A Modified Customization Strategy in a Dual-Channel Supply

Research ArticleA Modified Customization Strategy in a Dual-Channel SupplyChain Model with Price-Sensitive Stochastic Demand andDistribution-Free Approach

Ruchi Chauhan1 Varun Kumar1 Tapas Kumar Jana2 and Arunava Majumder 1

1Department of Mathematics Lovely Professional University Phagwara Punjab 144411 India2Department of Mathematics Ghatal Rabindra Satabarsiki Mahavidyalaya Ghatal West Bengal 721212 India

Correspondence should be addressed to Arunava Majumder amarunavamajumdergmailcom

Received 13 May 2021 Revised 15 August 2021 Accepted 20 August 2021 Published 23 September 2021

Academic Editor Vahid Kayvanfar

Copyright copy 2021 Ruchi Chauhan et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

With the advancement of technology many companies provide customization facilities to customers is facility provides a vastvariety to customers which enhances the level of customer satisfaction is approach helps various technologically advancedcompanies to increase their profit In this paper a dual-channel supply chain model is developed with the aforementionedcustomization strategy with the target of increasing the profit of the firm In dual-channel the core or standard product isprovided to the customer through a traditional retail channel whereas the customized product is made available through theonline channel is article incorporates a modification in the existing dual-channel policy on the number of customers thatswitch between the offline and online channels Moreover a preassigned threshold value is also assumed which signifies thedecrease in demand that takes place if the difference between the selling price of offline and online channels crosses a fixedspecified threshold value In addition to that due to fluctuation and uncertainty of demand both variability and randomness mayoccur simultaneously us the price-sensitive stochastic demand is considered to develop the dual-channel centralized supplychain model with customization A max-min distribution-free approach is applied to deal with the randomness and variability ofdemand e model is analyzed and validated with numerical experiments and graphical analysis Consequently the articleconcluded that it is better to adopt a dual-channel supply chain policy for better profitability than the traditional single-channelsupply chain as this firm will be able to provide customized products to customers Moreover if the difference between the sellingprices of the offline and online channels is greater than the preassigned threshold value then the shifting of customers takes placedepending upon the factor that which channelrsquos selling is less in comparison to another

1 Introduction

e growth of Internet has revolutionized the concept ofsupply chain management in a significant manner Severalstatistical reports of various groups show that e-commercehas accelerated the economic situation of modern marketingenvironment In 1985 Dell embraced the strategy of built-to-order computers on demand of customers is approachprovided Dell a sale of $70 million in that year and after fiveyears of adopting the strategy their revenue climbed to $500million By the end of 2000 Dellrsquos revenues had topped withan astounding of $25 billion [1] In this respect current

study compares the traditional retail channel with the onlinechannel under customization strategy Many manufacturerssell their standard products to the retailers through con-ventional retail channel and customized product directly tothe customers according to their requirement erefore inorder to achieve a smooth supply chain flow companies usedual-channel supply chain [2ndash4] and [5]

Furthermore the dual-channel supply chain modelprovides the customers with many options to choose eitherstandard products from the retailer or customized productsthrough online channel [6ndash8] In addition to that manu-facturers are adopting e-commerce to compete with their

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 5549882 25 pageshttpsdoiorg10115520215549882

opponents [9] and implementing green technology in themanufacturing industry for reduction of greenhouse gasemission [10 11] e retailerrsquos services and behavior greatlyinfluence the choices and demand of the customer [12 13]Many customers prefer online platform as it saves someamount of cost and time due to the direct interaction be-tween the manufacturer and the customer under onlineenvironment [14] Moreover in the online channel thecustomers are also provided with the feedback facility whichultimately helps the manufacturer to know their customerand improve their services accordingly [14]

Some well-known manufacturing and retailing indus-tries such as Nike started producing the customized prod-ucts on demand of customers through online channel etraditional retail channel is also utilized to deliver thestandard products without any customization [15 16] Astudy reveals that around 42 of the top brands are cateringthe customers directly through online channel [17] Priorlywith the acceptance of the online channel it would createconflict between the manufacturer and the retailer as itwould shrink the retailerrsquos market share However studieshave shown that whole sale prices reduce for product soldthrough the retailer and increase the profit of the firm withthe establishment of online channel [18 19] With the in-troduction of online channel along with offline channelhouse needs to define markup margin accurately as it in-fluences the wholesale prices

is paper put up to the studies in several aspects whichwere unnoticed in the previous literature e main con-tributions of this research lie under the consideration ofthreshold limit unequal shifting of customers multideliverypolicy uncertain demand and controllable lead time in thedual-channel supply chain model with customization policyHeretofore researchers developed several dual-channelmodels but with equal number of customer shifting from theonline to offline channel and vice versae concept of equalcustomer shifting between the channels may not be realisticas all customers who are unwilling to purchase items fromone channel are not forced to shift to another one Secondlya preassigned threshold limit is considered in this articlewhich was not assumed in previous research studies ethreshold limit is incorporated under the concept that acustomer is only willing to change the mode of purchase ifthe selling price difference between the two channels crossesto a specified price limit ese two important modificationson dual-channel supply chain management (SCM) are in-troduced in this study Moreover this study developed adual-channel supply chain model under customization withSSMD policy along with make-to-order uncertain demandcontrollable lead time and distribution-free approach Allthese attributes in a single model are rarely found in theexisting literature

e objective of this paper is to develop an efficient andimproved customization policy in dual-channel SCM eaim of this study is to obtain the managerial decision underdemand uncertainty and price sensitivity with distribution-free approach e competitions between online vs retailchannel dual-channel vs retail channel and dual-channelvs online channel are shown in this article to determine the

best possible channel to be adopted by the company An-other vital objective of this research is to analyze the effect ofmigration of customers between the channels on supplychain profitability under a fixed and specified thresholdvalue of selling price which was not yet studied by anyexisting literature e effect of the variation of selling priceon customer demand and supply chain profit is also animportant matter of concern

e proposed study retains some important researchquestions which are illustrated as follows

(a) What is the effect of the demand uncertainty on theprofit of the firm

(b) How to determine that which channel should beadopted so that profit of the firm maximizes

(c) How profitability of the firm is affected if sellingprices exceed the fixed and specified threshold limit

(d) How does the transfer rate of consumers from onlinechannel to retail channel or vice versa affect optimaldecision

e entire paper is split into several sections to deal withdifferent aspects such as description of study literaturereview and mathematical modelling Sections 1 and 2 de-scribe introduction and literature review respectivelySection 3 explains the important research questions and aimof study Section 4 depicts problem definition assumptionsand notation to develop the mathematical modelling Sec-tion 5 elaborates the entire mathematical model and ana-lytical expressions of decision variables along with somepropositions and corollaries Solution technique (algorithm)of the model is also described in this section Section 6 dealswith numerical experiments and model validation emanagerial decisions are obtained by using a set of inputparameters and results are analyzed with graphical repre-sentations and sensitivity analysis e important discus-sions about the outcomes are also described in this sectionLastly in Section 7 conclusions regarding the study aredepicted

2 Literature Review

21 Dual-Channel Supply Chain In recent years study ondual-channel supply chain has gained reasonable attentionamong the researchers throughout the globe However mostof the studies focused on pricing decisions and channelcoordination In this technologically advanced era everyplayer involved in the supply chain (customers manufac-turers and retailers) expects the availability of the infor-mation of product Consequently Tsay and Agrawal [2]Wilder [4] and Mukhopadhyay et al [20] stated that theseexpectations can be fulfilled if companies have smoothsupply chain flow Moreover Takahashi et al [6] expressedthat through the dual-channel supply chain customers areprovided with two options ordering of the standard productthrough retail channel and the customized product throughonline channel

From health care providers (Medlife Indiarsquos largeste-health platform which provides medicine through online

2 Mathematical Problems in Engineering

platform) to retailers (Wal-Mart RT-Mart and Metro)organizations are under great strain to meet the expectationsof the customers and to stay ahead of the competition whilecontrolling costs [9 21] Moreover online platform bringsmore simplicity flexibility and cost-effectiveness to satisfythe requirement for maximum profitability of the firm (Sana[22] and Hosseini-Motlagh et al [23]) In addition to thisonline sales provide customer feedback which is helpful tothe manufacturer in revealing emerging trends changingattributes and opportunities to enhance customer value[14] Furthermore to cater wide range of consumer groupsmultiple product variety needs to be provided by the in-dustry [24] Furthermore industrialists are encouraged touse green technologies for sustainable development [25 26]

22 Centralized Dual-Channel Supply Chain withCustomization Customization is primarily the action ofmodifying something (product) to suit a particular indi-vidual or task MyVirtualModel a Montreal-based tech-nology provider and its roster of partners (including AdidasBest Buy Levirsquos and Sears) enables consumers to build avertical model of themselves an ldquoavatarrdquo is means thatcustomers can try on clothes virtually Consumers of BMWrsquosMini Cooper can use an online toolkit to design the carrsquosroof which is then produced with an advanced digitalprinting system on a special foil (Salvador et al [27])Moreover many companies are adopting mass custom-ization technologies to satisfy the growing individualizationof demand [28ndash32] Henceforth Figure 1 exemplifies thecustomization policy in the centralized dual-channel supplychain model Customization policy could be implementedwith the introduction of dual-channel strategy which furtherleads to the complexity in SCM as companies need to re-design the supply chain structure according to the demandof customers [6]

23 Comparison between Online and Offline ChannelsAccording to Zhang et al [33] in a dual-channel policy thecompetition between online and retail channels and how tomitigate the effect of adopting an online channel on the retailchannel are important matters to be concerned In thisstream of research Kaya et al [34] Yan and Pei [12] andDan et al [17] studied the role of the retailer and traditional-online competition in a dual-channel supply chain InitiallyKaya et al [34] concluded that the optimal channel strategyis influenced by the choices of the customers between tra-ditional channel and online channel Later on Yan and Pei[12] observed that supply chain performance can be im-proved with the improvement of retail service which waslater on manifested by Dan et al [17] Furthermore Huaet al [35] scrutinized the optimal prices of a dual-channelsupply chain under centralized and decentralized environ-ment considering the delivery lead time In addition theyalso proved that the customerrsquos acceptance of online channelgreatly influences the supply chain Moreover Li et al [36]demonstrated the acceptance of the customersrsquo onlinechannel over retail channel along with customersrsquo sensitivitytowards product differences Customersrsquo choices have an

impactful influence on firmsrsquo decision of offering the cus-tomized products Batarfi et al [37] demonstrated thatadding a customized product through online channel in-creases the profit of centralized supply chain system thoughit creates a conflict due to competition between the retail andonline channels Moreover Huang et al [15] and He et al[38] examined centralized and decentralized supply chainand pricing decisions in a dual-channel supply chain modelunder uncertain demand

24 Dual-Channel Supply Chain under Demand UncertaintyIn past decades researchers have been working on stochasticdemand in the dual-channel supply chain model Chiang andMonahan [39] contemplated price independent stochasticdemand in the traditional and online channel Yao et al [40]proved that information sharing in the dual-channel supplychain model can be beneficial for the firm which yields betterprofitability then noncoordinating supply chain Further-more Modak and Kelle [41] adopt price and lead time-de-pendent random demand for the make-to-order and make-to-stock production system Dumrongsiri et al [42] con-cluded that demand variation greatly influences the equi-librium prices and the manufacturerrsquos inspiration in adoptingonline channel Moreover He et al [43] demonstrated howthe manufacturer can inspire the retailer to expand itsbusiness by adopting the dual-channel supply chain modelunder stochastic demand e article also depicted that pricesharing decision of the manufacturer can actually act as aboon for the retailer Furthermore Roy et al [44] developed adual-channel supply chain model under stochastic demand toascertain the optimal stock level prices and promotion effort

25 1e Distribution-Free Approach Occasionally randomdemand might follow a well-known distribution but if leaninformation about demand fluctuation over time is availablethe distribution-free approach might be a better tool fordecision-making According to Moon et al [45] there isalways inclination towards normal distribution in case ofrandom demand disregarding other distribution with thesame mean and variance Scarf [46] solved the newsvendor

Stan

dard

izat

ion Custom

ization

Manufacturer

CustomerCustomer

Retailer

Centralized System

Del

iver

yD

eliv

ery

Delivery

Order

Order

Ord

er

Figure 1 Dual-channel centralized system

Mathematical Problems in Engineering 3

problem with the distribution-free approach with a knownmean and variance e article considered the maximum ofthe lower bounds of the expected profit for all possibledistribution functions

Hua et al [35] and Xu et al [47] assumed deterministicdemand without inventory consideration in the dual-channelsupply chain model for considering decisions of delivery leadtime and prices Yang et al [48] introduced the newsvendormodel in dual-channel supply chain In this model theyscrutinized the switching behavior of the customersdepending upon availability of the product and lead time buteffects of price on demand were neglected Table 1 enlists theresearch which inspires this article to consider a preassignedthreshold value that is a limit beyond which if the selling priceincreases the customers migrate from one traditional channelto online channel or vice versa when the demand is assumedto be variable additive stochastic demand Moreover thiswork applies distribution-free approach

26 Research Gap e dual-channel supply chain man-agement with customization policy has been studying bymany researchers and scientists Previous studies considerthat customer shifting between the channels is invariant It isnot necessary that the all customers who left a channelshould mandatorily shift to the other channel Some cus-tomers may be reluctant to purchase also us equalshifting of customers may not be practical in modern marketscenario e condition of equal customer shifting is relaxedin this article Moreover there is a possibility that no cus-tomers may shift to the other channel is happens due tomany reasons such as inability to purchase due to highselling price us a specified threshold limit is considerede threshold limit is a specified positive numeric valueCustomer switch only happens if the difference between theselling prices of online and offline channels is greater thanthe threshold limit e concept of threshold value is in-corporated as a pioneer approach Moreover till now eithervariable demand or stochastic demand was used but price-sensitive stochastic demand is rarely assumed in literatureerefore this article extends the previous dual-channelmodels with unequal customer shipment specific thresholdlimit and price sensitive stochastic demand under distri-bution-free approach

3 Problem Definition Assumptionsand Notation

is section explains the problem definition assumptionsand notation used to formulate the mathematical model ofthis paper

31 ProblemDefinition A comparison between a traditionaloffline and online channel on the supply chain model with asingle manufacturer and single retailer is framed in thisstudy Most of the literature studies used either probabilisticuncertainty or demand variability but a few articles werefound which considered both of the characteristics simul-taneously in a demand function In addition to that this

paper uses max-min distribution-free approach to deal withthe stochastic uncertainty along with price sensitivityMoreover demand data or the exact mean and standarddeviation is required for estimating a probability distribu-tion of demand which is money as well as time consumingerefore to overcome all these constraints this studyconsiders distribution-free approach to obtain managerialdecisions with the help of known mean and standarddeviation

To implement an online channel the manufacturer andretailer should negotiate properly as a threshold value hasbeen incorporated in this article e threshold value sig-nifies that if the price difference between online and offlineexceeds beyond a specified limit then customers migratefrom retail channel to online channel or vice versa whichshould influence the profit of the channels individually alongwith the firm as a whole

32 Assumptions Following points enlist the various as-sumptions assumed for formulating the mathematicalmodel

(1) e company adopts the centralized dual-channelsupply chain model which is assumed with cus-tomization strategy Hence the product is availableto customers through a retailer channel and an In-ternet-based direct channel Furthermore demand isdeemed to be variable as well as random [41]

(2) e manufacturer has its own manufacturing housewhere standardized and customized products aremanufactured and does not need to invest inadopting a mass customization system [37]

(3) In case of retail channel which is selling standardproduct common cycle time is considered for themanufacturer and the retailer Moreover in case ofcustomization process the manufacturer has dif-ferent cycle time for production of core product usedin customization

(4) A proportion of the number of customer who refusesto purchase items through retail channel chooses topurchase customized product [41]

(5) Production rates for standard and customizedproducts are greater than the base demand that is noshortages are allowed

(6) Lead time is considered to be zero between themanufacturer and the retailer along with the supplier(who supplies the additional custom features) andthe manufacturer [37]

(7) Single-setup multiple-delivery policy is used to de-liver the product in retail channel whereas make-to-order policy is used for online channel

(8) When the selling price difference of the online andoffline channel falls within a fixed and preassignedlimit (threshold limit) no customer shifting wouldoccur

(9) All input parameters in this paper are positive


33 Notations Tables 2 and 3 enlist the notations used indeveloping the mathematical model

4 Mathematical Model

is section describes the demand function profit functiondistribution-free approach optimal decision of the supplychain and solution algorithm of this paper

41 Demand Function Customers are miscellaneous in na-ture in their inclination towards the standard or the customizedproductMany factors influence the customerrsquos intentions suchas price variety of products and incapability of customers toreach out to the retail stores Succeeding Huang et al [15] Yueand Liu [49] Zhang et al [53] and Raju and Roy [54] lineardemand functions for the standard and customized product areconsidered Following demand functions for retailer and onlinechannels are obtained by extending the work of Huang andSwaminathan [55] and Hua et al [35]

e demand function for offline channel is given by

Dr a1 minus β1pr + δ1 1113944

N

i1pdi minus pr

⎛⎝ ⎞⎠ (1)

e demand function for online channel is given by

Dd a2 minus β2 1113944

N

i1pdi minus δ2 1113944

N

i1pdi minus pr

⎛⎝ ⎞⎠ (2)

where the following points explain the various parametersused in demand function

Parameters a1 and a2 are the number of customerswithout having any certain and variable componentse price sensitivity coefficients β1 of the retailerrsquoschannel and β2 of the manufacturerrsquos channel representthe amount of decreaseincrease in market demandwhen both channels increasedecrease the price by onedollar us β1pr and β2 1113936

Ni1 pdi represent change in

customers because of price sensitivityδ1 and δ2 represent the shifting of customers fromthe offline channel to the online channel or viceversa depending upon the factor that which chan-nelrsquos product is having less cost in comparison toothersA house would usually generate the same return perdollar invested irrespective of the item sold thereforethe selling price of the standard product for the retaileris given by pr Cp(1 + m)2 (calculation of it is shownin Appendix A) and the selling price of customizedproduct for the manufacturer is given by pdi Cdi(1 +

m) where m is a markup margin Henceforthδ1(1113936

Ni1 pdi minus pr) and δ2(1113936

Ni1 pdi minus pr) indicate that

the change is the number of customers because of δ1and δ2

Assuming fixed markup margin for standard as well ascustomized products we get the following equations

Dr a1 minus β1Cp(1 + m) + δ1(1 + m) 1113944N

i1Cdi minus Cp

⎛⎝ ⎞⎠ (3)


Table 1 Related literature

Authorrsquos name DCSC SP CP UCS SSMD MTO DU DFA CLT MMChiang et al [21] radic radicTsay and Agrawal [2] radic radicChiang and Monahan [39] radic radicYue and Liu [49] radic radic radicKaya et al [34] radic radic radic radicYan and Pei [12] radic radicHua et al [35] radic radic radicTakahashi et al [6] radic radicDan et al [17] radic radicJing et al [14] radic radic radicShao [18] radic radic radic radic radicLi et al [36] radic radic radic radicZhang and Choi [3] radic radicBatarfi et al [37] radic radic radic radic radic radicMoon et al [45] radicYang et al [48] radic radic radic radicMajumder et al [50] radic radic radicModak and Kelle [41] radic radic radic radic radic radicMajumder et al [51] radic radicSarkar et al [52] radic radicZhou et al [13] radic radic radicis paper radic radic radic radic radic radic radic radic radic radicDCSC dual-channel supply chain SP standard product CP customized product UCS unequal customer shifting from online to offline and vice versaMTO make-to-order DU demand uncertainty DFA distribution-free approach CLT controllable lead time MM markup margin


Dd a2 minus β2(1 + m) 1113944N

i1Cdi

⎛⎝ ⎞⎠ minus δ2(1 + m) 1113944N

i1Cdi minus Cp

⎛⎝ ⎞⎠ (4)

42 Profit Functions is section includes profit equationsof the manufacturer and retailer for standard and cus-tomized products

421 Manufacturerrsquos Profit for Standard Product e profitof the manufacturer per unit of time from selling thestandard product through the retail channel is given by

nabla1 Revenue minus Setup cost minus Holding cost

minus Manufacturerrsquos production cost

nabla1 Cp(1 + m)Dr

minusSrDr

nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(5)

where the following points explain the various cost com-ponents used in equation (5)

Table 3 Parameters of the model

Parameters DescriptionCp Production cost for standard product ($unit)Cdi Production cost for customized product (i 1 2 N) ($unit)a1 Number of customers prefer retail channela2 Number of customers prefer online channel

Pdi

Production rate for the core product for eventualcustomization (Pdi gt a2) (positive number)

Pr Production rate for the standard product (Pr gt a1) (positive number)pr Retailerrsquos selling price of the standard product (pr gtCp) ($order)pdi Manufacturerrsquos selling price of the customized product i i 1 2 N ($order)Dr Variable demand of retail channel (unitsyear)Dd Variable demand of online channel (unitsyear)ϕdi Percentage of the core product stock used for customized product (i 1 2 N)Ar Ordering cost of the retailer per order ($order)Sr Manufacturerrsquos setup cost for standard product ($setup)Sd Manufacturerrsquos setup cost for core product for customized product ($setup)rv Holding cost rate of the manufacturer ($unitunit time)rb Holding cost rate of the retailer ($unitunit time)Cb Unit production cost paid by the retailer (buyer) ($unit)Cvr Unit production cost paid by the manufacturer (vendor) ($unit)β1 Price sensitivity in retail channel (customerday)β2 Price sensitivity in online channel (customerday)δ1 Number of customers switching from retail channel to online channelδ2 Number of customers switching from online channel to retail channelh1 Manufacturerrsquos holding cost which includes financial cost and storage cost ($unit)π Unit backlogging cost for the retailer ($unit)σ Standard deviation of demand per unit timem Markup margin (percentage)R Reorder point of the retailer (units)s Safety factor of the retailer (units)M Random lead time demand which has a cumulative distribution function (cdf ) F with mean DL and standard deviation σ

L

radic

E( ) Mathematical expectation

Table 2 Decision variables of the model

Decision variables DescriptionL Length of the lead time for the retailer (days)n Number of lots delivered from the manufacturer to the retailer in one production cycle a positive integerk Safety factor (units)Qr Quantity of the standard product ordered by the retailer (units)Qd Quantity of the core product ordered for customization (units)


Cp(1 + m)Dr is revenue and Cp(1 + m) is the sellingprice of the manufacturer for standard product(SrDrnQr) is setup cost for the retail channel and themanufacturer produces integer multiple of the retailerrsquosorder quantity Figure 2 displays that as Qr is the totalorder quantity of all retailers the manufacturer

produces nQr quantity where n is a positive integereexpected cycle length of the manufacturer thus be-comes (DrnQr) and the setup cost of the manufac-turer becomes (SrDrnQr)e average inventory of the manufacturer can becalculated as

Dr nQr QrPr( 1113857 + (n minus 1)QrDr(( 1113857( 1113857 minus n2Q

2r2Pr1113872 1113873 minus Q

2rDr1113872 1113873(1 + 2 + middot middot middot +(n minus 1))1113872 11138731113873

nQr

Qr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889

(6)

e holding cost of the manufacturer is(rvCpQr2)(n(1 minus (DrPr)) minus 1 + (2DrPr)) us theexpected holding cost per unit time per unit item is

rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889 (7)

CvrDr represents the manufacturerrsquos production costfor standard product

422 Manufacturerrsquos Profit for Customized ProductFigure 3 exhibits the online channel inventory of themanufacturer whereas the profit of the manufacturer perunit of time from selling the customized product through theonline channel is given by


minus Manufacturing cost

nabla2 1113944N

i1Cdi(1 + m)ϕdiDd

minusSdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦

(8)


1113936Ni1 Cdi(1 + m)ϕdiDd is revenue and 1113936

Ni1 Cdi

(1 + m)ϕdi is the selling price of the manufacturer forcustomized product

(SdDdQd) is setup cost as the manufacturer followsmake-to-order policy where he manufactures theproduct on demand by the customer Qd is the totalorder quantity of all customers e expected cyclelength of the manufacturer thus becomes (DdQd) andthe setup cost of themanufacturer becomes (SdDdQd)e average inventory of the manufacturer can becalculated as

average inventory Qd

21 minus

Dd

Pd

1113888 1113889 (9)

e holding cost of the manufacturer is(h1Qd2)(1 minus (DdPd))us the expected holding cost per unit time is

h1Qd

21 minus

Dd

Pd

1113888 1113889 (10)

Each customization is composed of a production cost1113936

Ni1 Cdi and total production cost of the vendor will be

1113936Ni1 CdiϕdiDd

423 Retailerrsquos Profit Figure 4 manifests the inventorypattern between the manufacturer and retailer whereas theprofit of the retailer per unit time from selling the standardproduct is given by

nabla3 Revenue minus Ordering cost minus Holding cost minus shortage cost minus Lead time crashing cost

nabla3 Cp(1 + m)2Dr minus

ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(11)



Cp(1 + m)2Dr represents revenue of the retailerCp(1 + m)2 is the selling price for the retailer forstandard product

e ordering cost per unit time for the retailer is(ArDrQr)When the level of inventory reaches the reorder pointR quantity Q is ordered by the retailer R minus DrL rep-resents the expected level of inventory before the orderis received by the retailer whereas Q + R minus DrL rep-resents expected inventory level immediately after thedelivery of quantity Q Consequently (Q2) + R minus DrL

is the average inventory over a cycle Henceforth theretailerrsquos expected holding cost per unit time isrbcb((Qr2) + R minus DrL)M is the stochastic lead time demand and R is thereorder point for the retailer then the expectedshortage at the end of the cycle is expressed asE(M minus R)+ for the retailer resulting into(πDrQr)E(M minus R)+ as shortage coste lead time crashing cost per unit time is (DrCLQr)

Total profit of the single-channel supply chainnablaS is givenby adding nabla1nabla3 We get

nablaS nabla1 + nabla3

nablaS Cp(1 + m)Dr minusSrDr

nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ Cp(1 + m)2Dr minus

ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(12)

43 Distribution-Free Approach We elaborate the distri-bution-free approach by using the following points

Any specific probability distribution should not beconsidered for any random variable in this approachA class of cumulative distribution function (cdf )

having mean DrL and standard deviation σL

radicis

considered

e max-min distribution-free approach is applied toobtain the values of the decision variables in thisapproach initially the worst possible case (ie the

Qua

ntity

G

A B

C

F E

O Time

nQr

nQr

Pr

QrPr

(n ndash 1)QrDr

Figure 2 Manufacturersquos inventory position

D

A

C

B

Man

ufac

ture

r(O

nlin

e cha

nnel

)

TimeTd

P d

P d ndash D d

1 minusQdDdPd

Dd

Figure 3 Manufacturersquos online channel inventory under the EPQmodel


expression for minimum profit is obtained which isthen maximized for the best profitability)emethod was first used by Gallego andMoon [56] asa mini-max approach to obtain the minimum value ofcost function is approach was utilized to deal withthe uncertain demand onlyIn this article we use a max-min method for profitmaximization which is the opposite of Gallego andMoonrsquos [56] min-max distribution-free approachMoreover instead of assuming only uncertainty thisarticle provides demand variability also erefore amodification is added in the inequality used by Moon

and Gallego [56] with a max-min approach and in-corporation of variability in demand functione lead time demand M is depending on Dr whichfurther depends on the selling price of the retailerCp(1 + m)2 which leads to variability in the demandMoreover randomness in demand is accomplished inadditive form (Petruzzi and Dada [57] Sarkar et al[58] and Modak and Kelle [41])

e inequality stated by Proposition 1 is applied to solvethe model

Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572

+ k2σ2L minus 2DrLkσ

L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)

Moreover the upper bound of the above inequality is tightIn the equation R DrL + kσ

L

radicis reorder point DrL is

the lead time demand kσL

radicis safety stock and k is a safety

factor

Proof See Appendix Bus using inequality (10) the expected total profit of

the centralized single-channel supply chain nablaS is given byadding nabla1 and nabla3 We get

Production and shipment Shipment

Accumulated inventory level of manufacturer

Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time

Accumulated inventory level

of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr

QrDr (n ndash 1) QrDr

Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L

Figure 4 Manufacturersquos and retailerrsquos inventory pattern


nablaS Cp(1 + m)Dr minusArDr

Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)

And expected total profit of the centralized dual-channelsupply chain nablaD is given by adding nabla1 nabla2 and nabla3 We get

nablaD Cp(1 + m)Dr minusSrDr

nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N

i1Cdi(1 + m)ϕdiDd minus

SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)

44 Optimal Decision of the Supply Chain Since equations(17) are nonlinear in nature so for a fixed positive integer m

we take partial derivative of the profit with respect to Qd Qrand k to obtain the optimal solution

znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L

radic1113969 minus σL

radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)

Now for fixed value of integer m the value of Qd Qrand k is obtained by equating equations (18)ndash(20) to zerothat is


znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr

minus DrCL + SrDr + ArDr + πDr2( 1113857

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic

rbCb2( 1113857 + rv2( 1113857 n 1 minus DrPr( 1113857( 1113857 minus 1 + 2 DrPr( 1113857( 1113857

11139741113972

(21)

klowast

minus σ

QrCbrb minus QrCbrb + πDr( 1113857

1113969

1113874 1113875 minus QrCbrb + πDr( 11138571113874 1113875 + Drπσ

QrCbrb minus QrCbrb + πDr( 1113857L

1113969

1113874 1113875 2QrCbrb πDr minus QrCbrb( 1113857( 11138571113874 1113875 + LDrL

radicσ

(22)

It can be clearly seen that the optimal solution for Qlowastr Qlowastd and klowast is dependent on one another Furthermore aclosed form expression for centralized profit function is hardto find erefore finding these optimal values requires theuse of a numerical procedure An iteration method is usedwith the following algorithm to find the managerialdecisions

45 Solution Algorithm

Step 1 assign the values of all parameters as specified inTable 4Step 2 set n 1Step 3 for each value of Li i 1 2 perform thefollowing steps

Step 3(a) obtain the value of Qd from (22)Step 3(b) obtain the value of Qr from (25)

Step 3(c) obtain the value of k from (26)Step 3(d) repeat Steps 3(a) to 3(c) until there are nochanges in the values of Qd Qr and k up to a specifiedaccuracy level

Step 4 use the values of Qd Qr and k to obtain nablaS fromequation (14)Step 5 use the value of nablaS to obtain nablaDStep 6 set n n + 1 and repeat Step 3 to 5Step 7 if nablaD(n)gtnablaD(n + 1) then repeat Step 2 to Step6 else stop

Proposition 2 If we represent Qlowastr Qlowastd and klowast as the optimalvalues of Qr Qd and k then for fixed values of n andL isin [Li Liminus 1] the dual-channel profit function nablaD attains itsglobal maximum at Qlowastr Qlowastd and klowast under the condition

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572

+ k2σ2L minus 2DrLσ

L

radic1113969 + πσ4L2

+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)


Proof See Appendix C

Proposition 3 If we representQlowastr klowast as the optimal values of

Qr k then for fixed values of n and L isin [Li Liminus 1] the single-

channel profit function nablaS attains its global maximum at Qlowastr klowast under the condition

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)

Proof See Appendix D

Corollary 1 From equation (3) it is observed that the termsassociated with Cp are negative in Dr equation which clearlyshows that as Cp increases Dr decreases ie

Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)

Table 4 Input parameters

Input parameter Value UnitCp 100 $unitCd1 80 $unitCd2 100 $unitCd3 120 $unitSr 800 $setupSd 1000 $setupCvr 100 $unitCb 120 $unitrv 02 ($unit)unittimerb 02 ($unit)unittimeAr 200 $orderPr 5000 UnitsyearPd 5000 Unitsyearπ 150 $unita1 1000 (mdash)a2 1000 (mdash)β1 075 $unitβ2 080 $unitδ1 02 (mdash)δ2 03 (mdash)ϕd1 025 ϕd2 03 ϕd3 045 h1 30 ($unit)yearL 4 Weeksσ 200 (mdash)m 07 (mdash)


Furthermore from equations (14) and (15) we get

nablaDpropDr

nablaSpropDr(27)

which implies that as Dr decreases because of increase Cp nablaD

andnablaS decrease as fast rate which we can observe fromTable 5

Corollary 2 From equation (4) it is observed that the termsassociated with 1113936

Ni1 Cdi are negative Dd equation which

clearly shows that as 1113936Ni1 Cdi increases Dd decreases ie

ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)

which implies that as 1113936Ni1 Cdi increases Dd decreases

Furthermore from equations (15) we get

nablaDpropDd (29)

which implies that as Dd decreases because of increase CdnablaD increases at slow rate in comparison to otherwise whichwe can observe from Table 6

5 Numerical Experimentation and Discussion

In Table 4 we have enlisted the input parameters used fornumerical experiments

51 Profit Analysis We consider an example to compare theprofit of the dual-channel and single-channel supply chainmanagement In case of the dual-channel supply chain modelthe manufacturer provides the standard product to the cus-tomer through offline channel and customized productthrough online channel whereas in case of the single-channelsupply chain model the manufacturer provides only standardproduct to the customer through offline channel e purposeof this section is to examine and analyze the managerial de-cisions under various circumstances We assume that threetypes of customizations are offered on standard product in caseof the dual-channel supply chain model to the customers

e values of parameters to obtain the numerical resultsare enlisted in Table 4 e threshold value is preassignedwith the value $20 for all cases throughout the paper

Using the values of parameters we obtain the optimalvalues of the decision variables illustrated by Table 7 It isobserved from Table 7 that maximum profits for both single-and dual-channel supply chain are obtained as 78267028and 82250859 respectively e maximum profits are ob-served at n 2 and the corresponding values ofk L Qr Qd Dr andDd are 558 4 105821 16622 490and 456 respectively

Moreover Table 8 shows that if the selling price of thestandard product is more than that of customized productthen shifting of customers from the retailer shop to online

platform is more ie δ1 gt δ2 Furthermore if the sellingprice of the standard product is less than that of customizedproduct then shifting of customers from the retailer shop toonline platform is less ie δ1 lt δ2 Moreover it also depictsthat in either of case profit of dual-channel is more incomparison to single channel

52 Price Sensitivity in Demand When Selling Price DifferencebetweenOnlineandOfflineModeIsGreater1anthePreassigned1reshold Limit To analyze the effect of changing selling priceof standard and customized product on the demand as well asprofit for both dual- and single-channel supply chain two casesare considered for price sensitivity analysis Case 1 representsthe effect of changing the selling price of standard product ondemand and profit (Table 7) whereas Case 2 shows the samefor varying selling price of customized product (Table 8)

Both cases are analyzed when the selling price differenceof retail and online channel is more than the preassignedthreshold limit ie | 1113936

3i1 Cdi minus Cp|gt 20

521 Case 1 In this case the selling prices of customizedproducts are kept as constant and the selling price ofstandard product is varyinge results are shown in Table 5as well as in Figure 5

522 Case 2 e selling price of standard product is kept asconstant and the selling price of customized products isvarying in this case We consider the variation of the sum ofselling prices of all customizations ie 1113936

3i1 Cdi e results

are shown in Table 6 as well as in Figure 6

53 Price Sensitivity in DemandWhen Selling Price DifferencebetweenOnline andOfflineMode Is Less1an the Preassigned1reshold Limit Similar analysis is performed by consid-ering two different cases as described in Section 63However the price difference between online and offlinemode is less than the fixed and specified threshold limit ie| 1113936

3i1 Cdi minus Cp|lt 20Two cases are set out as follows

531 Case 1 e selling prices of customized products areset as constant and the selling price of standard product ischanging e results are shown in Table 9 and in Figure 7

532 Case 2 e selling price of standard product is kept asconstant and the selling price of customized products isvarying in this case (as considered in Case 2 of Section 63)e results are shown in Table 10 as well as in Figure 8

54 Important Discussion and Significance of ResultsTable 5 and Figure 5 illustrate that if the selling price of thecustomized product increases it results in increasing dif-ference between the selling prices of the standard andcustomized product Henceforth more number of cus-tomers shifts from customized product to standard product


Table 5 Managerial decisions for | 1113936Ni1 Cdi minus Cp|gt 20 (Case 1)

Sr no Cd 11139363i1 Cdi Cp Dr Dd Qr Qd nablaD nablaS

1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894

Table 7 Optimum values of decision variables

n L (weeks) k Qr Qd Dr Dd nablaD nablaS

1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222

Table 8 Analysis of profit for customer shifting

Dr Dd Qr Qd k nablaD nablaS

δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407

300 360 420 480 540 600 660 665 667 668Selling price of customized product

Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12

Figure 5 Managerial decisions for | 1113936Ni1 Cdi minus CP|gt 20 (Case 1)


and ultimately dual-channel behaves like single-channelonly Furthermore Table 6 and Figure 6 present that if theselling price of the standard product increases the differencebetween the selling prices of the standard and customized

product also increases As a result firm slowly moves to-wards the loss because it leads to increase in price of cus-tomized product as it requires standard product forcustomization

460 520 580 640 700 740 742Selling price of standard product

Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS


Table 9 Managerial decisions for | 1113936Ni1 Cdi minus Cp|lt 20 (Case 1)

Sr no Cp Cd Cp minus Cd Dr Dd Qr Qd nablaD nablaS

1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207

-20 -15 -10 -5 0 5 10 15 20Difference between selling price of standard

and customized product

Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS

Figure 7 Managerial decisions for | 1113936Ni1 Cdi minus CP|lt 20 (Case 1)


Table 9 and Figure 7 characterize that though the sellingprice of the customized product increases if the differencebetween the selling prices of standard and customizedproduct is less than preassigned threshold value thenshifting of customers is inconspicuous Furthermore Ta-ble 10 and Figure 8 reflect that if the selling price of thestandard product increases but if the difference between theselling price of standard and customized product is less thanpreassigned threshold value then shifting of customers isunspectacular

Figure 9 draws the analysis of profit with markup pricefor both dual-channel (nablaD) and single-channel (nablaS) whenselling price of the standard product is more than cus-tomized product Moreover Figure 10 draws analysis ofprofit with markup price for both dual-channel (nablaD) andsingle-channel (nablaS) when selling price of the standardproduct is less than customized product It is observed thatin both the figures profit increases with the increase inmarkup margin

55 Sensitivity Analysis

551 Sensitivity Analysis of Offline Channel As similar tothe sensitivity analysis of the online channel the cost pa-rameters are differed by minus 10 minus 5 +5 and +10 to sift

the profit of the firm Each parameter is changed one at atime while keeping the other parameters fixed e effects ofthis change of the key parameters are illustrated in Table 11


Sr no Cp Cd Cd minus Cp Dr Dd Qr Qd nablaD nablaS

1 400 380 minus 20 490 4832 10608 17059 822558 78256572 400 385 minus 15 490 4764 10608 16951 8224075 78256573 400 390 minus 10 490 469 10608 16840 8220299 78256574 400 395 minus 5 490 4628 10608 16730 8218237 78256575 400 400 0 490 456 10608 16622 8216060 78256576 400 405 5 490 449 10608 1651 8213766 78256577 400 410 10 490 4424 10608 1639 8211357 78256578 400 415 15 490 4356 10608 1628 8208832 78256579 400 420 20 490 4288 10608 1617 8206191 7825657

-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )

Difference between selling price of standardand customized product

76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS

Figure 9 Analysis of profit with markup price when Cp gt 1113936 Cdi

Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10

Figure 10 Analysis of profit with markup price when Cp lt 1113936 Cdi


and Figure 11 Variations of key parameters Cp Sr Cv andCb are considered e outcomes of this experiment arediscussed as follows

(1) e manufacturerrsquos cost component is more sensi-tive than the retailerrsquos cost component

(2) Selling price of the standard product is the mostsensitive cost whereas that of setup cost of the re-tailer is the least sensitive

552 Sensitivity Analysis of Online Channel e cost pa-rameters are varied over minus 10 minus 5 +5 and +10 toexamine the effect of expected profit of the online channelEach parameter is changed one at a time while keeping theother parameters fixed e effects of this change of thekey parameters are illustrated in Table 11 and Figure 12

Variations of key parameters are Cd1 Cd2 Cd3 (sinceonly three customizationrsquos are considered) and Sd consid-ered For the sake of simplicity only three types of cus-tomizations are considered Scrutinizing the sensitivityanalysis following points are observed

(1) Cost components of the standard product are moresensitive than cost components of customized product

(2) e manufacturerrsquos cost components of standardproduct are more sensitive than the manufacturerrsquossetup cost

(3) Out of the three customizations third customizationcost Cd3 for customized product is the least sensitive

6 Managerial Implications

is article provides a dual-channel supply chainmodel witha single manufacturer and single retailer Managerial deci-sions shall be more realistic if more than one decisionvariables are considered instead of a single variable or anyconstant entity More pragmatic managerial decisions can beobtained if multiple decision variables are assumed usdecisions are made on the basis of many variable quantitiessuch as selling price of standard product selling price ofcustomized product profit of single channel and profit ofdual channel and lead time e managerial implications ofthis study are enlisted as follows

Table 11 Sensitivity analysis for different key parameters

Parameters Change (in ) Sensitivity (offline channel) Sensitivity (online channel)

Cd1

minus 10 mdash +191minus 5 mdash +097+5 mdash minus 101+10 mdash minus 205

Cd2


Cd3


Cp

minus 10 +339 mdashminus 5 +157 mdash+5 minus 133 mdash+10 minus 341 mdash

Sr


Cv


Cb


Sd



Providing the customization facility to the customersinfluences hike in demand which may increase the totalprofitability of the industry In 1985 Dell took up thisstrategy ie make-to-order of computers on demandof customers and this provided Dell a sale of $70million in that year and after five years of adopting thestrategy their revenue climbed to $500 million By theend of 2000 Dellrsquos revenues had topped with an as-tounding of $25 billion [1]e manager can increase the overall profit of the firmif the dual-channel supply chain strategy is appliedinstead of the single-channel supply chain model As aresult of this strategy the Dellrsquos revenue was able toreach to $25 billion within two decades [1]A centralized dual-channel supply chain model shouldbring forth increased profit to the firm due to theexchange of information between the manufacturerand retailere manager can reduce lead time and enhance cus-tomer service by incurring a lead time crashing costAn improved managerial decision can be obtained byassuming a preassigned threshold limit and unequalshifting of customers from one channel to anotherSince the manager is providing customized product inaddition to standard product so price of the customizedfeature should be arranged in such a way that overallcost of the customized product should not increasethreshold limit otherwise customers will shift back tostandard productemanager should not increase the selling price of thestandard product beyond a limit otherwise the profitfirm will slowly doom On account of the fact thatcustomization is implemented on the standard productand if the prices of the standard product increaseautomatically it will lead to an increase in the price ofcustomized product

7 Conclusions

emost important and original contribution of this study isthe modifications incorporated on the existing customizationstrategy with the inclusion of a fixed and specified thresholdlimit and unequal shifting of customers from one channel toanother which leads to an improved realistic scenario of dual-channel supply channel management Moreover due to thespeculative fluctuation of demand estimation becomes dif-ficult task for the decisionmakers Some factors such as sellingprice lead to a variation in demand along with uncertaintyerefore the decision makers should consider the proba-bilistic uncertainty with the sensitivity of various factors suchas selling price in demand expression

e study concluded that implementing the dual-channelsupply chain policy the company receives the personallycustomized demands of the customers along with standardproduct which increases the trust of the customers on firmMoreover if the firm adopts the centralized policy in thatcase it can manage the complete profit parameters that arerelated to product after studying the market which helped inincreasing the overall profit of the firm is paper clearlydepicted that if the selling price of standard product is ex-travagant when compared to customized product then thefirm slowly moves towards losses whereas if the selling priceof customized product is overpriced in comparison tostandard product then the dual-channel supply chain modelbehaves like the single-channel model As a result customersstart preferring standard product over customized productMoreover if the difference between the selling prices of thestandard product and customized product is less than pre-assigned threshold value then there is an indistinct shifting ofcustomers is paper also revealed that the selling price ofstandard product is the most sensitive cost parameter in themodel Accordingly the least sensitive parameters are thesetup costs for the manufacturer for both channels

Cost parameters (in )

097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity

Figure 12 Sensitivity analysis for different key parameters in caseof online channel


039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity

Figure 11 Sensitivity analysis for different key parameters in caseof offline channel


e present study is limited to some aspects such as absenceof multiple retailers strategy to reduce setup and ordering costsimplementation of vendor managed inventory (VMI) or con-signment policy integrating towards a sustainable developmentand many more erefore there is a significant scope of ex-tension of this study Considering a multiretailer dual-channelsupply chain system [51 52] a fair competition among all retailchannels can be shown e centralized supply chain profit canbe improved by reducing several cost components us largecost components such as setup or ordering cost may also bediminished by an efficient investment strategy [50]e study ofVMI and consignment contracts has been supervised by manyresearchers in recent years [37 58] is agreement allowsmanufacturers to take the ownership of its products even aftershifting to the retailerrsquos end A VMI or consignment contractunder dual-channel centralized supply chain with the modifiedcustomization policy can be an innovative research to beconcerned Furthermore considering a humanitarian supplychain and social and environmental sustainability the studies ofNandra et al [59] and Nandra et al [60] are one of the virtuousways of extension of this study

Appendix

A Derivation of the Selling Price ofStandard Product

Markup selling price(for retailer) minus cost price(for retailer)

cost price(for retailer)

m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)

B Proof of Proposition 1

E(M minus R)+

|M minus R| +(M minus R)

2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL

radic+ X summation of variability and

randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL

radicis a safety factor

E(M minus R)+ le

E(DL + X minus DL + kσL

radic)2

1113969

+ E(DL + X minus DL + kσL

radic)

2 (B3)

In this case we consider worst possible distribution ofrandom variable Dr with mean DL and standard deviationσ

L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)


C Proof of Proposition 2

e Hessian matrix H for dual-channel supply chain is asfollows

H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)

e first-order principal minor of |H1| is

H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868Qlowastr Qlowast

dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)


e second-order principal minor of |H1| is

H1( 1113857221113868111386811138681113868

1113868111386811138681113868 Qlowastr Qlowastdklowast( )

z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)

e third-order principal minor of |(H1)33|(Qlowastr Qlowastdklowast) is

H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 minusDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)


We see that all the principal minors of the Hessianmatrix are negative Hence the Hessian matrix H1 is neg-ative definite at (Qlowastr Qlowastd klowast) erefore the total expecteddual-channel profit function attains its global maximum at(Qlowastr Qlowastd klowast)

D Proof of Corollary 1

e Hessian matrix H for single-channel supply chain is asfollows

H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868

1113868111386811138681113868 Qlowastr klowast

( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868Qlowastr klowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868

1113868111386811138681113868 Qlowastr klowast( ) z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572

+ k2σ2L minus 2DrLσL


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)

We see that all the principal minors of the Hessianmatrix are negative Hence the Hessian matrix H1 is neg-ative definite at (Qlowastr klowast) erefore the total expectedsingle-channel profit function attains its global maximum at(Qlowastr klowast)

Data Availability

e data used to support the findings of this paper are freelyaccessible through the Internet and cited articles

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] MaRS ldquoCase study dellmdashdistribution and supply chaininnovationrdquo MaRS Startup Toolkit Canadarsquos InnovationCommunity Ottawa Canada 2021 httpslearnmarsddcomarticlecase-study-dell-distribution-and-supply-chain-innovation

[2] A Tsay and N Agrawal ldquoChannel conflict and coordinationin the e-commerce agerdquo Production and Operations Man-agement vol 13 no 1 pp 93ndash110 2004

[3] T Zhang and T M Choi ldquoOptimal consumer sales taxpolicies for online-offline retail operations with consumerreturnsrdquo Naval Research Logistics vol 68 no 6 pp 701ndash7202020

[4] C Wilder HPrsquos Online Push Information Week San Fran-cisco CA USA 1999

[5] A Barman R Das and P Kanti De ldquoOptimal pricing andgreening decision in a manufacturer retailer dual-channelsupply chainrdquo Materials Today Proceedings vol 42 no 2pp 870ndash875 2021

[6] K Takahashi T Aoi D Hirotani and K Morikawa ldquoIn-ventory control in a twoechelon dual-channel supply chainwith setup of production and deliveryrdquo International Journalof Production Economics vol 133 no 2 pp 403ndash415 2011

[7] S-M Hosseini-Motlagh M Nouri-Harzvili T-M Choi andS Ebrahimi ldquoReverse supply chain systems optimization withdual channel and demand disruptions sustainability CSRinvestment and pricing coordinationrdquo Information Sciencesvol 503 pp 606ndash634 2019

[8] P He Y He and H Xu ldquoBuy-online-and-deliver-from-storestrategy for a dual-channel supply chain considering retailerrsquoslocation advantagerdquo Transportation Research Part E vol 144Article ID 102127 2020

[9] D Littler and D Melanthiou ldquoConsumer perceptions of riskand uncertainty and the implications for behaviour towardsinnovative retail services the case of internet bankingrdquoJournal of Retailing and Consumer Services vol 13 no 6pp 431ndash443 2006

[10] M Bhattacharyya and S S Sana ldquoA mathematical model oneco-friendly manufacturing system under probabilistic de-mandrdquo RAIRO-Operations Research vol 53 no 5pp 1899ndash1913 2019

[11] N M Modak N Modak S Panda and S S Sana ldquoAnalyzingstructure of two-echelon closed-loop supply chain for pricingquality and recycling managementrdquo Journal of Cleaner Pro-duction vol 171 pp 512ndash528 2018

[12] R Yan and Z Pei ldquoRetail services and firm profit in a dual-channel marketrdquo Journal of Retailing and Consumer Servicesvol 16 no 4 pp 306ndash314 2009


[13] J Zhou R Zhao and B Wang ldquoBehavior-based price dis-crimination in a dual-channel supply chain with retailerrsquosinformation disclosurerdquo Electronic Commerce Research andApplications vol 39 Article ID 100916 2019

[14] C Jing Z Hui and S Ying ldquoImplementing coordinationcontracts in a manufacturer stackelberg dual-channel supplychainrdquo Omega vol 40 no 1 pp 571ndash583 2012

[15] S Huang C Yang and H Liu ldquoPricing and productiondecisions in a dual-channel supply chain when productioncosts are disruptedrdquo EconomicModelling vol 30 pp 521ndash5382013

[16] S Huang C Yang and X Zhang ldquoPricing and productiondecisions in dual-channel supply chains with demand dis-ruptionsrdquo Computers amp Industrial Engineering vol 62 no 1pp 70ndash80 2012

[17] B Dan G Xu and C Liu ldquoPricing policies in a dual-channelsupply chain with retail servicesrdquo International Journal ofProduction Economics vol 139 no 1 pp 312ndash320 2012

[18] S S Sana ldquoA structural mathematical model on two echelonsupply chain syatemrdquo Annals of operations Research 2021

[19] P Zhang Y He and C Shi ldquoRetailerrsquos channel structurechoice online channel offline channel or dual channelsrdquoInternational Journal of Production Economics vol 191pp 37ndash50 2017

[20] S K Mukhopadhyay X Zhu and X Yue ldquoOptimal contractdesign for mixed channels under information asymmetryrdquoProduction and Operations Management vol 17 no 6pp 641ndash650 2008

[21] W-y K Chiang D Chhajed and J D Hess ldquoDirect mar-keting indirect profits a strategic analysis of dual-channelsupply-chain designrdquo Management Science vol 49 no 1pp 1ndash20 2003

[22] S S Sana ldquoA structural mathematical model on two echelonsupply chain systemrdquo Annals of Operations Research 2021

[23] S-M Hosseini-Motlagh M Johari S Ebrahimi andP Rogetzer ldquoCompetitive channels coordination in a closed-loop supply chain based on energy-saving effort and cost-tariffcontractrdquo Computers amp Industrial Engineering vol 490Article ID 106763 2020

[24] M G Dekimpe K Gielens J Raju and J S omasldquoStrategic assortment decisions in information-intensive andturbulent environmentsrdquo Journal of Retailing vol 87 no 1pp S17ndashS28 2011

[25] S S Sana ldquoPrice competition between green and non greenproducts under corporate social responsible firmrdquo Journal ofRetailing and Consumer Services vol 55 Article ID 1021182020

[26] K Rana S R Singh S R Singh N Saxena and S S SanaldquoGrowing items inventory model for carbon emission underthe permissible delay in payment with partially backloggingrdquoGreen Finance vol 3 no 2 pp 153ndash174 2021

[27] F Salvador P M De Holan and F T Piller ldquoCracking thecode of mass customizationrdquoMIT SloanManagement Reviewvol 50 pp 71ndash78 2009

[28] M Comstock K Johansen and M Winroth ldquoFrom massproduction to mass customization enabling perspectivesfrom the Swedish mobile telephone industryrdquo ProductionPlanning amp Control vol 15 no 4 pp 362ndash372 2004

[29] B G C Dellaert and S Stremersch ldquoMarketing mass-cus-tomized products striking a balance between utility andcomplexityrdquo Journal of Marketing Research vol 42 no 2pp 219ndash227 2005

[30] H Ismail I Reid J Mooney J Poolton and I ArokiamldquoHow small and medium enterprises effectively participate in

the mass customization gamerdquo IEEE Transactions on Engi-neering Management vol 54 no 1 pp 86ndash97 2007

[31] M J Rungtusanatham and F Salvador ldquoFrom mass pro-duction to mass customization hindrance factors structuralinertia and transition hazardrdquo Production and OperationsManagement vol 17 no 3 pp 385ndash396 2008

[32] KMatsui ldquoOptimal bargaining timing of a wholesale price fora manufacturer with a retailer in a dual-channel supplychainrdquo European Journal of Operational Research vol 287no 1 pp 225ndash236 2020

[33] P Zhang Y He and X Zhao ldquoPreorder-online pickup-in-store strategy for a dual-channel retailerrdquo TransportationResearch Part E Logistics and Transportation Review vol 122pp 27ndash47 2019

[34] K Y Chen M Kaya and O Ozer ldquoDual sales channelmanagement with service competitionrdquo Manufacturing ampService Operations Management vol 10 no 4 pp 676ndash6912008

[35] G Hua S Wang and T C E Cheng ldquoPrice and lead timedecisions in dual-channel supply chainsrdquo European Journal ofOperational Research vol 205 no 1 pp 113ndash126 2010

[36] G Li F Huang T C E Cheng and P Ji ldquoCompetitionbetween manufacturerrsquos online customization channel andconventional retailerrdquo IEEE Transactions on EngineeringManagement vol 62 no 2 pp 150ndash157 2015

[37] R Batarfi M Y Jaber and S Zanoni ldquoDual-channel supplychain a strategy to maximize profitrdquo Applied MathematicalModelling vol 40 no 21-22 pp 9454ndash9473 2016

[38] Y He H Huang and D Li ldquoInventory and pricing decisionsfor a dual-channel supply chain with deteriorating productsrdquoOperational Research vol 20 no 3 pp 1461ndash1503 2018

[39] W-Y Kevin Chiang and G E Monahan ldquoManaging in-ventories in a two-echelon dual-channel supply chainrdquo Eu-ropean Journal of Operational Research vol 162 no 2pp 325ndash341 2005

[40] D-Q Yao X Yue X Wang and J J Liu ldquoe impact ofinformation sharing on a returns policy with the addition of adirect channelrdquo International Journal of Production Eco-nomics vol 97 no 2 pp 196ndash209 2005

[41] N M Modak and P Kelle ldquoManaging a dual-channel supplychain under price and delivery-time dependent stochasticdemandrdquo European Journal of Operational Research vol 272no 1 pp 147ndash161 2018

[42] A Dumrongsiri M Fan A Jain and K Moinzadeh ldquoAsupply chain model with direct and retail channelsrdquo EuropeanJournal of Operational Research vol 187 no 3 pp 691ndash7182008

[43] P He Y He and H Xu ldquoChannel structure and pricing in adual-channel closed-loop supply chain with governmentsubsidyrdquo International Journal of Production Economicsvol 213 pp 108ndash123 2018

[44] A Roy S S Sana and K Chaudhuri ldquoJoint decision on EOQand pricing strategy of a dual channel of mixed retail and e-tailcomprising of single manufacturer and retailer under sto-chastic demandrdquo Computers amp Industrial Engineeringvol 102 pp 423ndash434 2016

[45] I Moon D K Yoo and S Saha ldquoe distribution-freenewsboy problem with multiple discounts and upgradesrdquoMathematical Problems in Engineering vol 2016 Article ID2017253 11 pages 2016

[46] H Scarf ldquoA minndashmax solution of an inventory problemrdquo inStudies in 1e Mathematical 1eory of Inventory and Pro-duction K Arrow S Karlin and H Scarf Eds pp 201ndash209Stanford University Press Palo Alto CA USA 1958


[47] H Xu Z Z Liu and S H Zhang ldquoA strategic analysis of dual-channel supply chain design with price and delivery lead timeconsiderationsrdquo International Journal of Production Eco-nomics vol 139 no 2 pp 654ndash663 2012

[48] J Q Yang X M Zhang H Y Fu and C Liu ldquoInventorycompetition in a dual-channel supply chain with delivery leadtime considerationrdquo Applied Mathematical Modelling vol 42pp 675ndash692 2017

[49] X Yue and J Liu ldquoDemand forecast sharing in a dual-channelsupply chainrdquo European Journal of Operational Researchvol 174 no 1 pp 646ndash667 2006

[50] A Majumder R Guchhait and B Sarkar ldquoManufacturingquality improvement and setup cost reduction in a vendor-buyer supply chain modelrdquo European Journal of IndustrialEngineering vol 11 no 5 pp 588ndash612 2017

[51] A Majumder C K Jaggi and B Sarkar ldquoA multi-retailersupply chain model with backorder and variable productioncostrdquo RAIRO-Operations Research vol 52 no 3 pp 943ndash9542018

[52] B Sarkar C Zhang A Majumder M Sarkar and Y W SeoldquoA distribution free newsvendor model with consignmentpolicy and retailerrsquos royalty reductionrdquo International Journalof Production Research vol 56 no 15 pp 1ndash20 2018

[53] P Zhang Y Xiong and Z Xiong ldquoCoordination of a dual-channel supply chain after demand or production cost dis-ruptionsrdquo International Journal of Production Researchvol 53 no 10 pp 3141ndash3160 2015

[54] J S Raju and A Roy ldquoMarket information and firm per-formancerdquoManagement Science vol 46 no 8 pp 1075ndash10842000

[55] W Huang and J M Swaminathan ldquoIntroduction of a secondchannel implications for pricing and profitsrdquo EuropeanJournal of Operational Research vol 194 no 1 pp 258ndash2792009

[56] G Gallego and I Moon ldquoe distribution free newsboyproblem review and extensionsrdquo Journal of the OperationalResearch Society vol 44 no 8 pp 825ndash834 1993

[57] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a review with extensionsrdquo Operations Researchvol 47 no 2 pp 183ndash194 1999

[58] B Sarkar A Majumder M Sarkar N Kim and M UllahldquoEffects of variable production rate on quality of products in asingle-vendor multi-buyer supply chain managementrdquo 1eInternational Journal of Advanced Manufacturing Technologyvol 99 no 1-4 pp 567ndash581 2018

[59] R Nandra A Majumder and M Mishra ldquoA multi-retailersustainable supply chain model with coordination and qualitydeteriorationrdquo RAIRO-Operational Research vol 55pp S2773ndashS2794 2020

[60] K R Marak R Nandra B K Dey A Majumder and R KaurldquoEstablishing relation between production rate and productquality in a single-vendor multi-buyer supply chain modelrdquoInternational Journal of Services Operations and Informaticsvol 11 no 23 p 1 2021


opponents [9] and implementing green technology in themanufacturing industry for reduction of greenhouse gasemission [10 11] e retailerrsquos services and behavior greatlyinfluence the choices and demand of the customer [12 13]Many customers prefer online platform as it saves someamount of cost and time due to the direct interaction be-tween the manufacturer and the customer under onlineenvironment [14] Moreover in the online channel thecustomers are also provided with the feedback facility whichultimately helps the manufacturer to know their customerand improve their services accordingly [14]

Some well-known manufacturing and retailing indus-tries such as Nike started producing the customized prod-ucts on demand of customers through online channel etraditional retail channel is also utilized to deliver thestandard products without any customization [15 16] Astudy reveals that around 42 of the top brands are cateringthe customers directly through online channel [17] Priorlywith the acceptance of the online channel it would createconflict between the manufacturer and the retailer as itwould shrink the retailerrsquos market share However studieshave shown that whole sale prices reduce for product soldthrough the retailer and increase the profit of the firm withthe establishment of online channel [18 19] With the in-troduction of online channel along with offline channelhouse needs to define markup margin accurately as it in-fluences the wholesale prices

is paper put up to the studies in several aspects whichwere unnoticed in the previous literature e main con-tributions of this research lie under the consideration ofthreshold limit unequal shifting of customers multideliverypolicy uncertain demand and controllable lead time in thedual-channel supply chain model with customization policyHeretofore researchers developed several dual-channelmodels but with equal number of customer shifting from theonline to offline channel and vice versae concept of equalcustomer shifting between the channels may not be realisticas all customers who are unwilling to purchase items fromone channel are not forced to shift to another one Secondlya preassigned threshold limit is considered in this articlewhich was not assumed in previous research studies ethreshold limit is incorporated under the concept that acustomer is only willing to change the mode of purchase ifthe selling price difference between the two channels crossesto a specified price limit ese two important modificationson dual-channel supply chain management (SCM) are in-troduced in this study Moreover this study developed adual-channel supply chain model under customization withSSMD policy along with make-to-order uncertain demandcontrollable lead time and distribution-free approach Allthese attributes in a single model are rarely found in theexisting literature

e objective of this paper is to develop an efficient andimproved customization policy in dual-channel SCM eaim of this study is to obtain the managerial decision underdemand uncertainty and price sensitivity with distribution-free approach e competitions between online vs retailchannel dual-channel vs retail channel and dual-channelvs online channel are shown in this article to determine the

best possible channel to be adopted by the company An-other vital objective of this research is to analyze the effect ofmigration of customers between the channels on supplychain profitability under a fixed and specified thresholdvalue of selling price which was not yet studied by anyexisting literature e effect of the variation of selling priceon customer demand and supply chain profit is also animportant matter of concern

e proposed study retains some important researchquestions which are illustrated as follows

(a) What is the effect of the demand uncertainty on theprofit of the firm

(b) How to determine that which channel should beadopted so that profit of the firm maximizes

(c) How profitability of the firm is affected if sellingprices exceed the fixed and specified threshold limit

(d) How does the transfer rate of consumers from onlinechannel to retail channel or vice versa affect optimaldecision

e entire paper is split into several sections to deal withdifferent aspects such as description of study literaturereview and mathematical modelling Sections 1 and 2 de-scribe introduction and literature review respectivelySection 3 explains the important research questions and aimof study Section 4 depicts problem definition assumptionsand notation to develop the mathematical modelling Sec-tion 5 elaborates the entire mathematical model and ana-lytical expressions of decision variables along with somepropositions and corollaries Solution technique (algorithm)of the model is also described in this section Section 6 dealswith numerical experiments and model validation emanagerial decisions are obtained by using a set of inputparameters and results are analyzed with graphical repre-sentations and sensitivity analysis e important discus-sions about the outcomes are also described in this sectionLastly in Section 7 conclusions regarding the study aredepicted

2 Literature Review

21 Dual-Channel Supply Chain In recent years study ondual-channel supply chain has gained reasonable attentionamong the researchers throughout the globe However mostof the studies focused on pricing decisions and channelcoordination In this technologically advanced era everyplayer involved in the supply chain (customers manufac-turers and retailers) expects the availability of the infor-mation of product Consequently Tsay and Agrawal [2]Wilder [4] and Mukhopadhyay et al [20] stated that theseexpectations can be fulfilled if companies have smoothsupply chain flow Moreover Takahashi et al [6] expressedthat through the dual-channel supply chain customers areprovided with two options ordering of the standard productthrough retail channel and the customized product throughonline channel

From health care providers (Medlife Indiarsquos largeste-health platform which provides medicine through online








Stan

dard

izat

ion Custom

ization

Manufacturer

CustomerCustomer

Retailer

Centralized System

Del

iver

yD

eliv

ery

Delivery

Order

Order

Ord

er




























N

i1pdi minus pr

⎛⎝ ⎞⎠ (1)



N


N

i1pdi minus pr

⎛⎝ ⎞⎠ (2)











i1Cdi minus Cp

⎛⎝ ⎞⎠ (3)





Dd a2 minus β2(1 + m) 1113944N

i1Cdi

⎛⎝ ⎞⎠ minus δ2(1 + m) 1113944N

i1Cdi minus Cp

⎛⎝ ⎞⎠ (4)





nabla1 Cp(1 + m)Dr

minusSrDr

nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(5)




Pdi



L

radic








2r2Pr1113872 1113873 minus Q


nQr

Qr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889

(6)


rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889 (7)





nabla2 1113944N

i1Cdi(1 + m)ϕdiDd

minusSdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦

(8)



Ni1 Cdi




21 minus

Dd

Pd

1113888 1113889 (9)


h1Qd

21 minus

Dd

Pd

1113888 1113889 (10)







ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(11)









nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr


ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(12)




radicis

considered


Qua

ntity

G

A B

C

F E

O Time

nQr

nQr

Pr

QrPr

(n ndash 1)QrDr


D

A

C

B

Man

ufac

ture

r(O

nlin

e cha

nnel

)

TimeTd

P d

P d ndash D d

1 minusQdDdPd

Dd






Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)


L




factor





Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time


of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr


Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L




Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References







































































Stan

dard

izat

ion Custom

ization

Manufacturer

CustomerCustomer

Retailer

Centralized System

Del

iver

yD

eliv

ery

Delivery

Order

Order

Ord

er




























N

i1pdi minus pr

⎛⎝ ⎞⎠ (1)



N


N

i1pdi minus pr

⎛⎝ ⎞⎠ (2)











i1Cdi minus Cp

⎛⎝ ⎞⎠ (3)





Dd a2 minus β2(1 + m) 1113944N

i1Cdi

⎛⎝ ⎞⎠ minus δ2(1 + m) 1113944N

i1Cdi minus Cp

⎛⎝ ⎞⎠ (4)





nabla1 Cp(1 + m)Dr

minusSrDr

nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(5)




Pdi



L

radic








2r2Pr1113872 1113873 minus Q


nQr

Qr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889

(6)


rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889 (7)





nabla2 1113944N

i1Cdi(1 + m)ϕdiDd

minusSdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦

(8)



Ni1 Cdi




21 minus

Dd

Pd

1113888 1113889 (9)


h1Qd

21 minus

Dd

Pd

1113888 1113889 (10)







ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(11)









nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr


ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(12)




radicis

considered


Qua

ntity

G

A B

C

F E

O Time

nQr

nQr

Pr

QrPr

(n ndash 1)QrDr


D

A

C

B

Man

ufac

ture

r(O

nlin

e cha

nnel

)

TimeTd

P d

P d ndash D d

1 minusQdDdPd

Dd






Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)


L




factor





Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time


of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr


Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L




Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References


























































































N

i1pdi minus pr

⎛⎝ ⎞⎠ (1)



N


N

i1pdi minus pr

⎛⎝ ⎞⎠ (2)











i1Cdi minus Cp

⎛⎝ ⎞⎠ (3)





Dd a2 minus β2(1 + m) 1113944N

i1Cdi

⎛⎝ ⎞⎠ minus δ2(1 + m) 1113944N

i1Cdi minus Cp

⎛⎝ ⎞⎠ (4)





nabla1 Cp(1 + m)Dr

minusSrDr

nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(5)




Pdi



L

radic








2r2Pr1113872 1113873 minus Q


nQr

Qr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889

(6)


rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889 (7)





nabla2 1113944N

i1Cdi(1 + m)ϕdiDd

minusSdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦

(8)



Ni1 Cdi




21 minus

Dd

Pd

1113888 1113889 (9)


h1Qd

21 minus

Dd

Pd

1113888 1113889 (10)







ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(11)









nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr


ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(12)




radicis

considered


Qua

ntity

G

A B

C

F E

O Time

nQr

nQr

Pr

QrPr

(n ndash 1)QrDr


D

A

C

B

Man

ufac

ture

r(O

nlin

e cha

nnel

)

TimeTd

P d

P d ndash D d

1 minusQdDdPd

Dd






Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)


L




factor





Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time


of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr


Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L




Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References







































































N

i1pdi minus pr

⎛⎝ ⎞⎠ (1)



N


N

i1pdi minus pr

⎛⎝ ⎞⎠ (2)











i1Cdi minus Cp

⎛⎝ ⎞⎠ (3)





Dd a2 minus β2(1 + m) 1113944N

i1Cdi

⎛⎝ ⎞⎠ minus δ2(1 + m) 1113944N

i1Cdi minus Cp

⎛⎝ ⎞⎠ (4)





nabla1 Cp(1 + m)Dr

minusSrDr

nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(5)




Pdi



L

radic








2r2Pr1113872 1113873 minus Q


nQr

Qr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889

(6)


rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889 (7)





nabla2 1113944N

i1Cdi(1 + m)ϕdiDd

minusSdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦

(8)



Ni1 Cdi




21 minus

Dd

Pd

1113888 1113889 (9)


h1Qd

21 minus

Dd

Pd

1113888 1113889 (10)







ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(11)









nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr


ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(12)




radicis

considered


Qua

ntity

G

A B

C

F E

O Time

nQr

nQr

Pr

QrPr

(n ndash 1)QrDr


D

A

C

B

Man

ufac

ture

r(O

nlin

e cha

nnel

)

TimeTd

P d

P d ndash D d

1 minusQdDdPd

Dd






Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)


L




factor





Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time


of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr


Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L




Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References

































































Dd a2 minus β2(1 + m) 1113944N

i1Cdi

⎛⎝ ⎞⎠ minus δ2(1 + m) 1113944N

i1Cdi minus Cp

⎛⎝ ⎞⎠ (4)





nabla1 Cp(1 + m)Dr

minusSrDr

nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(5)




Pdi



L

radic








2r2Pr1113872 1113873 minus Q


nQr

Qr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889

(6)


rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889 (7)





nabla2 1113944N

i1Cdi(1 + m)ϕdiDd

minusSdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦

(8)



Ni1 Cdi




21 minus

Dd

Pd

1113888 1113889 (9)


h1Qd

21 minus

Dd

Pd

1113888 1113889 (10)







ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(11)









nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr


ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(12)




radicis

considered


Qua

ntity

G

A B

C

F E

O Time

nQr

nQr

Pr

QrPr

(n ndash 1)QrDr


D

A

C

B

Man

ufac

ture

r(O

nlin

e cha

nnel

)

TimeTd

P d

P d ndash D d

1 minusQdDdPd

Dd






Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)


L




factor





Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time


of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr


Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L




Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References




































































2r2Pr1113872 1113873 minus Q


nQr

Qr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889

(6)


rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113888 1113889 (7)





nabla2 1113944N

i1Cdi(1 + m)ϕdiDd

minusSdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦

(8)



Ni1 Cdi




21 minus

Dd

Pd

1113888 1113889 (9)


h1Qd

21 minus

Dd

Pd

1113888 1113889 (10)







ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(11)









nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr


ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(12)




radicis

considered


Qua

ntity

G

A B

C

F E

O Time

nQr

nQr

Pr

QrPr

(n ndash 1)QrDr


D

A

C

B

Man

ufac

ture

r(O

nlin

e cha

nnel

)

TimeTd

P d

P d ndash D d

1 minusQdDdPd

Dd






Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)


L




factor





Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time


of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr


Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L




Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References








































































nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr


ArDr

Qr

+ rbcb

Qr

2+ R minus DrL1113874 1113875 +

πDr

Qr

E(M minus R)+

+DrCL

Qr

1113890 1113891

(12)




radicis

considered


Qua

ntity

G

A B

C

F E

O Time

nQr

nQr

Pr

QrPr

(n ndash 1)QrDr


D

A

C

B

Man

ufac

ture

r(O

nlin

e cha

nnel

)

TimeTd

P d

P d ndash D d

1 minusQdDdPd

Dd






Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)


L




factor





Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time


of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr


Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L




Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References




































































Proposition 1

E(M minus R)+

E DrL + X( 1113857 minus R( 1113857+

le

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(13)


L




factor





Stoc

k of

man

ufac

ture

rSt

ock

of re

taile

r

r

s

Time

Time


of retailer

Qr

Qr

Qr

Qr Qr Qr Qr

nQrDr

QrPr QrDrQrDr QrDr


Qr

Qr

Qr Qr

Qr

Qr

Qr

Qr

L




Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References


































































Qr

minus rbcb

Qr

2+ kσ

L

radic1113874 1113875 minus

DrCL

Qr

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877


SrDr

Qrn+

rvQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

(14)



nQr

+rvCpQr

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 11138911113890 1113891 minus CvrDr

+ 1113944N


SdDd

Qd

+h1Qd

21113888 1113889 1 minus

Dd

Pd

1113888 1113889 + 1113944N

i1CdiϕdiDd

⎡⎣ ⎤⎦


ArDr

Qr

+ rbcb

Qr

2+ kσ

L

radic1113874 1113875 +

DrCL

Qr

1113890 1113891

minusπDr

Qr2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

(15)



znablaD

zk minus rbCbσ

L

radicminusπDr

2Qr

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

znablaD

zQr

ArDr

Q2r

minusrbcb

2+

SrDr

nQ2r

+DrCL

Q2r

minusrvCp

2n 1 minus

Dr

Pr

1113888 1113889 minus 1 +2Dr

Pr

1113890 1113891

+πDr

2Q2r

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 11138731113876 1113877

znablaD

zQd

SdDd

Q2d

minush1

21 minus

Dd

Pd

1113888 1113889

(16)



znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References

































































znablaD

zk 0 (17)

znablaD

zQr

0 (18)

znablaD

zQd

0 (19)

and we get

Qlowastd

SdDd

h12( 1113857 1 minus DdPd( 1113857( 1113857

1113971

(20)

Qlowastr


σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic


11139741113972

(21)

klowast

minus σ


1113969



1113969


radicσ

(22)








Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L


+πkσ3L

L

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(23)






Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References





































































Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(24)



Drprop1

Cp

(25)

⟹Cpprop1

Dr

(26)





nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References


































































nablaDpropDr

nablaSpropDr(27)






ie Ddprop1

1113936Ni1 Cdi

⟹1113944N

i1Cdiprop

1Dd

(28)



nablaDpropDd (29)














3i1 Cdi e results










1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References



































































1 400 300 4390 626 10120 19110 7395981 69980882 400 360 4696 524 10416 17680 7896011 74945563 400 420 5002 422 10702 16050 8368281 79912414 400 480 5308 320 10978 14130 8812994 84881195 400 540 5614 218 11246 11790 9230553 89851746 400 600 5920 116 11506 8690 9621954 94823877 400 660 6226 14 11759 3051 9992545 99797458 400 665 6252 55 11779 1914 1002417 10021209 400 667 6262 21 11788 1183 1003757 100377810 400 668 6267 04 11792 520 1004516 1004607



1 300 400 4560 643 102859 1933 7683447 72738752 360 400 4764 5308 104810 1778 8011956 76049123 420 400 4968 4186 106710 1600 8309837 79360444 480 400 5172 3064 108570 1385 8577384 82672625 540 400 5376 1942 110390 1116 8815199 85985616 600 400 558 820 112170 733 9025032 89299377 610 400 5614 633 112463 645 9058194 89851748 620 400 5648 446 112756 543 9090156 90404129 630 400 5682 259 113047 414 9122017 909565210 640 400 5716 72 113338 219 9155339 9150894



1 4 559 106079 16622 490 456 82240397 782565652 4 558 105821 16622 490 456 82250859 782670283 4 559 105566 16622 490 456 82249054 78265222



δ1 gt δ2 4968 4186 106709 15991 566 83069065 79360435δ1 lt δ2 4543 4628 102695 16732 521 80470127 76633407


Profi

t (times1

05 )

nablaD

nablaS

0

2

4

6

8

10

12






Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References




































































Profi

t (times1

05 )0

2

4

6

8

10

12

nablaD

nablaS




1 380 400 -20 5155 456 108415 16622 8164811 77664282 385 400 -15 50913 456 107837 16622 8183995 77856123 390 400 -10 50275 456 107260 16622 8200261 78018784 395 400 -5 49638 456 106670 16622 8213609 78152265 400 400 0 49000 456 106080 16622 8214464 78256576 405 400 5 48363 456 105490 16622 8231553 7833177 410 400 10 47725 456 10489 16622 8236149 78377668 415 400 15 47088 456 10428 16622 8236589 78394449 420 400 20 46450 456 10368 16622 8237827 7838207



Profi

t (times

105 )

76

77

78

79

8

81

82

83

nablaD

nablaS











-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References









































































-20 -15 -10 -5 0 5 10 15 20

Profi

t (times1

05 )


76

77

78

79

8

81

82

83

nablaD

nablaS


0

1

2

3

4

5

6

7

8

9

Profi

t (times1

05 )

01 02 03 04 05 06 07 08 09Markup margin

nablaD

nablaS


Prof

it (times

105 )

Markup margin01 02 03 04 05 06 07 08 09

nablaDnablaS

0

1

2

3

4

5

6

7

8

9

10















Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References













































































Cd1


Cd2


Cd3


Cp


Sr


Cv


Cb


Sd




7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References


































































7 Conclusions




097

-101

-205

191

099

-106

-219

026 022

-041

-102

065032

0-031

-061-10 -5 0 5 10

Cd1Cd2

Cd3Sd

-25

-2

-15

-1

-05

0

05

1

15

2

25

Prof

it se

nsiti

vity



039 057

-133

-341

002 001 -001 -002

063031

-031-063

077039

0-038

-077

-10 -5 0 5 10

Cp

Sr

Cv

Cb

-4

-35

-3

-25

-2

-15

-1

-05

0

05

1

Prof

it se

nsiti

vity




Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References


































































Appendix




m P minus Cr

Cr

mCr + Cr P Cr(1 + m) Cr(1 + m)2

P Cp(1 + m)2

(A1)


E(M minus R)+


2

E(M minus R)+ le

E(M minus R)2

1113969

+ E(M minus R)

2

(B1)

Consider M DL


randomness

E(M minus R)+ le

E(DL

radic+ E(X minus R)

21113969

+ E(DL

radic+ X minus R)

2

(B2)

where R DL + kσL


E(M minus R)+ le


radic)2

1113969


radic)

2 (B3)


L

radic We get

E(M minus R)+

E(X + kσL

radic)2

1113969

+ E((X + kσL

radic))

2

E X

2+ k

2σ2L + 2XkσL

radic1113872 1113873

1113969+ E(X + kσ

L

radic)

2

σ2L + DrL( 11138572


L

radic1113969

+ DrL minus kσL

radic1113872 1113873

2⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(B4)




H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References



































































H1

z2nablaD

zQ2r

z2nablaD

zQrzQd

z2nablaD

zQrzk

z2nablaD

zQdzQr

z2nablaD

zQ2d

z2nablaD

zQdzk

z2nablaD

zk zQr

z2nablaD

zk zQd

z2nablaD

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(C1)

where

z2nablaD

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaD

zQdzk 0

z2nablaD

zk zQd

0

z2nablaD

zQrzQd

0

z2nablaD

zQ2d

minus2SdDd

Q3d

z2nablaD

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaD

zQdzQr

0

z2nablaD

zk zQr

πDr

2Q2r

Kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(C2)


H111113868111386811138681113868

1113868111386811138681113868 z2nablaD

zQ2r

11138681113868111386811138681113868111386811138681113868


dklowast( )

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(C3)



H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References


































































H1( 1113857221113868111386811138681113868


z2nablaD

zQ2r

1113888 1113889z2nablaD

zQ2d

1113888 1113889

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889 minus2SdDd

Q3d

1113888 1113889gt 0

(C4)


H1( 1113857331113868111386811138681113868


z2nablaD

zQ2d

z2nablaS

zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2⎡⎣ ⎤⎦

2SdDd

Q3d

1113890 1113891πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(C5)





H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References




































































H2

z2nablaD

zQ2r

z2nablaS

zQrzk

z2nablaS

zk zQr

z2nablaS

zk2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(D1)

where

z2nablaS

zQ2r

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891

z2nablaS

zk zQr

πDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zk2 minus

πDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z2nablaS

zQrzkπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572


L


radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

(D2)


H111113868111386811138681113868


( 1113857 z2nablaS

zQ2r

11138681113868111386811138681113868111386811138681113868


minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 1113891lt 0

(D3)


H221113868111386811138681113868


zQ2r

1113888 1113889z2nablaS

zk21113888 1113889 minus

z2nablaS

zk zQr

1113888 1113889

2

minus2

Q3r

ArDr +SrDr

n+ DrCL + πDr

σ2L + DrL( 11138572


L

radic1113969

1113874 11138751113890 11138911113888 1113889

middot minusπDr

2Qr

σ4L2

σ2L + DrL( 11138572


L

radic1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

minusπDr

2Q2r

kσ2L minus DrLσL

radic

σ2L + DrL( 11138572



radic⎡⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

2


πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References

































































πD

2r

Q4r

Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L


2σ2

2

σ2L + DrL( 11138572


L

radic1113969

minusπ 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

gt 0

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+ πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969

+π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ2 minus 4Drσ3L2

Lradic

1113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

⟹ Ar +Sr

n+ CL1113874 1113875

σ4L2

σ2L + DrL( 11138572


L

radic1113969

+πσ4L2+

πkσ3LL

radic

2

σ2L + DrL( 11138572


L

radic1113969 +Drσ

3L2

Lradic

σ2L + DrL( 11138572


L

radic

gtDrL

2σ2

2

σ2L + DrL( 11138572


L

radic1113969 +π 2k

2σ4L2+ σ4L2

+ 2D2rL

3σ21113872 1113873

4 σ2L + DrL( 11138572


L

radic1113872 1113873

(D4)


Data Availability




References



































































































































Documents

A Modified Customization Strategy in a Dual-Channel Supply