Ahmed U Cakmak and A Shapiro ldquoCoherent risk mea-sures in inventory problemsrdquo European Journal of OperationalResearch vol 182 no 1 pp 226ndash238 2007

[28] Y Rekik E Sahin and Y Dallery ldquoInventory inaccuracy inretail stores due to theft an analysis of the benefits of RFIDrdquoInternational Journal of Production Economics vol 118 no 1 pp189ndash198 2009

[29] T-J Fan X-Y Chang C-H Gu J-J Yi and S Deng ldquoBenefitsof RFID technology for reducing inventory shrinkagerdquo Interna-tional Journal of Production Economics vol 147 pp 659ndash6652014

[30] T Fan F Tao S Deng and S Li ldquoImpact of RFID technologyon supply chain decisions with inventory inaccuraciesrdquo Inter-national Journal of Production Economics vol 159 pp 117ndash1252015

[31] R H Clarke D Twede J R Tazelaar and K K Boyer ldquoRadiofrequency identification (RFID) performance the effect of tagorientation and package contentsrdquo Packaging Technology andScience vol 19 no 1 pp 45ndash54 2006

[32] E Sahin A qualitative and quantitative analysis of the impact ofauto ID technology on the performance of supply chains [PhDthesis] Ecole Centrale Pairs 2004

[33] 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

[34] J H Goto M E Lewis andM L Puterman ldquoCoffee Tea orA Markov decision process model for airline meal provision-ingrdquo Transportation Science vol 38 no 1 pp 107ndash118 2004


[35] H Wang M Guo and J Efstathiou ldquoA game-theoretical coop-erative mechanism design for a two-echelon decentralized sup-ply chainrdquo European Journal of Operational Research vol 157no 2 pp 372ndash388 2004

[36] N C Petruzzi andMDada ldquoPricing and the newsvendor prob-lem a review with extensionsrdquo Operations Research vol 47 no2 pp 183ndash194 1999

[37] R T Rockafellar and S Uryasev ldquoOptimization of conditionalvalue-at-riskrdquo Journal of Risk vol 2 pp 21ndash42 2000

[38] R T Rockafellar and S Uryasev ldquoConditional value-at-risk forgeneral loss distributionsrdquo Journal of Banking and Finance vol26 no 7 pp 1443ndash1471 2002

[39] Y Wang L Jiang and Z-J Shen ldquoChannel performance underconsignment contract with revenue sharingrdquoManagement Sci-ence vol 50 no 1 pp 34ndash47 2004

[40] S Li and Z Hua ldquoA note on channel performance under con-signment contract with revenue sharingrdquo European Journal ofOperational Research vol 184 no 2 pp 793ndash796 2008

[41] G P Cachon ldquoSupply chain coordination with contractsrdquo inHandbooks in Operations Research and Management ScienceSupplyChainManagement S Graves andT deKok Eds North-Holland Publishing Amsterdam The Netherlands 2003

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

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

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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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


Nowadays however radio frequency identification(RFID) technology can be used to track products andmanageinventory based on its ability to higher visibility and it hasbeen seen as a promising solution for inventory shrinkage inthe supply chain arena [7] Generally RFID has two principaladvantages high-frequency monitoring and nonline ofsight reading which yield information that is both moreaccurate and timely [8] Recently in order to characterizemore practical features Hervert-Escobar et al [9] utilizedthe information obtained in consecutive read attempts tohelp identify a tag in RFID implementation and developed aheuristic method of selection based on Hamming distancecomputer simulation is used to illustrate the validity of theproposed method Talavera et al [10] also studied RFIDimplementation in the steel industry where RFID technologyshows that the shared inventory improvement and the rate ofobsolescence reduction are related to inventorymanagementAs information technologies continue to improve and theircosts continue to decrease obtaining more accurate real-time inventory information is becoming increasinglycost-effective As a result more and more large companies(such as Wal-Mart Proctor amp Gamble and Gillette) areguided by high-quality information in their daily operationsand decision making [11]

Motivated by the issue of RFID technology adoption inthe industry (especially in supply chain management) westudy the effect of item-level RFID on inventory shrinkagein the retail supply chain frame To get more general resultswe extend previous knowledge of inventory inaccuracy anditem-level RFID by incorporating risk-averse considerationAs mentioned above in reality it is difficult for some com-panies to bear losses stemming from inventory shrinkageWhen facing a higher inventory shrinkage rate and a greaterdemanduncertainty the revenue obtained by selling productscan not balance the increased loss supply chain managersmust take more risks caused by inventory shrinkage anddemand uncertainty and the results in the risk-neutral casemay be considered as unrealistic Therefore the questionwe are concerned about is how the risk aversion levelaffects their optimal decisions in supply chain system withor without item-level RFID under conditional value-at-risk(CVaR) criterion (see Section 31) In addition many existingstudies incorporated price-dependent stochastic demand orRFID technology into supply chain models but few of themexplored RFID technology for retail inventory shrinkageprice-dependent stochastic demand and risk issue simulta-neously Our paper will cover these gaps

(2) Related Literature

(i) RFID and Inventory Inaccuracy Although recentresearchers have given some literature reviews on RFIDtechnology [12 13] the related research on RFIDimplementation in inventory management is relatively newWe here only focus on reviewing some recent studies onanalyzing the impact of item-level RFID on the reductionof inventory inaccuracies De Kok et al [14] consideredcost-benefit trade-off between inventory costs and the costsof RFID regarding shrinkage and proved that this break-even

price is highly related to the value of the items that arelost the shrinkage fraction and the remaining shrinkageafter employing RFID Heese [15] studied inventory recordinaccuracy in a supply chain model with a manufacturerand a retailer and analyzed the impact of inventory recordinaccuracy on optimal stocking decisions and profits Bycontrasting optimal decisions in a decentralized supply chainwith those in an integrated supply chain they concluded thatinventory record inaccuracy exacerbates the inefficienciesresulting from double marginalization in decentralizedsupply chains Rekik et al [16] developed a newsvendormodel and analyzed the RFID adoption strategy withcoordination to improve the supply chainrsquos performanceunder retail inventory inaccuracy that is subject to errorsstemming from execution problems Based on imperfectinventory records and unobserved lost sales Mersereau[17] also discussed a periodic review inventory systemthat explores one- and two-period versions of the problemand demonstrated several mechanisms by which the errorprocess and associated record inaccuracy can impact optimalreplenishment Considering identical error distributions andcounting costs Kok and Shang [2] investigated how todesign cycle-count policies from the perspective of the entiresupply chain for a two-stage system provided a simplerecursion to evaluate the system cost proposed a heuristicto obtain effective base-stock levels and proved that it ismore effective to conduct more frequent cycle counts at thedownstream stage Taking into account inventory inaccuracystemming from shrinkage and delivery errors Sarac et al[18] investigated a simulation model with RFID technologyin a three-level supply chain and evaluated the qualitativeand quantitative impacts of RFID technologies on supplychain system performances and profits The aforementionedarticles all assumed demand to be a constant or a stochasticvariable To our knowledge there has been no work oninvestigating the impact of inventory shrinkage on supplychain with price-dependent stochastic demand In thispaper bearing in mind the fact that the demand of product isaffected by price we discuss a supply chain model with retailinventory shrinkage and assess the benefit of the item-levelRFID implementation

(ii) Risk Aversion Risk aversion issues in supply inventorymanagement have been extensively studied in the pastdecades There are mean variance (MV) value-at-risk (VaR)and conditional value-at-risk (CVaR) in traditional riskanalysis approaches Gan et al [19] used MV approach tostudy the supply chain coordination in a two-echelon supplychain with risk-averse agents Under MV framework Choiet al [20] considered two-echelon supply chain coordinationproblemwhen the partners take differentsame risk attitudesthe result shows that the whole supply chain coordinationdepends on how different the risk related thresholds betweenthe two supply chain agents are Choi [21] developed thesupply chain models for a multiperiod retail replenishmentproblem with and without RFID under the MV frameworkand analytically discussed the use of RFID under vendor-managed inventory (VMI) scheme in a two-echelon single-manufacturer single-retailer supply chain Considering












units

Salable

Order quantity


Shrinkage quantity

units

units

Recovery quantity

Shrinkage quantity


Salable

Shrinkage

Shrinkage

Q units

120572Qunits120572Q

(1 minus 120572)Q

units

+

120573(1 minus 120572)Q

(1 minus 120573)(1 minus 120572)Q






unit product(4) 119888119872


119872



119877







120578119877isin (0 1]






119862(sdot))




119862(sdot))





119877(sdot) = CVaR(120587No

119877(sdot))


level RFID(23) Π



CVaR(120587RF119877(sdot))





119872(sdot))





119872(sdot))














CVaR120578(120587 (x y)) = 119864 [120587 (x y) | 120587 (x y) le Θ

120578(y)]

=1

120578int120587(xy)leΘ120578

120587 (x y) 119892 (y) 119889y(1)




vector y that is



CVaR120578(120587 (x y))

= max120592isinR

120592 +1

120578119864 [min (120587 (x y) minus 120592 0)]

(3)




120587RF119862(119901 119897) = 119901119864 [min (1199021015840 119863 (119901))] minus (119888

119872+ 119888119877+ 119905) 119902 (4)



120587RF119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877+ 119905)119898119897 (5)










119897 le 119865minus1(120578119877) lt 1198611015840



Π(119901 119897 V)

= maxVisinR

V +1

120578119877

119864 [min (120587RF119862(119901 119897) minus V 0)]

(6)


Π(119901 119897 V) = V minus1

120578119877

int1198611015840

1198601015840


minus (119888119872+ 119888119877+ 119905)119898119897

+

119889119865 (119909)

(7)


Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

(8)


119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V

120597Π (119901 119897 V)120597V

= 1

(9)


(119888119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V minus1

120578119877

int(V+119889(119901)(119888119872+119888119877+119905)119898119897)119901119889(119901)

1198601015840

V minus 119889 (119901)

sdot [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

119865(V + 119889 (119901) (119888

119872+ 119888119877+ 119905)119898119897

119901119889 (119901))

(10)

In particular

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[1199011198601015840minus(119888119872+119888119877+119905)119898119897]= 1

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[119901119897minus(119888119872+119888119877+119905)119898119897]= 1 minus

1

120578119877

119865 (119897)

(11)


Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

lt 0

(12)




sdot [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]]

(13)


Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (14)



119877) then

Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (15)


Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) [[119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

minus119901

120578119877

int119897

1198601015840

(119897 minus 119909) 119889119865 (119909)]

(16)

(b) If 119897 ge 119865minus1(120578119877) then


and we have

Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) 119901119865minus1 (120578119877)

minus (119888119872+ 119888119877+ 119905)119898119897 minus

119901

120578119877

int119865minus1(120578119877)

1198601015840

119865 (119909) 119889119909

(18)



120597Π (119901 119897 Vlowast (119901 119897))120597119897

= minus119889 (119901) (119888119872+ 119888119877+ 119905)119898 lt 0 (19)




Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (20)



119872+ 119888119877+ 119905)119898119897] maximizes



Π(119901 119897) = 119889 (119901) [119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(21)

where Λ(119897) = int1198971198601015840(119897 minus 119909)119889119865(119909)





119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (22)


(119902119877119865

119862)lowast

= 119860119898(119897119877119865

119862)lowast[

[

(120578119877minus 119865 ((119897119877119865

119862)lowast

))

119898120578119877(119888119872+ 119888119877+ 119905)

]

]

119896

(23)



1119896


Π(119897 119902) = 119860(119902

119860119898119897)minus(1minus119896)119896

[119897 minus1

120578119877

Λ (119897)]

minus (119888119872+ 119888119877+ 119905) 119902

(24)


120597Π (119897 119902)

120597119897=11986011198961199021minus1119896

1198961205781198771198972minus1119896

[120578119877119897 minus 119896119897119865 (119897) + (119896 minus 1) Λ (119897)]

= 0

(25)









119877gt 0

and 119885(119865minus1(120578119877)) = (1 minus 119896)120578

119877119865minus1(120578

119877) + (119896 minus 1) int

119865minus1(120578119877)

1198601015840

119865(119909)119889119909

Since int119865minus1(120578119877)

1198601015840 119865(119909)119889119909 le 119865(119865minus1(120578

119877))(119865minus1(120578

119877) minus 1198601015840) =

120578119877(119865minus1(120578


119877)) le (1 minus 119896)1198601015840 lt 0




119877minus 119865(119897))(1 minus 119896119897ℎ(119897)) and 11988510158401015840(119897) =



1198851015840(119897)=0

lt





Π((119897RF119862)lowast

119902)

=1198961198601119896119898minus1+1119896 ((119897RF

119862)lowast

)1+1119896

1199021minus1119896 (120578119877minus 119865 ((119897RF

119862)lowast

))

(119896 minus 1) 120578119877

minus (119888119872+ 119888119877+ 119905) 119902

(26)


119889Π((119897RF119862)lowast

119902)

119889119902

=1198601119896119898minus1+1119896

((119897RF119862)lowast

)1+1119896

119902minus1119896

(120578119877minus 119865 ((119897

RF119862)lowast

))

120578119877

minus (119888119872+ 119888119877+ 119905)

(27)


119862)lowast

=


119862)lowast))119898120578

119877(119888119872+ 119888119877+ 119905)]119896 Meanwhile






the proof


119877= 1 and 120572 = 1





retail price

(119901RF119862)lowast

=119898120578119877(119888119872+ 119888119877+ 119905)

120578119877minus 119865 ((119897RF

119862)lowast) (28)




Π((119897RF119862)lowast

)

=119860 (119897RF119862)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

((120578119877minus 119865 ((119897RF

119862)lowast

))

120578119877

)

119896

(29)


120587No119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877)1198981015840119897 (30)






(119901119873119900

119862)lowast

=1198981015840120578119877(119888119872+ 119888119877)

120578119877minus 119865 ((119897119873119900

119862)lowast) (31)


119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (32)



(119902No119862)lowast

= 1198601198981015840(119897

No119862)lowast[

[

120578119877minus 119865 ((119897

No119862)lowast

)

1198981015840120578119877(119888119872+ 119888119877)]

]

119896

(33)


Π1015840((119897

No119862)lowast

)

=119860 (119897No119862)lowast

(119896 minus 1) [1198981015840 (119888119872 + 119888119877)]119896minus1

(120578119877minus 119865 ((119897No

119862)lowast

)

120578119877

)

119896

(34)



(1) (119897119877119865119862)lowast= (119897119873119900119862)lowast

(2) Both (119897119877119865119862)lowast and (119897119873119900



(3) Both (119901119877119865119862)lowast and (119901119873119900



(i) when 119866((119897119877119865119862)lowast) gt 0 (119901119877119865


119877

when 119866((119897119877119865119862)lowast) lt 0 (119901119877119865


119877

when119866((119897119877119865119862)lowast) = 0 (119901119877119865


119877

(ii) when119866((119897119873119900119862)lowast) gt 0 (119901119873119900-119877119865


119877

when 119866((119897119873119900119862)lowast) lt 0 (119901119873119900


119877

when119866((119897119873119900119862)lowast) = 0 (119901119873119900


119877


119862)lowast= (119897RF119862)lowast





119862)lowast) =




119860120578119877gt 0 119885(119865minus1(120578

119877)) le (1 minus 119896)119860 lt 0 and (119897RF






119862)lowast119889120578119877gt 0







119862)lowast119889120578119877

=


119862)lowast)[120578119877

minus



119862)lowast

1198851015840(119897)|119897=(119897


119862)lowast119889120578119877


119862)lowast119889120578119877=

(119898(119888119872

+ 119888119877+ 119905)[120578

119877minus 119865(119897RF




119866((119897RF119862)lowast) gt 0 (119901RF


119877 when

119866((119897RF119862)lowast) lt 0 (119901RF


119877 when

119866((119897RF119862)lowast) = 0 (119901RF


119877


119862)lowast














119862)lowast) or 119866((119897No






119877





ln[Π((119897RF119862)lowast)Π1015840((119897No


119862)lowast) and Π1015840((119897No

119862)lowast) are





Δ = (119896 minus 1) ln [120572 + 120573 (1 minus 120572)

120572sdot

119888119872+ 119888119877

119888119872+ 119888119877+ 119905

] (35)


119877 and





119877 it




then Δ le 0


119872+ 119888119877)(119888119872+ 119888119877+ 119905)) gt 1 (le1)














120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo




119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840


119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896


119896 minus 1

119888119877+ 119896119888119872



119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1



119896)

119896

Πlowast

(119896 minus 1

119896)

119896



119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast


1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)


119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896


120587RF119877(119901 119897)


119877+ 119905)119898119897

(36)


CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877


119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF




119877(119901 119897 V(119901 119897))


Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)






119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)




(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)



Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















units

Salable

Order quantity


Shrinkage quantity

units

units

Recovery quantity

Shrinkage quantity


Salable

Shrinkage

Shrinkage

Q units

120572Qunits120572Q

(1 minus 120572)Q

units

+

120573(1 minus 120572)Q

(1 minus 120573)(1 minus 120572)Q






unit product(4) 119888119872


119872



119877







120578119877isin (0 1]






119862(sdot))




119862(sdot))





119877(sdot) = CVaR(120587No

119877(sdot))


level RFID(23) Π



CVaR(120587RF119877(sdot))





119872(sdot))





119872(sdot))














CVaR120578(120587 (x y)) = 119864 [120587 (x y) | 120587 (x y) le Θ

120578(y)]

=1

120578int120587(xy)leΘ120578

120587 (x y) 119892 (y) 119889y(1)




vector y that is



CVaR120578(120587 (x y))

= max120592isinR

120592 +1

120578119864 [min (120587 (x y) minus 120592 0)]

(3)




120587RF119862(119901 119897) = 119901119864 [min (1199021015840 119863 (119901))] minus (119888

119872+ 119888119877+ 119905) 119902 (4)



120587RF119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877+ 119905)119898119897 (5)










119897 le 119865minus1(120578119877) lt 1198611015840



Π(119901 119897 V)

= maxVisinR

V +1

120578119877

119864 [min (120587RF119862(119901 119897) minus V 0)]

(6)


Π(119901 119897 V) = V minus1

120578119877

int1198611015840

1198601015840


minus (119888119872+ 119888119877+ 119905)119898119897

+

119889119865 (119909)

(7)


Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

(8)


119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V

120597Π (119901 119897 V)120597V

= 1

(9)


(119888119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V minus1

120578119877

int(V+119889(119901)(119888119872+119888119877+119905)119898119897)119901119889(119901)

1198601015840

V minus 119889 (119901)

sdot [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

119865(V + 119889 (119901) (119888

119872+ 119888119877+ 119905)119898119897

119901119889 (119901))

(10)

In particular

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[1199011198601015840minus(119888119872+119888119877+119905)119898119897]= 1

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[119901119897minus(119888119872+119888119877+119905)119898119897]= 1 minus

1

120578119877

119865 (119897)

(11)


Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

lt 0

(12)




sdot [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]]

(13)


Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (14)



119877) then

Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (15)


Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) [[119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

minus119901

120578119877

int119897

1198601015840

(119897 minus 119909) 119889119865 (119909)]

(16)

(b) If 119897 ge 119865minus1(120578119877) then


and we have

Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) 119901119865minus1 (120578119877)

minus (119888119872+ 119888119877+ 119905)119898119897 minus

119901

120578119877

int119865minus1(120578119877)

1198601015840

119865 (119909) 119889119909

(18)



120597Π (119901 119897 Vlowast (119901 119897))120597119897

= minus119889 (119901) (119888119872+ 119888119877+ 119905)119898 lt 0 (19)




Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (20)



119872+ 119888119877+ 119905)119898119897] maximizes



Π(119901 119897) = 119889 (119901) [119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(21)

where Λ(119897) = int1198971198601015840(119897 minus 119909)119889119865(119909)





119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (22)


(119902119877119865

119862)lowast

= 119860119898(119897119877119865

119862)lowast[

[

(120578119877minus 119865 ((119897119877119865

119862)lowast

))

119898120578119877(119888119872+ 119888119877+ 119905)

]

]

119896

(23)



1119896


Π(119897 119902) = 119860(119902

119860119898119897)minus(1minus119896)119896

[119897 minus1

120578119877

Λ (119897)]

minus (119888119872+ 119888119877+ 119905) 119902

(24)


120597Π (119897 119902)

120597119897=11986011198961199021minus1119896

1198961205781198771198972minus1119896

[120578119877119897 minus 119896119897119865 (119897) + (119896 minus 1) Λ (119897)]

= 0

(25)









119877gt 0

and 119885(119865minus1(120578119877)) = (1 minus 119896)120578

119877119865minus1(120578

119877) + (119896 minus 1) int

119865minus1(120578119877)

1198601015840

119865(119909)119889119909

Since int119865minus1(120578119877)

1198601015840 119865(119909)119889119909 le 119865(119865minus1(120578

119877))(119865minus1(120578

119877) minus 1198601015840) =

120578119877(119865minus1(120578


119877)) le (1 minus 119896)1198601015840 lt 0




119877minus 119865(119897))(1 minus 119896119897ℎ(119897)) and 11988510158401015840(119897) =



1198851015840(119897)=0

lt





Π((119897RF119862)lowast

119902)

=1198961198601119896119898minus1+1119896 ((119897RF

119862)lowast

)1+1119896

1199021minus1119896 (120578119877minus 119865 ((119897RF

119862)lowast

))

(119896 minus 1) 120578119877

minus (119888119872+ 119888119877+ 119905) 119902

(26)


119889Π((119897RF119862)lowast

119902)

119889119902

=1198601119896119898minus1+1119896

((119897RF119862)lowast

)1+1119896

119902minus1119896

(120578119877minus 119865 ((119897

RF119862)lowast

))

120578119877

minus (119888119872+ 119888119877+ 119905)

(27)


119862)lowast

=


119862)lowast))119898120578

119877(119888119872+ 119888119877+ 119905)]119896 Meanwhile






the proof


119877= 1 and 120572 = 1





retail price

(119901RF119862)lowast

=119898120578119877(119888119872+ 119888119877+ 119905)

120578119877minus 119865 ((119897RF

119862)lowast) (28)




Π((119897RF119862)lowast

)

=119860 (119897RF119862)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

((120578119877minus 119865 ((119897RF

119862)lowast

))

120578119877

)

119896

(29)


120587No119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877)1198981015840119897 (30)






(119901119873119900

119862)lowast

=1198981015840120578119877(119888119872+ 119888119877)

120578119877minus 119865 ((119897119873119900

119862)lowast) (31)


119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (32)



(119902No119862)lowast

= 1198601198981015840(119897

No119862)lowast[

[

120578119877minus 119865 ((119897

No119862)lowast

)

1198981015840120578119877(119888119872+ 119888119877)]

]

119896

(33)


Π1015840((119897

No119862)lowast

)

=119860 (119897No119862)lowast

(119896 minus 1) [1198981015840 (119888119872 + 119888119877)]119896minus1

(120578119877minus 119865 ((119897No

119862)lowast

)

120578119877

)

119896

(34)



(1) (119897119877119865119862)lowast= (119897119873119900119862)lowast

(2) Both (119897119877119865119862)lowast and (119897119873119900



(3) Both (119901119877119865119862)lowast and (119901119873119900



(i) when 119866((119897119877119865119862)lowast) gt 0 (119901119877119865


119877

when 119866((119897119877119865119862)lowast) lt 0 (119901119877119865


119877

when119866((119897119877119865119862)lowast) = 0 (119901119877119865


119877

(ii) when119866((119897119873119900119862)lowast) gt 0 (119901119873119900-119877119865


119877

when 119866((119897119873119900119862)lowast) lt 0 (119901119873119900


119877

when119866((119897119873119900119862)lowast) = 0 (119901119873119900


119877


119862)lowast= (119897RF119862)lowast





119862)lowast) =




119860120578119877gt 0 119885(119865minus1(120578

119877)) le (1 minus 119896)119860 lt 0 and (119897RF






119862)lowast119889120578119877gt 0







119862)lowast119889120578119877

=


119862)lowast)[120578119877

minus



119862)lowast

1198851015840(119897)|119897=(119897


119862)lowast119889120578119877


119862)lowast119889120578119877=

(119898(119888119872

+ 119888119877+ 119905)[120578

119877minus 119865(119897RF




119866((119897RF119862)lowast) gt 0 (119901RF


119877 when

119866((119897RF119862)lowast) lt 0 (119901RF


119877 when

119866((119897RF119862)lowast) = 0 (119901RF


119877


119862)lowast














119862)lowast) or 119866((119897No






119877





ln[Π((119897RF119862)lowast)Π1015840((119897No


119862)lowast) and Π1015840((119897No

119862)lowast) are





Δ = (119896 minus 1) ln [120572 + 120573 (1 minus 120572)

120572sdot

119888119872+ 119888119877

119888119872+ 119888119877+ 119905

] (35)


119877 and





119877 it




then Δ le 0


119872+ 119888119877)(119888119872+ 119888119877+ 119905)) gt 1 (le1)














120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo




119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840


119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896


119896 minus 1

119888119877+ 119896119888119872



119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1



119896)

119896

Πlowast

(119896 minus 1

119896)

119896



119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast


1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)


119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896


120587RF119877(119901 119897)


119877+ 119905)119898119897

(36)


CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877


119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF




119877(119901 119897 V(119901 119897))


Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)






119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)




(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)



Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















CVaR120578(120587 (x y)) = 119864 [120587 (x y) | 120587 (x y) le Θ

120578(y)]

=1

120578int120587(xy)leΘ120578

120587 (x y) 119892 (y) 119889y(1)




vector y that is



CVaR120578(120587 (x y))

= max120592isinR

120592 +1

120578119864 [min (120587 (x y) minus 120592 0)]

(3)




120587RF119862(119901 119897) = 119901119864 [min (1199021015840 119863 (119901))] minus (119888

119872+ 119888119877+ 119905) 119902 (4)



120587RF119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877+ 119905)119898119897 (5)










119897 le 119865minus1(120578119877) lt 1198611015840



Π(119901 119897 V)

= maxVisinR

V +1

120578119877

119864 [min (120587RF119862(119901 119897) minus V 0)]

(6)


Π(119901 119897 V) = V minus1

120578119877

int1198611015840

1198601015840


minus (119888119872+ 119888119877+ 119905)119898119897

+

119889119865 (119909)

(7)


Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

(8)


119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V

120597Π (119901 119897 V)120597V

= 1

(9)


(119888119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V minus1

120578119877

int(V+119889(119901)(119888119872+119888119877+119905)119898119897)119901119889(119901)

1198601015840

V minus 119889 (119901)

sdot [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

119865(V + 119889 (119901) (119888

119872+ 119888119877+ 119905)119898119897

119901119889 (119901))

(10)

In particular

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[1199011198601015840minus(119888119872+119888119877+119905)119898119897]= 1

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[119901119897minus(119888119872+119888119877+119905)119898119897]= 1 minus

1

120578119877

119865 (119897)

(11)


Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

lt 0

(12)




sdot [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]]

(13)


Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (14)



119877) then

Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (15)


Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) [[119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

minus119901

120578119877

int119897

1198601015840

(119897 minus 119909) 119889119865 (119909)]

(16)

(b) If 119897 ge 119865minus1(120578119877) then


and we have

Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) 119901119865minus1 (120578119877)

minus (119888119872+ 119888119877+ 119905)119898119897 minus

119901

120578119877

int119865minus1(120578119877)

1198601015840

119865 (119909) 119889119909

(18)



120597Π (119901 119897 Vlowast (119901 119897))120597119897

= minus119889 (119901) (119888119872+ 119888119877+ 119905)119898 lt 0 (19)




Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (20)



119872+ 119888119877+ 119905)119898119897] maximizes



Π(119901 119897) = 119889 (119901) [119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(21)

where Λ(119897) = int1198971198601015840(119897 minus 119909)119889119865(119909)





119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (22)


(119902119877119865

119862)lowast

= 119860119898(119897119877119865

119862)lowast[

[

(120578119877minus 119865 ((119897119877119865

119862)lowast

))

119898120578119877(119888119872+ 119888119877+ 119905)

]

]

119896

(23)



1119896


Π(119897 119902) = 119860(119902

119860119898119897)minus(1minus119896)119896

[119897 minus1

120578119877

Λ (119897)]

minus (119888119872+ 119888119877+ 119905) 119902

(24)


120597Π (119897 119902)

120597119897=11986011198961199021minus1119896

1198961205781198771198972minus1119896

[120578119877119897 minus 119896119897119865 (119897) + (119896 minus 1) Λ (119897)]

= 0

(25)









119877gt 0

and 119885(119865minus1(120578119877)) = (1 minus 119896)120578

119877119865minus1(120578

119877) + (119896 minus 1) int

119865minus1(120578119877)

1198601015840

119865(119909)119889119909

Since int119865minus1(120578119877)

1198601015840 119865(119909)119889119909 le 119865(119865minus1(120578

119877))(119865minus1(120578

119877) minus 1198601015840) =

120578119877(119865minus1(120578


119877)) le (1 minus 119896)1198601015840 lt 0




119877minus 119865(119897))(1 minus 119896119897ℎ(119897)) and 11988510158401015840(119897) =



1198851015840(119897)=0

lt





Π((119897RF119862)lowast

119902)

=1198961198601119896119898minus1+1119896 ((119897RF

119862)lowast

)1+1119896

1199021minus1119896 (120578119877minus 119865 ((119897RF

119862)lowast

))

(119896 minus 1) 120578119877

minus (119888119872+ 119888119877+ 119905) 119902

(26)


119889Π((119897RF119862)lowast

119902)

119889119902

=1198601119896119898minus1+1119896

((119897RF119862)lowast

)1+1119896

119902minus1119896

(120578119877minus 119865 ((119897

RF119862)lowast

))

120578119877

minus (119888119872+ 119888119877+ 119905)

(27)


119862)lowast

=


119862)lowast))119898120578

119877(119888119872+ 119888119877+ 119905)]119896 Meanwhile






the proof


119877= 1 and 120572 = 1





retail price

(119901RF119862)lowast

=119898120578119877(119888119872+ 119888119877+ 119905)

120578119877minus 119865 ((119897RF

119862)lowast) (28)




Π((119897RF119862)lowast

)

=119860 (119897RF119862)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

((120578119877minus 119865 ((119897RF

119862)lowast

))

120578119877

)

119896

(29)


120587No119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877)1198981015840119897 (30)






(119901119873119900

119862)lowast

=1198981015840120578119877(119888119872+ 119888119877)

120578119877minus 119865 ((119897119873119900

119862)lowast) (31)


119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (32)



(119902No119862)lowast

= 1198601198981015840(119897

No119862)lowast[

[

120578119877minus 119865 ((119897

No119862)lowast

)

1198981015840120578119877(119888119872+ 119888119877)]

]

119896

(33)


Π1015840((119897

No119862)lowast

)

=119860 (119897No119862)lowast

(119896 minus 1) [1198981015840 (119888119872 + 119888119877)]119896minus1

(120578119877minus 119865 ((119897No

119862)lowast

)

120578119877

)

119896

(34)



(1) (119897119877119865119862)lowast= (119897119873119900119862)lowast

(2) Both (119897119877119865119862)lowast and (119897119873119900



(3) Both (119901119877119865119862)lowast and (119901119873119900



(i) when 119866((119897119877119865119862)lowast) gt 0 (119901119877119865


119877

when 119866((119897119877119865119862)lowast) lt 0 (119901119877119865


119877

when119866((119897119877119865119862)lowast) = 0 (119901119877119865


119877

(ii) when119866((119897119873119900119862)lowast) gt 0 (119901119873119900-119877119865


119877

when 119866((119897119873119900119862)lowast) lt 0 (119901119873119900


119877

when119866((119897119873119900119862)lowast) = 0 (119901119873119900


119877


119862)lowast= (119897RF119862)lowast





119862)lowast) =




119860120578119877gt 0 119885(119865minus1(120578

119877)) le (1 minus 119896)119860 lt 0 and (119897RF






119862)lowast119889120578119877gt 0







119862)lowast119889120578119877

=


119862)lowast)[120578119877

minus



119862)lowast

1198851015840(119897)|119897=(119897


119862)lowast119889120578119877


119862)lowast119889120578119877=

(119898(119888119872

+ 119888119877+ 119905)[120578

119877minus 119865(119897RF




119866((119897RF119862)lowast) gt 0 (119901RF


119877 when

119866((119897RF119862)lowast) lt 0 (119901RF


119877 when

119866((119897RF119862)lowast) = 0 (119901RF


119877


119862)lowast














119862)lowast) or 119866((119897No






119877





ln[Π((119897RF119862)lowast)Π1015840((119897No


119862)lowast) and Π1015840((119897No

119862)lowast) are





Δ = (119896 minus 1) ln [120572 + 120573 (1 minus 120572)

120572sdot

119888119872+ 119888119877

119888119872+ 119888119877+ 119905

] (35)


119877 and





119877 it




then Δ le 0


119872+ 119888119877)(119888119872+ 119888119877+ 119905)) gt 1 (le1)














120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo




119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840


119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896


119896 minus 1

119888119877+ 119896119888119872



119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1



119896)

119896

Πlowast

(119896 minus 1

119896)

119896



119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast


1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)


119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896


120587RF119877(119901 119897)


119877+ 119905)119898119897

(36)


CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877


119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF




119877(119901 119897 V(119901 119897))


Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)






119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)




(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)



Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119897 le 119865minus1(120578119877) lt 1198611015840



Π(119901 119897 V)

= maxVisinR

V +1

120578119877

119864 [min (120587RF119862(119901 119897) minus V 0)]

(6)


Π(119901 119897 V) = V minus1

120578119877

int1198611015840

1198601015840


minus (119888119872+ 119888119877+ 119905)119898119897

+

119889119865 (119909)

(7)


Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

+119889119865 (119909)

(8)


119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V

120597Π (119901 119897 V)120597V

= 1

(9)


(119888119872+ 119888119877+ 119905)119898119897] then

Π(119901 119897 V) = V minus1

120578119877

int(V+119889(119901)(119888119872+119888119877+119905)119898119897)119901119889(119901)

1198601015840

V minus 119889 (119901)

sdot [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

119865(V + 119889 (119901) (119888

119872+ 119888119877+ 119905)119898119897

119901119889 (119901))

(10)

In particular

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[1199011198601015840minus(119888119872+119888119877+119905)119898119897]= 1

120597Π (119901 119897 V)120597V

100381610038161003816100381610038161003816100381610038161003816V=119889(119901)[119901119897minus(119888119872+119888119877+119905)119898119897]= 1 minus

1

120578119877

119865 (119897)

(11)


Π(119901 119897 V) = V minus1

120578119877

sdot int119897

1198601015840

V minus 119889 (119901) [119901119909 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

minus1

120578119877

sdot int1198611015840

119897

V minus 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] 119889119865 (119909)

120597Π (119901 119897 V)120597V

= 1 minus1

120578119877

lt 0

(12)




sdot [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]]

(13)


Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (14)



119877) then

Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (15)


Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) [[119901119897 minus (119888119872+ 119888119877+ 119905)119898119897]

minus119901

120578119877

int119897

1198601015840

(119897 minus 119909) 119889119865 (119909)]

(16)

(b) If 119897 ge 119865minus1(120578119877) then


and we have

Π(119901 119897 Vlowast (119901 119897)) = 119889 (119901) 119901119865minus1 (120578119877)

minus (119888119872+ 119888119877+ 119905)119898119897 minus

119901

120578119877

int119865minus1(120578119877)

1198601015840

119865 (119909) 119889119909

(18)



120597Π (119901 119897 Vlowast (119901 119897))120597119897

= minus119889 (119901) (119888119872+ 119888119877+ 119905)119898 lt 0 (19)




Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (20)



119872+ 119888119877+ 119905)119898119897] maximizes



Π(119901 119897) = 119889 (119901) [119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(21)

where Λ(119897) = int1198971198601015840(119897 minus 119909)119889119865(119909)





119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (22)


(119902119877119865

119862)lowast

= 119860119898(119897119877119865

119862)lowast[

[

(120578119877minus 119865 ((119897119877119865

119862)lowast

))

119898120578119877(119888119872+ 119888119877+ 119905)

]

]

119896

(23)



1119896


Π(119897 119902) = 119860(119902

119860119898119897)minus(1minus119896)119896

[119897 minus1

120578119877

Λ (119897)]

minus (119888119872+ 119888119877+ 119905) 119902

(24)


120597Π (119897 119902)

120597119897=11986011198961199021minus1119896

1198961205781198771198972minus1119896

[120578119877119897 minus 119896119897119865 (119897) + (119896 minus 1) Λ (119897)]

= 0

(25)









119877gt 0

and 119885(119865minus1(120578119877)) = (1 minus 119896)120578

119877119865minus1(120578

119877) + (119896 minus 1) int

119865minus1(120578119877)

1198601015840

119865(119909)119889119909

Since int119865minus1(120578119877)

1198601015840 119865(119909)119889119909 le 119865(119865minus1(120578

119877))(119865minus1(120578

119877) minus 1198601015840) =

120578119877(119865minus1(120578


119877)) le (1 minus 119896)1198601015840 lt 0




119877minus 119865(119897))(1 minus 119896119897ℎ(119897)) and 11988510158401015840(119897) =



1198851015840(119897)=0

lt





Π((119897RF119862)lowast

119902)

=1198961198601119896119898minus1+1119896 ((119897RF

119862)lowast

)1+1119896

1199021minus1119896 (120578119877minus 119865 ((119897RF

119862)lowast

))

(119896 minus 1) 120578119877

minus (119888119872+ 119888119877+ 119905) 119902

(26)


119889Π((119897RF119862)lowast

119902)

119889119902

=1198601119896119898minus1+1119896

((119897RF119862)lowast

)1+1119896

119902minus1119896

(120578119877minus 119865 ((119897

RF119862)lowast

))

120578119877

minus (119888119872+ 119888119877+ 119905)

(27)


119862)lowast

=


119862)lowast))119898120578

119877(119888119872+ 119888119877+ 119905)]119896 Meanwhile






the proof


119877= 1 and 120572 = 1





retail price

(119901RF119862)lowast

=119898120578119877(119888119872+ 119888119877+ 119905)

120578119877minus 119865 ((119897RF

119862)lowast) (28)




Π((119897RF119862)lowast

)

=119860 (119897RF119862)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

((120578119877minus 119865 ((119897RF

119862)lowast

))

120578119877

)

119896

(29)


120587No119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877)1198981015840119897 (30)






(119901119873119900

119862)lowast

=1198981015840120578119877(119888119872+ 119888119877)

120578119877minus 119865 ((119897119873119900

119862)lowast) (31)


119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (32)



(119902No119862)lowast

= 1198601198981015840(119897

No119862)lowast[

[

120578119877minus 119865 ((119897

No119862)lowast

)

1198981015840120578119877(119888119872+ 119888119877)]

]

119896

(33)


Π1015840((119897

No119862)lowast

)

=119860 (119897No119862)lowast

(119896 minus 1) [1198981015840 (119888119872 + 119888119877)]119896minus1

(120578119877minus 119865 ((119897No

119862)lowast

)

120578119877

)

119896

(34)



(1) (119897119877119865119862)lowast= (119897119873119900119862)lowast

(2) Both (119897119877119865119862)lowast and (119897119873119900



(3) Both (119901119877119865119862)lowast and (119901119873119900



(i) when 119866((119897119877119865119862)lowast) gt 0 (119901119877119865


119877

when 119866((119897119877119865119862)lowast) lt 0 (119901119877119865


119877

when119866((119897119877119865119862)lowast) = 0 (119901119877119865


119877

(ii) when119866((119897119873119900119862)lowast) gt 0 (119901119873119900-119877119865


119877

when 119866((119897119873119900119862)lowast) lt 0 (119901119873119900


119877

when119866((119897119873119900119862)lowast) = 0 (119901119873119900


119877


119862)lowast= (119897RF119862)lowast





119862)lowast) =




119860120578119877gt 0 119885(119865minus1(120578

119877)) le (1 minus 119896)119860 lt 0 and (119897RF






119862)lowast119889120578119877gt 0







119862)lowast119889120578119877

=


119862)lowast)[120578119877

minus



119862)lowast

1198851015840(119897)|119897=(119897


119862)lowast119889120578119877


119862)lowast119889120578119877=

(119898(119888119872

+ 119888119877+ 119905)[120578

119877minus 119865(119897RF




119866((119897RF119862)lowast) gt 0 (119901RF


119877 when

119866((119897RF119862)lowast) lt 0 (119901RF


119877 when

119866((119897RF119862)lowast) = 0 (119901RF


119877


119862)lowast














119862)lowast) or 119866((119897No






119877





ln[Π((119897RF119862)lowast)Π1015840((119897No


119862)lowast) and Π1015840((119897No

119862)lowast) are





Δ = (119896 minus 1) ln [120572 + 120573 (1 minus 120572)

120572sdot

119888119872+ 119888119877

119888119872+ 119888119877+ 119905

] (35)


119877 and





119877 it




then Δ le 0


119872+ 119888119877)(119888119872+ 119888119877+ 119905)) gt 1 (le1)














120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo




119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840


119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896


119896 minus 1

119888119877+ 119896119888119872



119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1



119896)

119896

Πlowast

(119896 minus 1

119896)

119896



119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast


1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)


119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896


120587RF119877(119901 119897)


119877+ 119905)119898119897

(36)


CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877


119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF




119877(119901 119897 V(119901 119897))


Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)






119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)




(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)



Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597Π (119901 119897 Vlowast (119901 119897))120597119897

= minus119889 (119901) (119888119872+ 119888119877+ 119905)119898 lt 0 (19)




Vlowast (119901 119897) = 119889 (119901) [119901119897 minus (119888119872+ 119888119877+ 119905)119898119897] (20)



119872+ 119888119877+ 119905)119898119897] maximizes



Π(119901 119897) = 119889 (119901) [119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(21)

where Λ(119897) = int1198971198601015840(119897 minus 119909)119889119865(119909)





119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (22)


(119902119877119865

119862)lowast

= 119860119898(119897119877119865

119862)lowast[

[

(120578119877minus 119865 ((119897119877119865

119862)lowast

))

119898120578119877(119888119872+ 119888119877+ 119905)

]

]

119896

(23)



1119896


Π(119897 119902) = 119860(119902

119860119898119897)minus(1minus119896)119896

[119897 minus1

120578119877

Λ (119897)]

minus (119888119872+ 119888119877+ 119905) 119902

(24)


120597Π (119897 119902)

120597119897=11986011198961199021minus1119896

1198961205781198771198972minus1119896

[120578119877119897 minus 119896119897119865 (119897) + (119896 minus 1) Λ (119897)]

= 0

(25)









119877gt 0

and 119885(119865minus1(120578119877)) = (1 minus 119896)120578

119877119865minus1(120578

119877) + (119896 minus 1) int

119865minus1(120578119877)

1198601015840

119865(119909)119889119909

Since int119865minus1(120578119877)

1198601015840 119865(119909)119889119909 le 119865(119865minus1(120578

119877))(119865minus1(120578

119877) minus 1198601015840) =

120578119877(119865minus1(120578


119877)) le (1 minus 119896)1198601015840 lt 0




119877minus 119865(119897))(1 minus 119896119897ℎ(119897)) and 11988510158401015840(119897) =



1198851015840(119897)=0

lt





Π((119897RF119862)lowast

119902)

=1198961198601119896119898minus1+1119896 ((119897RF

119862)lowast

)1+1119896

1199021minus1119896 (120578119877minus 119865 ((119897RF

119862)lowast

))

(119896 minus 1) 120578119877

minus (119888119872+ 119888119877+ 119905) 119902

(26)


119889Π((119897RF119862)lowast

119902)

119889119902

=1198601119896119898minus1+1119896

((119897RF119862)lowast

)1+1119896

119902minus1119896

(120578119877minus 119865 ((119897

RF119862)lowast

))

120578119877

minus (119888119872+ 119888119877+ 119905)

(27)


119862)lowast

=


119862)lowast))119898120578

119877(119888119872+ 119888119877+ 119905)]119896 Meanwhile






the proof


119877= 1 and 120572 = 1





retail price

(119901RF119862)lowast

=119898120578119877(119888119872+ 119888119877+ 119905)

120578119877minus 119865 ((119897RF

119862)lowast) (28)




Π((119897RF119862)lowast

)

=119860 (119897RF119862)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

((120578119877minus 119865 ((119897RF

119862)lowast

))

120578119877

)

119896

(29)


120587No119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877)1198981015840119897 (30)






(119901119873119900

119862)lowast

=1198981015840120578119877(119888119872+ 119888119877)

120578119877minus 119865 ((119897119873119900

119862)lowast) (31)


119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (32)



(119902No119862)lowast

= 1198601198981015840(119897

No119862)lowast[

[

120578119877minus 119865 ((119897

No119862)lowast

)

1198981015840120578119877(119888119872+ 119888119877)]

]

119896

(33)


Π1015840((119897

No119862)lowast

)

=119860 (119897No119862)lowast

(119896 minus 1) [1198981015840 (119888119872 + 119888119877)]119896minus1

(120578119877minus 119865 ((119897No

119862)lowast

)

120578119877

)

119896

(34)



(1) (119897119877119865119862)lowast= (119897119873119900119862)lowast

(2) Both (119897119877119865119862)lowast and (119897119873119900



(3) Both (119901119877119865119862)lowast and (119901119873119900



(i) when 119866((119897119877119865119862)lowast) gt 0 (119901119877119865


119877

when 119866((119897119877119865119862)lowast) lt 0 (119901119877119865


119877

when119866((119897119877119865119862)lowast) = 0 (119901119877119865


119877

(ii) when119866((119897119873119900119862)lowast) gt 0 (119901119873119900-119877119865


119877

when 119866((119897119873119900119862)lowast) lt 0 (119901119873119900


119877

when119866((119897119873119900119862)lowast) = 0 (119901119873119900


119877


119862)lowast= (119897RF119862)lowast





119862)lowast) =




119860120578119877gt 0 119885(119865minus1(120578

119877)) le (1 minus 119896)119860 lt 0 and (119897RF






119862)lowast119889120578119877gt 0







119862)lowast119889120578119877

=


119862)lowast)[120578119877

minus



119862)lowast

1198851015840(119897)|119897=(119897


119862)lowast119889120578119877


119862)lowast119889120578119877=

(119898(119888119872

+ 119888119877+ 119905)[120578

119877minus 119865(119897RF




119866((119897RF119862)lowast) gt 0 (119901RF


119877 when

119866((119897RF119862)lowast) lt 0 (119901RF


119877 when

119866((119897RF119862)lowast) = 0 (119901RF


119877


119862)lowast














119862)lowast) or 119866((119897No






119877





ln[Π((119897RF119862)lowast)Π1015840((119897No


119862)lowast) and Π1015840((119897No

119862)lowast) are





Δ = (119896 minus 1) ln [120572 + 120573 (1 minus 120572)

120572sdot

119888119872+ 119888119877

119888119872+ 119888119877+ 119905

] (35)


119877 and





119877 it




then Δ le 0


119872+ 119888119877)(119888119872+ 119888119877+ 119905)) gt 1 (le1)














120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo




119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840


119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896


119896 minus 1

119888119877+ 119896119888119872



119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1



119896)

119896

Πlowast

(119896 minus 1

119896)

119896



119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast


1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)


119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896


120587RF119877(119901 119897)


119877+ 119905)119898119897

(36)


CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877


119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF




119877(119901 119897 V(119901 119897))


Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)






119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)




(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)



Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











retail price

(119901RF119862)lowast

=119898120578119877(119888119872+ 119888119877+ 119905)

120578119877minus 119865 ((119897RF

119862)lowast) (28)




Π((119897RF119862)lowast

)

=119860 (119897RF119862)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

((120578119877minus 119865 ((119897RF

119862)lowast

))

120578119877

)

119896

(29)


120587No119862(119901 119897) = 119889 (119901) 119901 [119897 minus (119897 minus 119909)

+] minus (119888119872+ 119888119877)1198981015840119897 (30)






(119901119873119900

119862)lowast

=1198981015840120578119877(119888119872+ 119888119877)

120578119877minus 119865 ((119897119873119900

119862)lowast) (31)


119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897 (32)



(119902No119862)lowast

= 1198601198981015840(119897

No119862)lowast[

[

120578119877minus 119865 ((119897

No119862)lowast

)

1198981015840120578119877(119888119872+ 119888119877)]

]

119896

(33)


Π1015840((119897

No119862)lowast

)

=119860 (119897No119862)lowast

(119896 minus 1) [1198981015840 (119888119872 + 119888119877)]119896minus1

(120578119877minus 119865 ((119897No

119862)lowast

)

120578119877

)

119896

(34)



(1) (119897119877119865119862)lowast= (119897119873119900119862)lowast

(2) Both (119897119877119865119862)lowast and (119897119873119900



(3) Both (119901119877119865119862)lowast and (119901119873119900



(i) when 119866((119897119877119865119862)lowast) gt 0 (119901119877119865


119877

when 119866((119897119877119865119862)lowast) lt 0 (119901119877119865


119877

when119866((119897119877119865119862)lowast) = 0 (119901119877119865


119877

(ii) when119866((119897119873119900119862)lowast) gt 0 (119901119873119900-119877119865


119877

when 119866((119897119873119900119862)lowast) lt 0 (119901119873119900


119877

when119866((119897119873119900119862)lowast) = 0 (119901119873119900


119877


119862)lowast= (119897RF119862)lowast





119862)lowast) =




119860120578119877gt 0 119885(119865minus1(120578

119877)) le (1 minus 119896)119860 lt 0 and (119897RF






119862)lowast119889120578119877gt 0







119862)lowast119889120578119877

=


119862)lowast)[120578119877

minus



119862)lowast

1198851015840(119897)|119897=(119897


119862)lowast119889120578119877


119862)lowast119889120578119877=

(119898(119888119872

+ 119888119877+ 119905)[120578

119877minus 119865(119897RF




119866((119897RF119862)lowast) gt 0 (119901RF


119877 when

119866((119897RF119862)lowast) lt 0 (119901RF


119877 when

119866((119897RF119862)lowast) = 0 (119901RF


119877


119862)lowast














119862)lowast) or 119866((119897No






119877





ln[Π((119897RF119862)lowast)Π1015840((119897No


119862)lowast) and Π1015840((119897No

119862)lowast) are





Δ = (119896 minus 1) ln [120572 + 120573 (1 minus 120572)

120572sdot

119888119872+ 119888119877

119888119872+ 119888119877+ 119905

] (35)


119877 and





119877 it




then Δ le 0


119872+ 119888119877)(119888119872+ 119888119877+ 119905)) gt 1 (le1)














120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo




119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840


119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896


119896 minus 1

119888119877+ 119896119888119872



119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1



119896)

119896

Πlowast

(119896 minus 1

119896)

119896



119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast


1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)


119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896


120587RF119877(119901 119897)


119877+ 119905)119898119897

(36)


CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877


119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF




119877(119901 119897 V(119901 119897))


Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)






119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)




(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)



Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















119862)lowast) or 119866((119897No






119877





ln[Π((119897RF119862)lowast)Π1015840((119897No


119862)lowast) and Π1015840((119897No

119862)lowast) are





Δ = (119896 minus 1) ln [120572 + 120573 (1 minus 120572)

120572sdot

119888119872+ 119888119877

119888119872+ 119888119877+ 119905

] (35)


119877 and





119877 it




then Δ le 0


119872+ 119888119877)(119888119872+ 119888119877+ 119905)) gt 1 (le1)














120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo




119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840


119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896


119896 minus 1

119888119877+ 119896119888119872



119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1



119896)

119896

Πlowast

(119896 minus 1

119896)

119896



119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast


1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)


119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896


120587RF119877(119901 119897)


119877+ 119905)119898119897

(36)


CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877


119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF




119877(119901 119897 V(119901 119897))


Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)






119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)




(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)



Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120572 120572 =120573 (119888119872+ 119888119877)

120573 (119888119872+ 119888119877) + 119905

120572 isin (0 120572)

120572 isin [120572 +infin)

gt0le0

YesNo

120573 120573 =120572119905

(1 minus 120572) (119888119872+ 119888119877)

120573 isin (0 120573)

120573 isin [120573 +infin)

lt0ge0

NoYes

119905 119905 =120573 (1 minus 120572) (119888

119872+ 119888119877)

120572

119905 isin (0 119905)

119905 isin [119905 +infin)

gt0le0

YesNo




119896 minus 1119898119872119873

119896

119896 minus 1119898101584011987210158401198731015840 119898119872119873 119898101584011987210158401198731015840


119896)

119896 119860119898(119897RF119877)lowast

(119898119872119873)119896

(119896 minus 1

119896)

119896 1198601198981015840(119897

No119877)lowast

(11989810158401198721198731015840)119896

119860119898(119897RF119862)lowast

(119898119872119873)119896

1198601198981015840(119897

No119862)lowast

(11989810158401198721198731015840)119896


119896 minus 1

119888119877+ 119896119888119872



119896)

119896minus1

Πlowast

(119896 minus 1

119896)

119896minus1



119896)

119896

Πlowast

(119896 minus 1

119896)

119896



119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Πlowast

((119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

)Π1015840lowast

Πlowast

Π1015840lowast


1015840= 119888119872 + 119888119877 Π

lowast= 119860(119897

RF119862)lowast(119896 minus 1)(119898119873)


119862)lowast(119896 minus 1)(119898

10158401198731015840)119896minus1119872119896


120587RF119877(119901 119897)


119877+ 119905)119898119897

(36)


CVaR (120587RF119877(119901 119897))

= maxV119877isinR

V119877+1

120578119877


119877 0)]

(37)

Let Π119877(119901 119897 V(119901 119897)) = CVaR(120587RF




119877(119901 119897 V(119901 119897))


Π119877(119901 119897) = 119860119901

minus119896[119901 minus (119908 + 119888

119877+ 119905)119898] 119897 minus

119901Λ (119897)

120578119877

(38)






119865 (119897) =120578119877119897 + (119896 minus 1) Λ (119897)

119896119897

(119901119877119865

119877)lowast

=119898120578119877(119908 + 119888

119877+ 119905)

120578119877minus 119865 ((119897119877119865

119877)lowast)

(39)




(119902RF119877)lowast

= 119860119898(119897RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(40)



Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119908 + 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

(41)


119877)lowast= (119897

RF119862)lowast= (119897





Π119872 (119908)

= (119908 minus 119888119872) 119860119898 (119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

(42)



119877+ 119896119888119872)(119896 minus 1)



119872(119908) is

119889Π119872 (119908)

119889119908= 119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot [1 minus119896 (119908 minus 119888

119872)

119908 + 119888119877+ 119905

] = 0

(43)


119872)(119908 + 119888

119877+ 119905)] = 0 After


Furthermore

119889Π119872 (119908)

119889119908

10038161003816100381610038161003816100381610038161003816119908=119908lowast= minus119860119898(119897

RF119877)lowast[

[

120578119877minus 119865 ((119897RF

119877)lowast

)

119898120578119877(119908 + 119888

119877+ 119905)

]

]

119896

sdot119896 (119905 + 119888

119872+ 119888119877)

(119908 + 119888119877+ 119905)2lt 0

(44)




Π119877((119897

RF119877)lowast

)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896minus1

(45)

Π119872(119908lowast)

=119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

(120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896


119896)

119896

(46)


Π119877((119897

RF119877)lowast

) + Π119872(119908lowast)

= [(119896 minus 1

119896)

119896minus1

+ (119896 minus 1

119896)

119896

]

sdot119860 (119897RF119877)lowast

(119896 minus 1) [119898 (119888119872+ 119888119877+ 119905)]119896minus1

sdot (120578119877minus 119865 ((119897RF

119877)lowast

)

120578119877

)

119896

lt2

119890sdot Π ((119897

RF119862)lowast

)


)

(47)



119862) and (119902RF




there exists 119908 = 119888119872





Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Π119877(119901 119897 120574)

= 119889 (119901) [120574119901 minus (119888119872+ 119888119877+ 119905)119898] 119897 minus

120574119901Λ (119897)

120578119877

(48)


Π119872(119908 120574) = 119889 (119901)


(1 minus 120574) 119901Λ (119897)

120578119877

(49)


119872+ (120574 minus 1)(119888




119877+119905) and 119889(119901) = 119860119901minus119896


Π119877(119901 119897 120574) = 120574Π (119901 119897)

Π119872(119908 120574) = (1 minus 120574)Π (119901 119897)

(50)



119872(119908 120574)



that case




119872+ (120574 minus 1)(119888

119877+ 119905) and 120574 isin (((119896 minus

1)119896)119896minus1

1 minus ((119896 minus 1)119896)119896) then

120574Π((119897119877119865

119862)lowast

) gt Π119877((119897119877119865

119877)lowast

)

(1 minus 120574)Π ((119897119877119865

119862)lowast

) gt Π119872(119908lowast)

(51)





119862= 120578119877= 02 119888

119872= 06 and 119888

119877= 06



119877


lowast















120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572 120573 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0607 831959 3062 125 811160 540774 2041 2807863 185211108 909311 2928 125 886578 590052 1952 3068924 199480109 990100 2806 125 965348 643565 1871 3341588 2172033

0707 889651 2961 125 867409 578273 1974 3002571 195167108 949276 2866 125 925544 617029 1911 3203806 208247409 1010835 2778 125 985564 657043 1852 3411567 2217519

0807 949276 2866 125 925544 617029 1911 3203806 208247408 990100 2806 125 965348 643565 1871 3341588 217203309 1031784 2749 125 1005990 670660 1833 3482272 2263477

0907 1010835 2778 125 985564 657043 1852 3411567 221751908 1031784 2749 125 1005990 670660 1833 3482272 226347709 1052948 2721 125 1026625 684416 1814 3553701 2309906


120572 119905 119902lowast

119877119901lowast

119877119908lowast

Πlowast

119877Πlowast

119872119901lowast

119862119902lowast

119862Πlowast

06 0 638398 3799 115 526678 351119 2533 2154592 1185025

0601 1156109 2703 120 1040498 693665 1802 3901867 234112003 728045 3154 130 764447 509632 2102 2457152 172000705 487773 3604 140 584280 390187 2403 1646100 1316880

0701 1206921 2646 120 1086229 724152 1764 4073357 244401403 760043 3087 130 798046 532030 2058 2565146 179560205 50917 3528 140 611004 407336 2352 1718448 1374758

0801 1258825 2591 120 1132943 755295 1727 4248536 254912103 79273 3022 130 832366 554911 2015 2675463 187282405 531067 3454 140 637280 424854 2303 1792351 1433881

0901 1311823 2538 120 1180641 787094 1692 4427402 265644103 826104 2961 130 867409 578273 1974 2788102 195167105 553425 3384 140 664110 442740 2256 1867810 1494248


119905 120573 119902lowast119877

119901lowast119877


Πlowast119872

119901lowast119862

119902lowast119862

Πlowast

0 0 638398 3799 115 526678 351119 2533 2154592 1185025

0107 1057763 2826 120 951987 634658 1884 3569950 214187008 1156109 2703 120 1040498 693665 1802 3901867 234112009 1258825 2591 120 1132943 755295 1727 4248536 2549121

0307 666113 3297 130 699419 466279 2198 2248132 157369208 728045 3154 130 764447 509632 2102 2457152 172000709 792730 3022 130 832366 554911 2015 2675463 1872824

0507 446244 3768 140 535493 356995 2512 1506073 120485808 487733 3604 140 585280 390187 2403 1646100 131688009 531067 3454 140 637280 424854 2303 1792351 1433881

0707 313411 4239 150 423015 282070 2826 1057763 95198708 342551 4055 150 462444 308296 2703 1156109 104049809 372985 3886 150 503530 335687 2591 1258825 1132943





119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119877all





119877increase







1 2 3 4 5 6 7 8 9 10154015601580160016201640166016801700

The t

otal

supp

ly ch

ainrsquos

pro

fit

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

120590 with 120572 = 05 and k = 3

with

coor

dina

tion


119877

005 01 015 02 025 03 035 04

120578C = 120578R = 02120578C = 120578R = 04

120578C = 120578R = 06

120578C = 120578R = 08120578C = 120578R = 10

1 minus 120572 with 120590 = 5 and k = 2

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n5100

4900

4700

4500


119877

21 22 23 24 25 26 27 28 29100015002000250030003500400045005000

The t

otal

supp

ly ch

ainrsquos

pro

fitw

ith co

ordi

natio

n

k with 120572 = 06 and 120590 = 5

t = 00t = 01

t = 02

t = 03

t = 04






119862lt 1)









119877= 1





Competing Interests


Acknowledgments



References



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119862lt 1)









119877= 1





Competing Interests


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









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