decision and coordination of low-carbon e-commerce...

Research ArticleDecision and Coordination of Low-Carbon E-Commerce SupplyChain with Government Carbon Subsidies and Fairness Concerns

Qiang Han 1 Yuyan Wang 1 Liang Shen2 and Wenquan Dong 34

1School of Management Science and Engineering Shandong University of Finance and Economics JinanShandong 250014 China2School of Public Finance and Taxation Shandong University of Finance and Economics Jinan Shandong 250014 China3Institute for a Secure and Sustainable Environment (e University of Tennessee Knoxville Tenn 37996 USA4Department of Industrial and Systems Engineering (e University of Tennessee Knoxville Tenn 37996 USA

Correspondence should be addressed to Yuyan Wang wangyuyan1224126com

Received 31 May 2020 Revised 16 July 2020 Accepted 30 July 2020 Published 24 August 2020

Academic Editor Juan Carlos Cortes

Copyright copy 2020 Qiang Han et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

More low-carbon products help fight climate change and environmental problems Governments consider encouraging themanufacturerrsquos initiative of producing low-carbon products by providing subsidies However when the manufacturer sells low-carbon products through the e-commerce platform fairness concerns arise because of the profit difference So this paper buildsgamemodels to study decision behavior in the low-carbon e-commerce supply chain when the manufacturer receives governmentcarbon subsidies and has fairness concerns Our findings show that consumersrsquo preference for low-carbon products will beconducive to the operation of the supply chain So it is necessary to popularize low-carbon products +e effect of governmentsubsidies on supply chain decisions is different from fairness concerns Government subsidies are positive factors in the supplychain operation which can stimulate themanufacturer tomake low-carbon products as expected and choose the high quality-highprice development mode +is will help improve the profit of enterprises in the supply chain but cannot effectively stimulate thee-commerce platform to increase its service level By contrast the manufacturerrsquos fairness concerns are negative factors whichmake the manufacturer prefer to adopt a low quality-low price development mode to improve their utility+is offsets the positiveeffect of government subsidies It turns out that the profit of both node enterprises and the supply chain system has declined Butfairness concerns are an important way to express the manufacturerrsquos demand Finally the joint allocation contract of cost andprofit designed by comprehensively considering the effect of government subsidies and fairness concerns can make the supplychain coordinated However even as positive factors only within a specific range do government subsidies help coordinate thesupply chain but not the more the better

1 Introduction

With the environmental problems being more and moreconcerned low-carbon products that can save energy andreduce carbon emission have gradually become consump-tion trends for example energy-saving light inverter andair conditioner But producing low-carbon products re-quires an update of the production process and equipmentwhich leads to an increase in manufacturersrsquo cost For en-couraging manufacturers to produce low-carbon productsthe government generally gives them some carbon subsidiesthereby effectively increasing the production of low-carbon

products [1 2] In the 2013 home appliance sales of Chinaenergy-saving subsidy policy drove the promotion of morethan 327 million energy-efficient home appliances as muchas USD 189 billion (httpwwwchinairncomnews20130523100849491html)

Since the 1990s the rapid development of the networktechnology and express delivery industry has attracted moreand more consumers to purchase online A number ofe-commerce platforms (abbreviated as platform in the fol-lowing if no ambiguity) come into being for instance AmazonJDcom andTmall In 2019 online sales of Black Friday reachedUSD 74 billion up 192 percent from 2018 and setting a new

HindawiComplexityVolume 2020 Article ID 1974942 19 pageshttpsdoiorg10115520201974942

high (httpswwwcnbccom20191129black-friday-online-sales-up-19percent-by-9-am-thanksgiving-sales-hit-record-onlinehtml) total sales of Tmall Double 11 Shopping Festivalpulled in USD 384 billion from 220 countries and regionsworldwide and 299merchants have total revenues of over USD143 million each (httpsalibabagroupcomennewsarticlenewsp191112) More and more manufacturers sell productsthrough e-commerce platforms and these manufacturers aremainly small and medium sized For example in 2017 morethan 20000 Amazon online sellers across the world exceededUSD 1 million in sales (httpswwwmarketplacepulsecomarticlestwenty-thousand-amazon-millionaire-sellers)

+e e-commerce platform provides manufacturers withnew sale channels to sell low-carbon products+erefore theplatform and low-carbon product manufacturers form alow-carbon e-commerce supply chain +is cooperationseems to be mutually beneficial However there are stillsome discordant factors in the operation process +ee-commerce supply chain is very different from the tradi-tional one that the platform because of its vast networktraffic becomes the leader +e platform formulates rules formanufacturersrsquo entry and controls the information flowgoods flow and capital flow In contrast manufacturers notonly have to pay a specific fixed fee to the platform forentering it but also share the sales profit as a commissionwith the platform according to an established proportion Inpractice Tmall JDcom and Dangdangcom charge theirsettled manufactures in this way Moreover as a sharedmedium the platform usually attracts many manufacturerssuch that it can achieve substantial profit so sometimeswould ignore the manufacturersrsquo interests by using itsdominant position In particular small- and medium-sizedmanufacturers are even more affected For example duringthe 6ndash18 promotion of 2017 many manufacturers haveexpressed strong dissatisfaction with Tmall and JDcombecause these sellers were forced to participate in large-scalediscount activities (httpswwwchinainternetwatchcom29999double-11-2019) As a result the decline in statusand the gap in profitability in the supply chain has ledmanufacturers to reexamine their relationship with theplatform which is called fairness concerns +e larger theplatformrsquos size the more significant the income gap and thestronger the concerns of small- and medium-sized manu-facturers in the platform

+is kind of fairness concerns of manufacturers is aninstinct response protecting their interests +en when theyhave fairness concerns how do manufacturers react to theplatform and what impact will they have on the decision ofthe low-carbon e-commerce supply chain system Whengovernment subsidies and fairness concerns coexist whatwill happen to the decision of the supply chain systemWhen manufacturers have fairness concerns how could thesystem coordination be achieved+e current research fruitsare still not able to address the above problem well Atpresent the research about fairness concern behavior ismainly focused on the traditional offline supply chain [3ndash5]Even if the online sale is considered it is only regarded asone of the distribution channels which does not reflect thekey features of a platform [6ndash8] As the leader of a low-

carbon e-commerce supply chain the platform is respon-sible for providing consumers with logistics informationand other services besides general services and consumersdemand much more from the platform than from offlinesellers +erefore the operation of the platform requirescosts to maintain and it is also necessary to take the plat-formrsquos service cost and service level into account whenmaking decisions Moreover manufacturersrsquo carbon sub-sidies obtained from the government make manufacturersrealize the importance of their carbon emission reductionbehavior which will intensify manufacturersrsquo desire forfairness +ese are rarely addressed in the existing research

+is paper will study how the low-carbon e-commercesupply chain will make decisions and be coordinated if themanufacturer receives carbon subsidies from the govern-ment and has fairness concerns Our contributions are asfollows

(1) To determine whether the carbon subsidies are ef-fective to promote the low-carbon products by an-alyzing the effect of the carbon subsidies on themanufacturerrsquos sales price and carbon emission re-duction level the platformrsquos service level and theprofit of the low-carbon e-commerce supply chain

(2) To explore the transmission mechanisms of themanufacturerrsquos fairness concerns by analyzing thedouble effects of both carbon subsidies and themanufacturerrsquos fairness concerns on the service levelthe sales price and the profit

(3) To coordinate the supply chain with governmentcarbon subsidies and fairness concerns by designingthe joint allocation contract of cost and profit toeliminate the double marginalization under decen-tralized decision modes

+e rest of this paper is organized as follows Literaturereview is given in Section 2 Some basic work for the modelsis provided in Section 3 We model four different decisionmodes in Section 4 and then analyze and compare differentdecision results in Section 5 In Section 6 the coordinationmechanism for the supply chain system is designed InSection 7 we do the example analysis In Section 8 we offerconcluding comments

2 Related Literature

+is paper mainly involves such three streams of literatureincentive mechanism of government carbon subsidiessupply chain fairness concerns and supply chaincoordination

21 Incentive Mechanism of Government Carbon SubsidiesFor protecting our environment researches about closed-loop supply chain have been conducted [9ndash11] As carbonemission reduction is concerned manufacturers have adirect relation to assume this responsibility and also manageto invest in this area [12] In fact economic and environ-mental benefits can be achieved simultaneously [13] andexternal factors could do something to guide the

2 Complexity

manufactures and the supply chain [14 15] Recently forpromoting the production and consumption of low-carbonproducts governments of the world are concerned abouthow the supply chain decision-making will be influenced bydifferent carbon emission reduction measures [16] Carbonsubsidies often adopted by governments to promoteproduct development can boost profit growth increasemarket demand and improve the benefits of supply chainmembers [17] Currently carbon subsidies are mainly usedin the field of recycling or remanufacturing of electronicproducts [18] Li et al [19] considered the application ofcarbon subsidy policies in the remanufacturing supply chainand analyzed when and how the government should usecarbon subsidies to stimulate enterprisesrsquo initiative to reducecarbon emissions In practice there are two departmentsformal and informal sections which can recycle electronicproducts In order to accelerate the development of therecycling industry subsidies should be regulated by thegovernment to limit the waste quality to a higher level If thegovernment subsidies are not sufficient and the waste qualityis high then the informal production sector has a greatercompetitive advantage [20] As the subsidy recipient isconcerned the government could subsidize the recycler orremanufacturer and provide subsidies according to the re-manufacturersrsquo different service levels which is helpful toexpand the market scale and remanufacturing service level[21]

+e carbon subsidy strategy studied in the existing lit-erature is a passive way mainly aiming at the remanu-facturing and waste recycling in the traditional supply chainbackground However under the e-commerce environmentthe transmission mechanism of government carbon sub-sidies in the supply chain will be quite different where themanufacturer directly sells products to consumers on theplatform and the low-carbon product manufacturer as themain contributor to manufacturing low-carbon productscan set the product price +erefore the government givescarbon subsidies to the low-carbon product manufacturerAfter obtaining carbon subsidies the manufacturerrsquos profitwill be raised and then this influence will be passed onthrough the supply chain which will further affect somedecisions of the platform +is means that the existingmethodology is no longer applicable So from the per-spective that the government actively encourages themanufacturer to produce low-carbon products by providingcarbon subsidies this paper introduces carbon emissionreduction costs and carbon subsidies into themanufacturerrsquosprofit function Carbon subsidies will be given proportionalto the carbon emission level We aim to drive the productionof low-carbon products from the source

22 Supply Chain Fairness Concerns Besides how muchprofit they earn companies or individuals in economicactivities will also pay close attention to the profit gap be-tween the trade parties that is the fairness of profit allo-cation [22ndash24] At present the concerns about fairness areparticularly reflected in the offline supply chain thatmaintains long-term partnerships [25ndash27] Cui et al [28]

used fairness concerns to express dissatisfaction with thestatus in the supply chain and explore how the transactionsbetween retailers and manufacturers would be affectedespecially by quantity discounts and two-part tariffs Zhaoet al [29] studied a product and service supply chain andcompared its optimal pricing strategy in two cases based onwhether fairness concerns of the retailer are consideredfinding that if the retailer in the supply chain has fairnessconcerns all the enterprises could adopt differentiatedservice pricing strategies but the retailerrsquos fairness concernswould do harm to the manufacturerrsquos profit If both man-ufacturers and retailers in the supply chain have fairnessconcerns retailersrsquo fairness concerns are more likely to resultin a narrower supply chain stable area than manufacturerswhile manufacturersrsquo fairness concerns can expand thesystemrsquos stable area [30 31]

+e existing research fruits mainly study the fairnessconcerns of retailers and manufacturers in the supply chainof ordinary products+e research subject of this paper is thefuse of the e-commerce supply chain and low-carbon supplychain Manufacturers produce low-carbon products andneed to invest in additional RampD costs +e e-commerceplatform in the supply chain is the dominant and coreenterprise which puts manufacturers at a disadvantagestatus in the profit allocation thereby making manufacturershave fairness concerns Moreover fair concerns in thissituation have their particularity On the one hand theplatform could get commissions from numerous small- andmedium-sized manufacturers that have settled in it and onthe other hand manufacturers think that they should getmore benefits because of paying for low-carbon productioncosts and receiving government subsidies So even thoughthe manufacturerrsquos profit is no less than what the e-com-merce platform can get from this manufacturer the man-ufacturer will still have fairness concerns +us themechanism of how fairness concerns come into being in thispaper is different from the existing research fruits

23 Supply Chain Coordination Classic coordination con-tracts include sharing revenue contract buy-back contractquantity-flexibility contract wholesale pricing contract andquantity discount contract [32] Now researchers have toadd some other factors into the coordination contract suchas environmental economic and social dimensions [33ndash40]For the two-tier reverse supply chain Heydari et al [41]proposed a coordination mechanism combining quantitydiscounts with increasing fee adding government subsidiesinto the contract as incentives so that each member hasenough initiative to participate in the coordination processwhich favors retailers much more For a low-carbon supplychain with carbon emission cap-and-trade Peng et al [42]combined subsidy on emission reduction with the revenue-sharing contract and then perfectly achieved both the supplychain coordination and the carbon emission reduction

For the supply chain coordination problems with fair-ness concerns many scholars have also designed effectivecoordination contracts according to practical problems[43ndash45] Cui et al [28] designed the whole price contract and

Complexity 3

used this contract to coordinate fairness concerns fromretailers only or from both retailers andmanufacturers in thesupply chain with linear demand +en Caliskan-Demiragaet al [46] extended the linear demand function to the ex-ponential one and found that when retailers have fairnessconcerns more relaxed conditions are needed to achievecoordination Zheng et al [47] coordinated a closed-loopsupply chain by variable-weighted Shapley values which isbetter than the traditional Shapley value coordinationmethod when remanufactured goods are accepted by cus-tomers at a lower rate

One supply chain contract cannot apply to all coor-dination problems and we need to design them accordingto the actual ones +e existing literature about supplychain coordination has not considered the existence ofboth fairness concerns and carbon subsidies In additionthe difference in the problem addressed here from theexisting literature lies in that it is necessary to take fourfactors into account that is the manufacturerrsquos low-carbon cost the platformrsquos service cost the commissioncharged by the platform and government carbon sub-sidies which are critical for achieving coordination Wehave to balance the four factors and explore how thecoordination condition is influenced by manufacturersrsquofairness concerns

Based on the above literature review the existing researchhas built a solid base for the related fields however there arestill research gaps to fill First most works studied carbonsubsidy strategy for recycling or remanufacturing of electronicproducts [18ndash21] although several studies have investigatedsupply chains under the governmentsrsquo carbon subsidy [12 1415] they are not applied to the e-commerce supply chainSecond fair concerns exist in different backgrounds but mostof them studied that between the retailers andmanufacturers inthe supply chain of ordinary products [24 27 29 31] eventhough there are researches related to online or low-carbonbackground they do not consider them together [23 30]+ird various coordination contracts have been designedaiming to reduce the carbon emission or protect the envi-ronment [32 36] and some of them have taken governmentrole [41] low-carbon preference [42] or fairness concern [44]into account but no works addressed the problem with all ofthem under the e-commerce background Wang et al [48]coordinated the supply chain when the manufacturer hasfairness concerns but no carbon subsidy is considered So inthis paper our contributions are to formulate the game modelfor low-carbon e-commerce supply chain with governmentcarbon subsidies and fairness concerns exploring the effect ofgovernment carbon subsidies and fairness concerns on thesupply chain determining what the manufacturer and theplatform will react achieving the coordination of the supplychain by allocating the carbon subsidies commission and thecost as a whole

3 Preliminaries of Models

We consider a supply chain composed of one e-commerceplatform and one small- and medium-sized manufacturer Inthis supply chain themanufacturer is responsible for producing

only one low-carbon product while the platform offers a directonline shop to themanufacturer Because the fixed fee pays onlyonce when themanufacturer enters the platform we do not takeit into account in the following models

+e platform and the manufacturer are independentmaking decisions for maximizing their profit respectively+ey form Stackelberg games As stated in Section 1 thispaper assumes that the platform is the leader with thepriority to make decisions while the manufacturer is thefollower +e platform first determines its service level Onobtaining the platformrsquos decision the manufacturer thendetermines sales price of the low-carbon product and itscarbon emission reduction level

Some notations are listed in Table 1 which will be usedfrequently in this paper

Generally carbon emission reduction involves moreadvanced technology and management so the manufacturerhas to spend more extra cost for producing the low-carbonproduct than general ones +e higher the carbon emissionreduction level the higher the cost Based on the relation putforward by Nair and Narasimhan [49] the manufacturerrsquoscarbon emission reduction cost is as follows

Cm(h) th

2

2 (1)

Because carbon emission reduction is conducive toenergy conservation and environmental protection thegovernment will provide the manufacturer with subsidies toencourage its positive behavior Suppose the unit emissionreduction subsidy received by the manufacturer is T(Tgt 0)

According to Yao and Liu [50] the demanded quantityof low-carbon product depends on its sales price themanufacturerrsquos carbon emission reduction level and theplatformrsquos service level which can be formulated as

q D minus d1p + d2s + d3h (2)

Suppose D is a large enough constant number Supposed1 gtd3 suggesting that the productrsquos price influencesconsumers much more than its low-carbon level

Generally the higher the platformrsquos service level themore the cost Based on Shenrsquos research [51] the platformrsquosservice cost is functioned as

Ce(s) ks

2

2 (3)

Suppose ρltp minus c representing that the commissionshould be less than the net profit of the product sales

To make the following analysis hold we further assumetd1 lt d2

3 lt 2td1 and td22 + kd2

3 lt 2ktd1 +e former equiva-lent to 2d1Cm(h)lt (d3h)2 lt 4d1Cm(h) means that the in-fluence of the manufacturerrsquos carbon emission reductionlevel on the demanded quantity of the low-carbon productdepends on the carbon emission reduction cost and theinfluence is limited neither too large nor too little while thelatter represents that the sales price has greater influence onthe demanded quantity than the combined effect of themanufacturerrsquos carbon emission reduction level and theplatformrsquos service level

4 Complexity

4 Modeling Framework

In this section we model different decision modes of thee-commerce supply chain according to whether there aregovernment carbon subsidies and manufacturerrsquos fairnessconcerns or not

41 Model I Benchmark Model Starting from the mostfundamental case we build the benchmark model in whichthe government does not provide any carbon subsidies andthe manufacturer does not have fairness concerns We canalso call it the decentralized decision model without gov-ernment carbon subsidies and manufacturerrsquos fairnessconcerns +en the manufacturerrsquos profit is

πm pq minus ρq minus cq minusth

2

2 (4)

+e platformrsquos profit is

πe ρq minusks

2

2 (5)

+e supply chainrsquos profit is

π pq minus cq minusth

2

2minus

ks2

2 (6)

By backward induction the optimal solution of thismodel could be achieved as follows

Theorem 1 In the benchmark model the optimal salesprice is

pblowast

(c + ρ) + d1 middottd2

2td1 minus d23

1113888 1113889

2

middotρk

+k D minus d1(c + ρ)1113858 1113859 minus ρd

22

k 2td1 minus d231113872 1113873

middot t

(7)

(e optimal carbon emission reduction level is

hblowast

t middot d1 middot d3 middotρk

middotd2

2td1 minus d23

1113888 1113889

2

+ d3 middotk D minus d1(c + ρ)1113858 1113859 minus d

22ρ

k 2td1 minus d231113872 1113873

(8)

(e optimal service level is

sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (9)

(e manufacturerrsquos optimal profit is

πblowastm

t

2middot

D minus d1(c + ρ)1113858 11138592

2td1 minus d23

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(10)

(e platformrsquos optimal profit is

πblowaste tρd1 middot

D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(11)

(e supply chainrsquos optimal profit is

πblowast t middot

D minus d1(c minus ρ)1113858 1113859 middot D minus d1(c + ρ)1113858 1113859

2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(12)

Proof See Appendix A

42 Model II Decentralized DecisionModel with GovernmentCarbon Subsidies and without Manufacturerrsquos FairnessConcerns Based on the benchmark model we turn to thecase that the government subsidies the manufacturer for itscarbon emission reduction behavior and the manufacturerdoes not have fairness concerns +en the manufacturerrsquosprofit is

πm pq + Th minus ρq minus cq minusth

2

2 (13)


πe ρq minusks

2

2 (14)

Table 1 Mathematical notations

Notations Meaningc Unit cost for producing the low-carbon product (in dollars)h Carbon emission reduction level of the manufacturep Sales price of the low-carbon product (in dollars)q Market demand for the low-carbon products Service level of the platformT Unit government subsidy for the manufacture (in dollars)ρ Commission charged by the platform for unit sales of products (in dollars)t Carbon emission reduction cost coefficientD Maximum quantity demanded of the low-carbon productd1 Influence coefficient of the low-carbon product sales price on demandd2 Influence coefficient of the platformrsquos service level on demandd3 Influence coefficient of the manufacturerrsquos carbon emission reduction level on demandk Service cost coefficient

Complexity 5


π pq + Th minus cq minusth

2

2minus

ks2

2 (15)

Similar to the solution process of +eorem 1 we reachthe following theorem

Theorem 2 In the decentralized decision model with gov-ernment carbon subsidies the optimal product sales price is

pslowast

(c + ρ) +ρd1

kmiddot

td2

2td1 minus d23

1113888 1113889

2

+k Dt + d3T minus td1(c + ρ)1113858 1113859 minus ρtd

22

k 2td1 minus d231113872 1113873

(16)


hslowast

ρk

middotd1d3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3

k

middotk D + 2Td1( 1113857 d3 minus d1(c + ρ)( 11138571113858 1113859 minus ρd

22

2td1 minus d23

(17)


sslowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (18)


πslowastm

t middot D minus d1(c + ρ)1113858 11138592

+ 2d3T D minus d1(c + ρ)1113858 1113859 + 2d1T2

2 2td1 minus d231113872 1113873

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(19)


πslowaste d1 middot

tρ D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(20)


πslowast

t middot D minus d1(c + ρ)1113858 1113859 D minus d1c minus ρ1113858 1113859 + 2T Td1 + D minus cd1( 1113857d31113858 1113859

2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(21)

By comparing the decision results of model II withmodel I we can find the effect of government carbonsubsidies on the supply chain decision

Proposition 1 pblowast ltpslowast hblowast lt hslowast sblowast sslowast πblowastm lt πslowast

m πblowast

e lt πslowaste and πblowast lt πslowast

Proof See Appendix B

Proposition 1 shows that after the government providessubsidies product sales price carbon emission reductionlevel the manufacturerrsquos profit and the platformrsquos profithave all increased +e proof process in Appendix B alsosuggests that these changes are related to subsidies whichreflects the effectiveness of government subsidies Howeverthe platformrsquos service level stays the same On the one handsince subsidies are directly given to the manufacturer theyhave no impact on the platformrsquos service level On the otherhand if the platform can have its profit increased evenwithout the necessity to improve its service level it does nothave any initiative to make efforts at all +is reflects theshortcoming of government subsidies since they cannotstimulate the platform to improve its sales service

Government subsidies express their emphasis and rec-ognition on the manufacturerrsquos production of low-carbonproducts which can become a guide to social consumption

habits From the consumersrsquo perspective their low-carbonawareness will also increase and they will pay much moreattention to the low-carbon level of products when making apurchase Even if the price is higher than nonlow-carbonproducts consumers will be more willing to buy From theperspective of the manufacturer government subsidies havegiven it the motivation and responsibility to continue de-veloping low-carbon products and its low-carbon level hasbeen constantly improved +e modest increase in the priceof the product by the manufacturer mainly reflects a dif-ferentiating strategy which is different from previousproducts and will eventually make itself more profitable

On January 16 2017 JDcom and Midea Group signed aUSD29 billion comprehensive strategic cooperation agreement(httpnews163com17011710CAVN0MMQ00018AOQhtml) JDcom has witnessed that prices of low-carbon en-ergy-saving products have generally increased by about USD871 Midea Group received government subsidies of USD 02billion in 2017 (httpmoneyfinancesinacomcncorpviewvCB_AllBulletinDetailphpstockid000333ampid4185007)and achieved a year-on-year net profit increase of 1733(httpbaijiahaobaiducomsid1596592651173428121ampwfrspiderampforpc) JDcomrsquos full-year net profit in 2017was USD 073 billion a substantial increase of 140year-on-year (httptechqqcoma20180302030924html)

6 Complexity

43Model IIIDecentralizedDecisionModelwithGovernmentCarbon Subsidies and Manufacturerrsquos Fairness ConcernsAs mentioned in Introduction the capital input of carbonemission reduction the dissatisfaction caused by the plat-form neglecting its interests and the support of govern-mentrsquos carbon subsidies and other factors will cause themanufacturerrsquos concern over the profit gap between it andthe platform thus creating fairness concerns +erefore webring the manufacturerrsquos fairness concern behavior into ourmodel

According to Fehr and Schmidt [52] and Katok et al[53] when having fairness concerns the manufacturer willtake δ times the platformrsquos profit as the reference point Sowe write the manufacturerrsquos utility function as

Um πm minus θ δπe minus πm( 1113857 (22)

where 0le θ le 1 denotes the manufacturerrsquos fairness concerncoefficient When θ is closer to 0 it means the manufac-turerrsquos fairness concern degree is getting weaker Otherwiseθ closer to 1 means the manufacturerrsquos fairness concerndegree is getting stronger

Substituting πm and πe in Um we express the manu-facturerrsquos utility function as

Um (1 + θ)pq + Th minus ρq minus cq minus th

2

21113888 1113889 + θδ

ks2

2minus ρq1113888 1113889

(23)

In this case the manufacturerrsquos decision is not based onprofit but utility +e manufacturer tries to maximize its

utility By backward induction we obtain the optimal so-lution of this model as follows

Theorem 3 In the decentralized decision model with gov-ernment carbon subsidies and manufacturerrsquos fairness con-cerns the optimal product sales price is

psflowast

(c + ρ) + d23 minus td11113872 1113873 middot t middot

ρk

middotd2

2td1 minus d23

1113888 1113889

2

+t D minus d1(c + ρ)1113858 1113859 + d3T

2td1 minus d23

minus δρ middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

(24)


hsflowast

d3 middotρk

middot d23 minus td11113872 1113873 middot

d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23

minusd1d3δρ2td1 minus d

23

middotθ

1 + θ

(25)


ssflowast

d2 middotρk

middotd23 minus td1

2tβ minus d23 (26)

(e manufacturerrsquos optimal utility is

Usflowastm T middot

d3 (1 + θ) D minus cd1 minus ρd1( 1113857 minus θd1ρδ1113858 1113859 + d1(1 + θ)T

2td1 minus d23

+t

2(1 + θ)middot

(1 + θ) D minus cd1 minus ρd1( 1113857 minus θd1ρδ1113858 11138592

2td1 minus d23

+ ρd2( 11138572

middotδθ2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2td1 minus d23

middot1 + θ2k

2 middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(27)


πsflowaste

ρd2( 11138572

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minusθ

1 + θmiddott(1 + δ) ρd1( 1113857

2

2td1 minus d23

(28)


πsflowastm

t D minus d1(c + ρ)1113858 1113859221113872 1113873 + Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

+ ρ2d22 middot

δk

middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minusθ

1 + θ1113888 1113889

2

middottδ2ρ2d2

1

2 2td1 minus d231113872 1113873

(29)


Complexity 7

πsflowast

t D minus d1(c + ρ)1113858 11138592

+ d3 + 2ρd1t( 1113857 D minus d1(c + ρ)1113858 1113859 + d1T 1 + 2ρd3( 1113857

2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2kmiddot 1 + 2δ middot

θ1 + θ

1113888 1113889 middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

22 ρd2( 1113857

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

θ1 + θ

middotd1ρ( 1113857

2

2td1 minus d23

2(1 + δ) + δ2θ

1 + θ1113890 1113891

(30)

Proposition 2 ssflowastis independent of θ while psflowast hsflowastπsflowast

m πsflowaste and πsflowast all decrease as θ increases

Proof See Appendix C

+e manufacturer employs fairness concern behavior toensure its utility not damaged with such available means as theproduct price and the carbon emission reduction levelDominant in their relationship the platform is reluctant toadjust its service level to ease up the manufacturerrsquos concernsand will pay the price of reduced profit for this With fairnessconcerns increasing the manufacturer may make decisionsirrationally aiming to increase its profit +erefore the man-ufacturer intends to decrease its production cost no longerexerting effort to improve the low-carbon level of the productAt the same time the manufacturer reduces the product salesprice expecting to increase the sales volume and tries to take thelow quality-low price route As a result the effect is coun-terproductive which eventually leads to the profit decline of theplatform the manufacturer and the supply chain +usProposition 2 suggests that the manufacturerrsquos fairness con-cerns go against the whole supply chain and its members

In a supply chain system the manufacturer who pays toomuch attention to the profit gap and its utility will get theopposite effect +erefore during the actual operation as longas its profit can be increased it is not necessary to pursue theabsolute profit fairness among members On the one hand thelow-carbon product manufacturer should actively work closelytogether with the platform play its role and keep improvingthe low-carbon level of products the platform on the otherhand should guarantee the service level and customer size andso is the manufacturerrsquos profit

44 Model IV Centralized Decision Model with GovernmentCarbon Subsidies Now we consider the case when themanufacturer and the platform can cooperate to maximizethe supply chainrsquos overall profit which forms a centralizeddecision mode Under this mode with governmentrsquos carbonsubsidies the supply chainrsquos profit is

π(p h s) (Th + pq) minus cq +th

2

2+

ks2

21113888 1113889 (31)

And its optimal solution is given below

Theorem 4 In the centralized decision model with gov-ernment carbon subsidies the optimal product retail price is

pclowast

kt D + cd1( 1113857 + kTd3 minus c td

22 + kd

231113872 1113873

2ktd1 minus td22 + kd

231113872 1113873

(32)


hclowast

kd3 D minus cd1( 1113857 + T 2kd1 minus d

221113872 1113873


231113872 1113873

(33)


sclowast

d2 middott D minus cd1( 1113857 + Td3


231113872 1113873

(34)


πclowast

kt D minus cd1( 11138572

+ T2 2kd1 minus d

221113872 1113873 + 2kTd3 D minus cd1( 1113857

4ktd1 minus 2 td22 + kd

231113872 1113873

(35)

5 Managerial Analysis

In this section we show how government carbon subsidiesand the manufacturerrsquos fairness concerns would affect thesupply chain decisions by comparison of models in Prop-osition 3 through Proposition 7

Proposition 3 sclowast gt sslowast ssflowast Both sslowastand ssflowast increasewith ρ sclowast sslowast and ssflowast increase with d3

Proof See Appendix D

Proposition 3 implies that since the service level is onlydepended on the platform it stays the same no matter thelow-carbon product manufacturer has fairness concerns ornot As the platformrsquos profit increases it has more availablemoney to improve the service level Correspondinglyconsumersrsquo preference for low-carbon products helps therise in service level in all three cases +is inspires themanufacturer that they should try to promote low-carbonproducts and let consumers favor low-carbon productswhich will also drive the platform to improve its service Inthis way the manufacturer even a follower in the supplychain still can have effect on the leader to some extent

Similar to the proof of Proposition 3 we give Propo-sitions 4ndash6 without extra proofs

Proposition 4 hclowast gt hslowast gt hsflowast Both hslowastand hsflowast decreasewith the increase in ρ while hclowast hslowast and hsflowastincrease with d3

8 Complexity

Proposition 4 shows that the e-commerce supply chainsystem has the highest carbon emission reduction level in thecentralized decisionmodel and thatwhen themanufacturer hasfairness concerns its excessive focus on profit allocation hasaffected the motivation and enthusiasm to reduce carbonemission As the profit gained by the platform increases thelow-carbon product manufacturer will cut down its carbonemission reduction level due to the decrease in obtained profit+us in order to raise the carbon emission reduction level theplatform should reduce the commissionwhichwill help the saleof low-carbon product However as consumersrsquo preference forlow-carbon products increases no matter under the decen-tralized decision models with or without the manufacturerrsquosfairness concerns or the centralized decision model efforts willbe made to improve carbon emission reduction levels

Proposition 5 pclowast gtpslowast gtpsflowast Both pslowastand psflowast decreasewith the increase in ρ while pclowast pslowast and psflowast increase withd3

Proposition 5 expresses that the highest price can be set inthe centralized decisionmodel because this mode has both thehighest carbon emission level and the highest service levelwhich can be accepted by consumers +is result reflects theapproval by consumers for the cost of the supply chain systemto provide high-quality products Among the three decisionmodels the manufacturer is most sensitive to the profit whenit has fairness concerns When the manufacturerrsquos profit isaffected it will prefer the low-end development way notactively improving the carbon emission level while theplatformrsquos service level will not get improved either so themanufacturer can only choose the low-price strategy In thedecentralized decision models with the platformrsquos profitincreasing the manufacturer expects to increase its profit byreducing prices It will not actually but will reduce themanufacturerrsquos profit +is also exposes the disadvantages ofdecentralized decision and further discloses the fact that theplatformrsquos attempt to obtain profit by collecting too muchcommission does harm to the supply chain With the rise inconsumersrsquo preference for low-carbon products it is a goodsituation for low-carbon sales which can promote the rise incommodity prices under three decision modes

Proposition 6 πclowast gt πslowast gt πsflowast Both πslowastand πsflowast decreasewith the increase in ρ while πclowast πslowast and πsflowast increase withd3

Proposition 6 tells us that the supply chainrsquos profit is thelargest in the centralized decision model the smallest whenthe manufacturer has the fairness concerns and in betweenwhen the manufacturer does not have any fairness concerns+is result demonstrates that the manufacturer is moresensitive to profit allocation when having fairness concernsand its performance in the supply chain is affectedWhen theplatformrsquos profit rises the supply chainrsquos one will decline inthe decentralized decision models (whether with the man-ufacturerrsquos fairness concerns or not) +is relation furtherreflects the manufacturerrsquos desire to share the profit rea-sonably Otherwise it will adopt a more uncooperative way

in the supply chain As consumers prefer low-carbonproducts more the production and sales of low-carbonproducts are more supported and then the supply chainrsquosprofit will increase in the end

Hence the manufacturerrsquos irrational fairness concernsgo against the operation and coordination of the supplychain system In practice manufacturers should be guided togive up this irrational behavior and closely cooperate withthe operation of the system

Proposition 7 Usflowastm increases as θ increases

Proof See Appendix E

As stated in Proposition 7 the manufacturerrsquos utility willincrease with the fairness concern coefficient Fairness con-cerns in some sense are thereby proven to have positive effectin the supply chain decision When the low-carbon productmanufacturer focuses on the fairness of profit allocation theplatformwill take themanufacturerrsquos needs into account in itsdecision process after obtaining this information +is guidesone way for the weak enterprise in the supply chain how tostruggle for its benefit with the strong one

6 Low-Carbon E-Commerce Supply ChainCoordination Based on GovernmentCarbon Subsidies

It can be seen from the comparison in previous section thatthe government carbon subsidymechanism helps increase thesupply chainrsquos profit When government subsidies are ap-plied the centralized decision is better than other decen-tralized ones regardless of with or without the manufacturerrsquosfairness concerns +e centralized decision mode achieves theoptimal profit for the system while decentralized decisionmodes result in loss of efficiency due to the double mar-ginalization effect+us a specific coordinationmechanism isnecessary to be applied in this system

Since cost and benefit are two key factors connecting themanufacture and the platform we adopt indicators aboutthem as coordination means and design the joint allocationcontract of cost and profit to coordinate the system+e ideaof this contract is to coordinate the benefits of both sides inthis way the platform adjusts the commission by themanufacturerrsquos sales price and production cost while themanufacturer pays part of government subsidies to theplatform besides that the manufacturer and the platformappropriately share the costs for each other

Under the joint allocation contract of cost and profit onthe one hand we assume that the platform adjusts thecommission to ρ pays the proportion y(0leyle 1) of theservice cost and leaves the remaining (1 minus y) to the man-ufacturer on the other hand we assume that the manu-facturer keeps the proportion z(0le zle 1) of governmentsubsidies giving others to the platform and pays the pro-portion x(0le xle 1) of the low-carbon cost leaving theremaining (1 minus x) to the platform +en the profit of themanufacturer is

Complexity 9

πcm (p minus ρ minus c)q + zTh minus x

th2

2minus (1 minus y)

ks2

2 (36)

+e profit of the platform is

πce ρq +(1 minus z)Th minus (1 minus x)

th2

2minus

yks2

2 (37)

Theorem 5 (e joint allocation contract of cost and profitcan coordinate the e-commerce supply chain if (ρ x y z)

satisfies

ρ (1 minus ϕ)(p minus c)

x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)

Proof See Appendix F

When the supply chain achieves coordination the ratioof the manufacturerrsquos profit to the platformrsquos one isϕ(1 minus ϕ) With this we call ϕ profit allocation coefficientϕ(0ltϕlt 1) reflects the manufacturerrsquos ability to negotiatewith the platform +e larger the ϕ is the more profit themanufacturer could obtain the less the platform is and viceversa So ϕ cannot be too large or too small To make thecoordination mechanism effective we analyze its feasiblecondition as follows

(1) In model II the condition for the platform and themanufacturer to agree on the joint allocation con-tract is that their profit after coordination should notbe less than before respectively +is condition canbe formulated as

πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)

which is equivalent to

πcm ϕπc

lowast ge πslowastm

πce (1 minus ϕ)πc

lowast ge πslowaste

⎧⎨

⎩ (40)

So ϕ has to satisfy

πslowastm

πclowast leϕle 1 minus

πslowaste

πclowast (41)

(2) In model III two conditions are needed for theplatform and the manufacturer to agree on the jointallocation contract One is that their profit shouldnot decrease after coordination Another is themanufacturerrsquos utility should not fall because ofcoordination +is condition can be formulated as

πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc

lowast ge πsflowastm

πce (1 minus ϕ)π lowastc ge π

sflowaste

Ucm πc

m minus θ δπce minus πc

m( 1113857 ϕπclowast

minus θ δ(1 minus ϕ)πclowast

minus ϕπclowast

1113858 1113859ge πsflowastm minus θ δπsflowast

e minus πsflowastm1113872 1113873

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)

Since the difference of the lower bound and the upperone of ϕ is

πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1

θ(δ + 1) + 1middotπclowast

minus πsflowast

πclowast

(45)

we have the following proposition

Proposition 8 (e value range of ϕ becomes larger as θincreases

Proposition 8 shows that the manufacturerrsquos fairnessconcerns are conducive to the coordination of the supplychain so that both the platform and the manufacturer canhave much more choice during coordination

7 Numerical Examples

In the following we will put the coordination mechanisminto practice by an example and analyze how this mecha-nism is influenced by other factors

Suppose c 200 T 1000 D 10000 d1 20 d2 2d3 10 k 5 and t 4 which will be used in this section

10 Complexity

71 Effectiveness Verification of the CoordinationMechanismFirst suppose ρ 20 θ 01 and δ 100 Refer Table 2 forthe decision results before the coordination mechanism isapplied

In model II the coordination mechanism (ρ x y z)

satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)

where 0866le ϕle 0943In model III the coordination mechanism (ρ x y z)

satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)

where 0913le ϕle 0956+en we use ϕ as the independent variable to compare

the difference before and after coordination (see Figure 1)As shown in Figure 1 in these two cases the joint al-

location contract of cost and profit has coordinated thesupply chain system and achieved win-win cooperation

+e profit allocation coefficient ϕ is the core parameterof the coordination mechanism (ρ x y z) In the followingthree subsections we will analyze how the coordinationmechanism is influenced by other factors of the supply chainsystem and illustrate this influence through ϕ

72 Influence of ρ on Coordination Mechanism For the casewithout fairness concerns we further suppose ρ isin [1 20]

and then obtain the relation between the bounds of ϕ and ρin this coordination mechanism as demonstrated inFigure 2(a) Similarly for the case with the manufacturerrsquosfairness concerns after supposing θ 01 and δ 100 wecan also depict this relation as demonstrated in Figure 2(b)

Figures 2(a) and 2(b) indicate that both the lower boundand the upper one of ϕ are negatively correlated to thecommission no matter whether the manufacturer hasfairness concerns or not As a result the more commissionsthe platform collects the less themanufacturerrsquos profit shareand the more profit proportion the platform gets which willbe more and more beneficial to the platform At the sametime it is known from +eorem 5 that the carbon emissionreduction cost and platform service cost undertaken by themanufacturer will decrease while the supply chain costassumed by the platform will rise +is reflects the value ofcore enterprises of the supply chain Besides that as thecommission increases the span between the upper and lowerbounds of ϕ becomes larger and larger which facilitates

coordination +is means that as the commission increasesthe platform gets more profit which gives the manufacturermore choices

73 Influence of θ and ρ on Coordination Mechanism +efeasible condition of our joint allocation contract is muchmore complicated when the manufacturer has fairnessconcerns because this mechanismwill be influenced by θ andδ besides ρ Suppose ρ 10 and δ 100 then Figure 3(a)describes how the bounds of ϕ varies with θ isin [005 02]Moreover suppose ρ 10 and θ 01 then the relationbetween the bounds of ϕ and δ isin [100 200] is shown inFigure 3(b)

Figures 3(a) and 3(b) show that in the case with fairnessconcerns both the upper and lower bound of ϕ increase withθ or δ indicating that the manufacturerrsquos fairness concernsplay a role in the coordination In contrast to this the spanbetween the upper and lower bound of ϕ is shrinking as θ orδ increases which means fairness concerns of the manu-facturer do not contribute to the design of the coordinationcontract so that the space for the platform and manufacturerto coordinate is getting smaller +e lower bound of ϕ in-creases rapidly with θ or δ so fairness concerns can sig-nificantly increase the manufacturerrsquos minimum profit Stillthe platform as the leader in the low-carbon e-commercesupply chain will also protect its own profit margin As far asθ and δ are concerned the influence of former is moreprominent that is to say the fairness concerns coefficient ofthe manufacturer has a bigger effect on the coordinationmechanism

74 Influence of T on Coordination Mechanism In thissection suppose ρ 10 θ 01 and δ 100

+e relation between ϕ and T with or without fairnessconcerns is shown in Figures 4(a) and 4(b) respectivelyAlso Figures 5(a) and 5(b) demonstrate how the span of ϕbetween its upper and lower bound changes with Trespectively

Figures 4(a) and 4(b) tell us that both the upper andlower bounds of ϕ increase with T which is not related tofairness concerns of the manufacturer +is indicates thatgovernment carbon subsidies have a positive effect in thesupply chain system increasing the manufacturerrsquos profitshare However Figures 5(a) and 5(b) show that the span ofϕ between its upper and lower bound has a trend of rise-fallwhich reflects that more government subsidies are not al-ways the better In the beginning government subsidies areconducive to supply chain coordination but when subsidiesexceed a certain limit it does not help coordinate the supplychain system any more +is is because with the continuousincrease in ϕ the manufacturer occupies the more pro-portion of government subsidies which leads to unfair al-location +erefore the platform will not cooperate asbefore Furthermore the comparison between Figures 5(a)and 5(b) also indicates clearly the manufacturerrsquos fairness

Complexity 11

Table 2 Optimal decisions of different decision modes before coordination

Price Carbon emission level Service levelProfit

Manufacturer Platform Supply chainModel II 76036 160089 267 231times 106 153times105 246times106

Model III 69975 99483 267 140times106 118times105 152times106

Model IV 79860 174648 23944 mdash mdash 267times106

25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)

Figure 1 Comparison before and after coordination (a) without fairness concerns (b) with fairness concerns

090

088

092

094

096

098

φ

5 10 15 20

Lower bound of φUpper bound of φ

ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)

Figure 2 Change in bounds of ϕ with ρ (a) the case without fairness concerns (b) the case with fairness concerns

12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)

Figure 5 Change in span of ϕ with T (a) case without fairness concerns (b) case with fairness concerns

006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)

Figure 3 Change in bounds of ϕ in the mode with fairness concerns (a) change in bounds of ϕ with θ (b) change in bounds of ϕ with δ

600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)

Figure 4 Change in bounds of ϕ with T (a) case without fairness concerns (b) case with fairness concerns

Complexity 13

concerns can push the maximum span of ϕ rightward whichis a decision favorable to itself made by the manufacturerusing fairness concerns

8 Concluding Comments

In this paper we address the decision and coordination ofthe e-commerce supply chain system composed of onesmall- and medium-sized low-carbon product manufacturerand one e-commerce platform According to whether thereare fairness concerns or government subsidies differentdecision models are formulated We reach the optimaldecision for each model design the contract to achieve thecoordination of the supply chain system and use numericalexamples to analyze these decision models and the coor-dination strategy We also reach the following constructiveconclusions

First the promotion of low-carbon products will havewin-win effects on the environment and economy On theone hand we have reached a consensus that carbon emissionreduction helps protect the environment One the otherhand consumersrsquo preference for low-carbon products willbe conducive to the operation of the low-carbon e-com-merce supply chain If consumers prefer low-carbonproducts the service level the carbon emission reductionlevel the sales price and the profit of the supply chain will allgo up

Second government subsidies have positive effects whenmanufacturers do not have fairness concerns +e govern-ment gives some subsidies to low-carbon product manu-facturers which can get benefited obviously Manufacturersmay use the governmental support of carbon subsidies as anattractive feature to attract consumers As a good result theproduct price and carbon emission reduction level can besignificantly improved and this can also lead to the greenconsumption trend However because the platform does notobtain direct subsidies its service level has not been im-proved Yet the profit improvement of manufacturers fromgovernment subsidies can be passed on to the platform

+ird manufacturersrsquo fairness concerns are an effectiveway for them to advocate interests After manufacturersexpress their fairness concerns their utility will get betterwith the increase in the fairness concerns +is means thatthe platform can receive the fairness concern informationreleased by manufacturers and will consider manufacturersrsquorequirements in the decision process

Fourth when manufacturers have fairness concernstheir negative effects will offset the positive effects of gov-ernment subsidies Fairness concerns make manufacturerspay too much attention to the profit difference between theirown and the platform which will make them reduce costsand choose to take the low quality-low price shortcut Butthe result of this choice is counterproductive It significantlyaffects the benefits of the supply chain leading to the failureof both manufacturers and the platform So as the leader inthe supply chain the platform should properly adjust the

commission and share parts of the manufacturerrsquos low-carbon cost In this way the manufacturerrsquos fairness con-cerns can be pacified

Last government subsidies are only conducive to supplychain coordination to a specific extent During the process ofcoordination manufacturers give some of the governmentsubsidies to the platform for facilitating centralized decisionHowever when government subsidies exceed a specific limitmanufacturers earn too much share +is will lead to thedecline of the cooperation from the platform and bring somedifficulties to the supply chain coordination

At present we only take manufacturersrsquo fairness con-cerns into account but in practice the platform will alsohave fairness concerns to strive for reasonable profitMoreover from the last proposition we can see that gov-ernment subsidies are not enough to make the supply chainsmoothly coordinated +erefore we can consider the fulluse of carbon tax and other means Research related to thesetopics will be further addressed in future studies

Appendix

A Proof of Theorem 1

As has been assumed in the model background in theStackelberg game the platform first determines its servicelevel then based on the platformrsquos decision the manu-facturer gives the low-carbon product sales price and carbonemission reduction level

+e Hessian matrix of the manufacturerrsquos profit withrespect to sales price and carbon emission reduction level is

Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)

Since d23 lt 2td1 Hm is negatively definite

So the response function of p and h can be obtained

pblowast

D + cd1 + sd2 + ρd1( 1113857t minus (c + ρ)d

23

2td1 minus d23

hblowast

λ D minus cd1 + sd2 minus d1ρ( 1113857

2td1 minus d23

(A2)

After updating equation (5) with these two expressionsand solving dπeds 0 the authors reach the platformrsquosoptimal service level

sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)

After substituting sblowast in the expression of pblowast and hblowastthe authors have the optimal product sales price

14 Complexity

pblowast

(c + ρ) + d1 middotρk

middottd2

2td1 minus d23

1113888 1113889

2

+ t middotk D minus d1(c + ρ)1113858 1113859 minus d

22ρ

k 2td1 minus d231113872 1113873

(A4)

And the optimal carbon emission reduction level of themanufacturer

hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3

kmiddotk D minus d1(c + ρ)1113858 1113859 minus d

22ρ

2td1 minus d23

(A5)

Substituting pblowast hblowast and sblowast in equation (5) the authorsfind the manufacturerrsquos optimal profit is

πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)

+e platformrsquos optimal profit is


D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)

+e supply chainrsquos optimal profit is

πblowast t middot

D minus d1(c + ρ)1113858 1113859 D minus d1(c minus ρ)1113858 1113859

2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)

B Proof of Proposition 1

It can be obtained from +eorems 1 and 2 that

pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0

πslowastm minus πblowast

m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

πslowaste minus πblowast

e Td1d3

2td1 minus d23

πslowastminus πblowast


m1113872 1113873 + πslowaste minus πblowast

e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)

Since 2td1 gt d23 and D is a large enough number

pblowast ltpslowast hblowast lt hslowast sblowast sslowast πblowastm lt πslowast

m πblowaste lt πslowast

e andπblowast lt πslowast Proposition 1 holds

C Proof of Proposition 2

(1) Because ssflowast d2 middot (ρk) middot (d23 minus td12td1 minus d2

3) thereis no θ in the expression of ssflowast So ssflowast is inde-pendent of θ

(2) Since (ddθ)(θ1 + θ) (1(1 + θ)2)gt 0 (θ1 + θ)

increases with θ+us

psflowast

(c + ρ) + d23 minus td11113872 1113873 middot

ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)

decreases with θ(3) As (θ1 + θ)increases with θ

hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)

decreases with θ(4) Let f(θ) (θ1 + θ) then πsflowast

m can be expressed as

πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)

where A represents the items independent of θSo πsflowast

m is a downward parabola with f If θ gt 0 f willbe on the right of the symmetry axis of this parabola

Complexity 15

which is the falling part +erefore πsflowastm decreases

with f Furthermore πsflowastm decreases with θ

(5) As (θ1 + θ) increases with θ

πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)

decreases with θ(6) Since both πsflowast

m and πsflowaste decrease with θ so does

πsflowast

D Proof of Proposition 3

(1) Since sslowast ssflowast and sclowast is the platformrsquos optimalservice level in the centralized decision modelsclowast gt sslowast ssflowast

sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)

(2) +us sslowast and ssflowast are direct proportional to ρ +at isto say both sslowast and ssflowast increase with ρ

(3) In the expression of sclowast d2 middot (t(D minus cd1)+

Td3)(2ktd1 minus (td22 + kd2

3)) td2(D minus cd1) + Td2d3increases with d3 while 2ktd1 minus (td2

2 + kd23) de-

creases with d3 +us sclowast increases with d3

Also (dsslowastdd3) (dssflowastdd3) d2 middot ρk middot ((8d1d3)(d2

3 minus 8d1)2)gt 0 +erefore sslowastand ssflowast increase with d3

E Proof of Proposition 7

Since

Usflowastm

(1 + θ)Td1 + (1 + θ) D minus cd1 minus ρd1( 1113857 minus ρδθd11113858 1113859d3

2td1 minus d23

middot T +(1 + θ) D minus cd1 minus ρd1( 1113857 minus ρδθd11113858 1113859

2

2(1 + θ) 2td1 minus d231113872 1113873

middot t +δθ ρd2( 1113857

2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ

Td1 + d3 D minus cd1 minus ρd1( 1113857 minus d1ρδ1113858 1113859

2td1 minus d23

middot T

+D(1 + θ) minus d1(1 + θ)(ρ + c) minus d1ρδθ1113858 1113859 middot D(1 + θ) minus d1(1 + θ)(ρ + c) minus d1(ρδθ + 2δ)1113858 1113859

2(1 + θ)2 2td1 minus d

231113872 1113873

middot t +δ2k

middot ρd2( 11138572

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

T middotTd1 + d3 D minus cd1 minus ρd1( 1113857 minus d1ρδ1113858 1113859

2td1 minus d23

+ t middotD(1 + θ) minus d1(1 + θ)(ρ + c) minus d1ρδθ1113858 1113859 middot D(1 + θ) minus d1(1 + θ)(ρ + c) minus d1(ρδθ + 2ρδ)1113858 1113859


231113872 1113873

+ρd2( 1113857

2

2kmiddotkδ 2td1 minus d

231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gtT middotTd1 + d3 D minus cd1 minus ρd1( 1113857 minus d1ρδ1113858 1113859

2td1 minus d23



231113872 1113873

+ρd2( 1113857

2

2kmiddot2ktd1 minus kd

23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity

+us Usflowastm increases as θ increases

F Proof of Theorem 5

In the joint allocation contract of cost and profit if(ρ x y z) satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)

the manufacturerrsquos profit is expressed as

πm ϕ(pq minus cq) minus ϕth

2

2minus ϕ

ks2

2+ ϕTh

ϕ (p minus c)q minusth

2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)

And the platformrsquos profit is

πe (1 minus ϕ)(pq minus cq) minus (1 minus ϕ)th

2

2minus (1 minus ϕ)

ks2

2

+(1 minus ϕ)Th (1 minus ϕ) (p minus c)q minusth

2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)

Obviously the profit of the manufacturer and platform isthe affine function of that of the supply chain +ereforeafter this contract is applied the supply chain system mustbe coordinated

Data Availability

+e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is paper was supported financially by the National NaturalScience Foundation of China (71971129 and 71501111)Science and Technology Support Program for Youth In-novation of Colleges and Universities in Shandong Province(2019RWG017) and Philosophy and Social Science Projectof Shandong Province (19BJCJ12)

References

[1] X Zhu M Ren W Chu and R Chiong ldquoRemanufacturingsubsidy or carbon regulation An alternative toward sus-tainable productionrdquo Journal of Cleaner Production vol 239p 15 Article ID 117988 2019

[2] J Li W Du F Yang and G Hua ldquo+e carbon subsidyanalysis in remanufacturing closed-Loop supply chainrdquoSustainability vol 6 no 6 pp 3861ndash3877 2014

[3] F Qin F Mai M J Fry and A S Raturi ldquoSupply-chainperformance anomalies fairness concerns under private costinformationrdquo European Journal of Operational Researchvol 252 no 1 pp 170ndash182 2016

[4] T Li J Xie X Zhao and J Tang ldquoOn supplier encroachmentwith retailerrsquos fairness concernsrdquo Computers amp IndustrialEngineering vol 98 pp 499ndash512 2016

[5] P Jokela and A Soderman ldquoRe-examining the link betweenfairness and commitment in buyer-supplier relationshipsrdquoJournal of Purchasing and Supply Management vol 23 no 4pp 268ndash279 2017

[6] J Xie L Liang L Liu and P Ieromonachou ldquoCoordinationcontracts of dual-channel with cooperation advertising inclosed-loop supply chainsrdquo International Journal of Produc-tion Economics vol 183 pp 528ndash538 2017

[7] Q Lu and N Liu ldquoPricing games of mixed conventional ande-commerce distribution channelsrdquo Computers amp IndustrialEngineering vol 64 no 1 pp 122ndash132 2013

[8] Q Lu and N Liu ldquoEffects of e-commerce channel entry in atwo-echelon supply chain a comparative analysis of single-and dual-channel distribution systemsrdquo International Journalof Production Economics vol 165 pp 100ndash111 2015

[9] Z Liu K W Li B-Y Li J Huang and J Tang ldquoImpact ofproduct-design strategies on the operations of a closed-loopsupply chainrdquo Transportation Research Part E Logistics andTransportation Review vol 124 pp 75ndash91 2019

[10] J Tang B Y Li K W Li Z Liu and J Huang ldquoPricing andwarranty decisions in a two-period closed-loop supply chainrdquoInternational Journal of Production Research vol 58 no 6pp 1688ndash1704 2020

[11] Y Su and W Sun ldquoAnalyzing a closed-loop supply chainconsidering environmental pollution using the NSGA-IIrdquoIEEE Transactions on Fuzzy Systems vol 27 no 5pp 1066ndash1074 2019

[12] S-M Hosseini-Motlagh S Ebrahimi and A Jokar ldquoSus-tainable supply chain coordination under competition andgreen effort schemerdquo Journal of the Operational ResearchSociety vol 2019 p 16 Article ID 1671152 2019

[13] J Jian Y Guo L Jiang Y An and J Su ldquoA multi-objectiveoptimization model for green supply chain considering en-vironmental benefitsrdquo Sustainability vol 11 no 21 p 59112019

[14] S-M Hosseini-Motlagh M Johari and R ZirakpourdehkordildquoGrain production management to reduce global warmingpotential under financial constraints and time value of moneyusing evolutionary game theoryrdquo International Journal ofProduction Research vol 2020 Article ID 1773562 22 pages2020

[15] 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

[16] K Lamba S P Singh and N Mishra ldquoIntegrated decisionsfor supplier selection and lot-sizing considering differentcarbon emission regulations in big data environmentrdquoComputers amp Industrial Engineering vol 128 pp 1052ndash10622019

[17] H Peng and T Pang ldquoOptimal strategies for a three-levelcontract-farming supply chain with subsidyrdquo InternationalJournal of Production Economics vol 216 pp 274ndash286 2019

Complexity 17

[18] Z Li J Zhang Q Meng W Zheng and J Du ldquoInfluence ofgovernment subsidy on remanufacturing decision underdifferent market modelsrdquo Mathematical Problems in Engi-neering vol 2019 Article ID 9460315 16 pages 2019

[19] J Li W Du F Yang and G Hua ldquo+e carbon subsidyanalysis in remanufacturing closed-loop supply chainrdquo Sus-tainability vol 6 no 6 pp 3861ndash3877 2014

[20] H Liu M Lei H Deng G Keong Leong and T Huang ldquoAdual channel quality-based price competition model for theWEEE recycling market with government subsidyrdquo Omegavol 59 pp 290ndash302 2016

[21] L Deng W Li and Z Liu ldquoDifferent subsidiesrsquo impact onequilibrium decision-making of closed-loop supply chainrdquoJournal of Industrial Engineering and Management vol 7no 5 pp 1061ndash1075 2014

[22] D Kahneman J L Knetsch and R H +aler ldquoFairness andthe assumptions of economicsrdquo Journal of Business vol 59no 4 pp 285ndash300 1986

[23] X Qin Q Su S H Huang U J Wiersma and M LiuldquoService quality coordination contracts for online shoppingservice supply chain with competing service providers in-tegrating fairness and individual rationalityrdquo OperationalResearch vol 19 no 1 pp 269ndash296 2019

[24] L Dai X Wang X Liu and L Wei ldquoPricing strategies indual-channel supply chain with a fair caring retailerrdquo Com-plexity vol 2019 p 23 Article ID 1484372 2019

[25] Z Guan T Ye and R Yin ldquoChannel coordination underNash bargaining fairness concerns in differential games ofgoodwill accumulationrdquo European Journal of OperationalResearch vol 285 no 3 pp 916ndash930 2020

[26] X-X Zheng Z Liu K W Li J Huang and J Chen ldquoCo-operative game approaches to coordinating a three-echelonclosed-loop supply chain with fairness concernsrdquo Interna-tional Journal of Production Economics vol 212 pp 92ndash1102019

[27] Q Li and B Li ldquoDual-channel supply chain equilibriumproblems regarding retail services and fairness concernsrdquoApplied Mathematical Modelling vol 40 no 15-16pp 7349ndash7367 2016

[28] T H Cui J S Raju and Z J Zhang ldquoFairness and channelcoordinationrdquo Management Science vol 53 no 8pp 1303ndash1314 2007

[29] D Zhao X Zhang T Ren and H Fu ldquoOptimal pricingstrategies in a product and service supply chain with extendedwarranty service competition considering retailer fairnessconcernrdquo Mathematical Problems in Engineering vol 2019p 15 Article ID 8657463 2019

[30] Q Li X Chen and Y Huang ldquo+e stability and complexityanalysis of a low-carbon supply chain considering fairnessconcern behavior and sales servicerdquo International Journal ofEnvironmental Research and Public Health vol 16 no 15p 2711 2019

[31] Q Li Y Zhang and Y Huang ldquo+e complexity analysis indual-channel supply chain based on fairness concern anddifferent business objectivesrdquo Complexity vol 2018 p 13Article ID 4752765 2018

[32] Q Han and Y Wang ldquoldquoDecision and coordination in a low-carbon e-supply chain considering the manufacturerrsquos carbonemission reduction behaviorrdquo Sustainability vol 10 no 5pp 1ndash23 2018

[33] S-M Hosseini-Motlagh M Nouri-Harzvili M Johari andB R Sarker ldquoCoordinating economic incentives customerservice and pricing decisions in a competitive closed-loop

supply chainrdquo Journal of Cleaner Production vol 255 p 41Article ID 120241 2020

[34] M Johari and S-M Hosseini-Motlagh ldquoCoordination ofsocial welfare collecting recycling and pricing decisions in acompetitive sustainable closed-loop supply chain a case forlead-acid batteryrdquo Annals of Operations Research vol 2019Article ID 3292-1 36 pages 2019

[35] M Johari S-M Hosseini-Motlagh and M NematollahildquoSimultaneous coordination of review period and order-up-to-level in a manufacturer-retailer chainrdquo Journal of Indus-trial and Systems Engineering vol 10 pp 1ndash17 2017

[36] S-M Hosseini-Motlagh M R Nematollahi M Johari andT-M Choi ldquoReverse supply chain systems coordinationacross multiple links with duopolistic third party collectorsrdquoIEEE Transactions on Systems Man and Cybernetics Systemsvol 2019 p 12 Article ID 2911644 2019

[37] S-MHosseini-Motlagh S Ebrahimi and R ZirakpourdehkordildquoCoordination of dual-function acquisition price and corporatesocial responsibility in a sustainable closed-loop supply chainrdquoJournal of Cleaner Production vol 251 p 28 Article ID 1196292020

[38] S-M Hosseini-Motlagh M Nematollahi M Johari andB R Sarker ldquoA collaborative model for coordination ofmonopolistic manufacturerrsquos promotional efforts and com-peting duopolistic retailersrsquo trade creditsrdquo InternationalJournal of Production Economics vol 204 pp 108ndash122 2018

[39] M Johari S-M Hosseini-Motlagh M Nematollahi M Gohand J Ignatius ldquoBi-level credit period coordination for pe-riodic review inventory system with price-credit dependentdemand under time value of moneyrdquo Transportation ResearchPart E Logistics and Transportation Review vol 114pp 270ndash291 2018

[40] J Su C Li Q Zeng J Yang and J Zhang ldquoA green closed-loop supply chain coordination mechanism based on third-party recyclingrdquo Sustainability vol 11 no 19 p 5335 2019

[41] J Heydari K Govindan and A Jafari ldquoReverse and closedloop supply chain coordination by considering governmentrolerdquo Transportation Research Part D Transport and Envi-ronment vol 52 pp 379ndash398 2017

[42] H Peng T Pang and J Cong ldquoCoordination contracts for asupply chain with yield uncertainty and low-carbon prefer-encerdquo Journal of Cleaner Production vol 205 pp 291ndash3022018

[43] T Nie and S Du ldquoDual-fairness supply chain with quantitydiscount contractsrdquo European Journal of Operational Re-search vol 258 no 2 pp 491ndash500 2017

[44] X Liu X Wang L Dai and Y Pan ldquoResearch on supplychain performance based on retailersrsquo fairness concernswholesale prices versus cost sharing of effortsrdquo PLoS Onevol 13 no 10 Article ID e0204482 2018

[45] J Jian Y Zhang L Jiang and J Su ldquoCoordination of supplychains with competing manufacturers considering fairnessconcernsrdquo Complexity vol 2020 Article ID 437260315 pages 2020

[46] O Caliskan-Demirag Y Chen and J Li ldquoChannel coordi-nation under fairness concerns and nonlinear demandrdquoEuropean Journal of Operational Research vol 207 no 3pp 1321ndash1326 2010

[47] X-X Zheng D-F Li Z Liu F Jia and J-B Sheu ldquoCoor-dinating a closed-loop supply chain with fairness concernsthrough variable-weighted Shapley valuesrdquo TransportationResearch Part E Logistics and Transportation Review vol 126pp 227ndash253 2019

18 Complexity

[48] Y Wang Z Yu and L Shen ldquoStudy on the decision-makingand coordination of an e-commerce supply chain withmanufacturer fairness concernsrdquo International Journal ofProduction Research vol 57 no 9 pp 2788ndash2808 2019

[49] A Nair and R Narasimhan ldquoDynamics of competing withquality- and advertising-based goodwillrdquo European Journal ofOperational Research vol 175 no 1 pp 462ndash474 2006

[50] D Yao and J Liu ldquoCompetitive pricing of mixed retail ande-tail distribution channelsrdquo Omega vol 33 no 3pp 235ndash247 2005

[51] B Shen R Qian and T-M Choi ldquoSelling luxury fashiononline with social influences considerations demand changesand supply chain coordinationrdquo International Journal ofProduction Economics vol 185 pp 89ndash99 2017

[52] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo (e Quarterly Journal of Economicsvol 114 no 3 pp 817ndash868 1999

[53] E Katok T Olsen and V Pavlov ldquoWholesale pricing undermild and privately known concerns for fairnessrdquo Productionand OperationsManagement vol 23 no 2 pp 285ndash302 2014

Complexity 19

high (httpswwwcnbccom20191129black-friday-online-sales-up-19percent-by-9-am-thanksgiving-sales-hit-record-onlinehtml) total sales of Tmall Double 11 Shopping Festivalpulled in USD 384 billion from 220 countries and regionsworldwide and 299merchants have total revenues of over USD143 million each (httpsalibabagroupcomennewsarticlenewsp191112) More and more manufacturers sell productsthrough e-commerce platforms and these manufacturers aremainly small and medium sized For example in 2017 morethan 20000 Amazon online sellers across the world exceededUSD 1 million in sales (httpswwwmarketplacepulsecomarticlestwenty-thousand-amazon-millionaire-sellers)

+e e-commerce platform provides manufacturers withnew sale channels to sell low-carbon products+erefore theplatform and low-carbon product manufacturers form alow-carbon e-commerce supply chain +is cooperationseems to be mutually beneficial However there are stillsome discordant factors in the operation process +ee-commerce supply chain is very different from the tradi-tional one that the platform because of its vast networktraffic becomes the leader +e platform formulates rules formanufacturersrsquo entry and controls the information flowgoods flow and capital flow In contrast manufacturers notonly have to pay a specific fixed fee to the platform forentering it but also share the sales profit as a commissionwith the platform according to an established proportion Inpractice Tmall JDcom and Dangdangcom charge theirsettled manufactures in this way Moreover as a sharedmedium the platform usually attracts many manufacturerssuch that it can achieve substantial profit so sometimeswould ignore the manufacturersrsquo interests by using itsdominant position In particular small- and medium-sizedmanufacturers are even more affected For example duringthe 6ndash18 promotion of 2017 many manufacturers haveexpressed strong dissatisfaction with Tmall and JDcombecause these sellers were forced to participate in large-scalediscount activities (httpswwwchinainternetwatchcom29999double-11-2019) As a result the decline in statusand the gap in profitability in the supply chain has ledmanufacturers to reexamine their relationship with theplatform which is called fairness concerns +e larger theplatformrsquos size the more significant the income gap and thestronger the concerns of small- and medium-sized manu-facturers in the platform

+is kind of fairness concerns of manufacturers is aninstinct response protecting their interests +en when theyhave fairness concerns how do manufacturers react to theplatform and what impact will they have on the decision ofthe low-carbon e-commerce supply chain system Whengovernment subsidies and fairness concerns coexist whatwill happen to the decision of the supply chain systemWhen manufacturers have fairness concerns how could thesystem coordination be achieved+e current research fruitsare still not able to address the above problem well Atpresent the research about fairness concern behavior ismainly focused on the traditional offline supply chain [3ndash5]Even if the online sale is considered it is only regarded asone of the distribution channels which does not reflect thekey features of a platform [6ndash8] As the leader of a low-

carbon e-commerce supply chain the platform is respon-sible for providing consumers with logistics informationand other services besides general services and consumersdemand much more from the platform than from offlinesellers +erefore the operation of the platform requirescosts to maintain and it is also necessary to take the plat-formrsquos service cost and service level into account whenmaking decisions Moreover manufacturersrsquo carbon sub-sidies obtained from the government make manufacturersrealize the importance of their carbon emission reductionbehavior which will intensify manufacturersrsquo desire forfairness +ese are rarely addressed in the existing research

+is paper will study how the low-carbon e-commercesupply chain will make decisions and be coordinated if themanufacturer receives carbon subsidies from the govern-ment and has fairness concerns Our contributions are asfollows

(1) To determine whether the carbon subsidies are ef-fective to promote the low-carbon products by an-alyzing the effect of the carbon subsidies on themanufacturerrsquos sales price and carbon emission re-duction level the platformrsquos service level and theprofit of the low-carbon e-commerce supply chain

(2) To explore the transmission mechanisms of themanufacturerrsquos fairness concerns by analyzing thedouble effects of both carbon subsidies and themanufacturerrsquos fairness concerns on the service levelthe sales price and the profit

(3) To coordinate the supply chain with governmentcarbon subsidies and fairness concerns by designingthe joint allocation contract of cost and profit toeliminate the double marginalization under decen-tralized decision modes

+e rest of this paper is organized as follows Literaturereview is given in Section 2 Some basic work for the modelsis provided in Section 3 We model four different decisionmodes in Section 4 and then analyze and compare differentdecision results in Section 5 In Section 6 the coordinationmechanism for the supply chain system is designed InSection 7 we do the example analysis In Section 8 we offerconcluding comments

2 Related Literature

+is paper mainly involves such three streams of literatureincentive mechanism of government carbon subsidiessupply chain fairness concerns and supply chaincoordination

21 Incentive Mechanism of Government Carbon SubsidiesFor protecting our environment researches about closed-loop supply chain have been conducted [9ndash11] As carbonemission reduction is concerned manufacturers have adirect relation to assume this responsibility and also manageto invest in this area [12] In fact economic and environ-mental benefits can be achieved simultaneously [13] andexternal factors could do something to guide the

2 Complexity








Complexity 3










Cm(h) th

2

2 (1)






Ce(s) ks

2

2 (3)





4 Complexity





2

2 (4)


πe ρq minusks

2

2 (5)



2

2minus

ks2

2 (6)



pblowast


2td1 minus d23

1113888 1113889

2

middotρk


22

k 2td1 minus d231113872 1113873

middot t

(7)


hblowast


middotd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(8)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (9)


πblowastm

t

2middot

D minus d1(c + ρ)1113858 11138592

2td1 minus d23

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(10)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(11)


πblowast t middot


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(12)




2

2 (13)


πe ρq minusks

2

2 (14)



Complexity 5



2

2minus

ks2

2 (15)



pslowast

(c + ρ) +ρd1

kmiddot

td2

2td1 minus d23

1113888 1113889

2


22

k 2td1 minus d231113872 1113873

(16)


hslowast

ρk

middotd1d3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3

k


22

2td1 minus d23

(17)


sslowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (18)


πslowastm


+ 2d3T D minus d1(c + ρ)1113858 1113859 + 2d1T2

2 2td1 minus d231113872 1113873

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(19)



tρ D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(20)


πslowast


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(21)



m πblowast







6 Complexity







2

21113888 1113889 + θδ

ks2

2minus ρq1113888 1113889

(23)




psflowast


ρk

middotd2

2td1 minus d23

1113888 1113889

2

+t D minus d1(c + ρ)1113858 1113859 + d3T

2td1 minus d23

minus δρ middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

(24)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(25)


ssflowast

d2 middotρk

middotd23 minus td1

2tβ minus d23 (26)


Usflowastm T middot


2td1 minus d23

+t

2(1 + θ)middot


2td1 minus d23

+ ρd2( 11138572

middotδθ2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2td1 minus d23

middot1 + θ2k


2td1 minus d23

1113888 1113889

2

(27)


πsflowaste

ρd2( 11138572

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minusθ

1 + θmiddott(1 + δ) ρd1( 1113857

2

2td1 minus d23

(28)


πsflowastm


2d1

2td1 minus d23

+ ρ2d22 middot

δk

middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minusθ

1 + θ1113888 1113889

2

middottδ2ρ2d2

1

2 2td1 minus d231113872 1113873

(29)


Complexity 7

πsflowast

t D minus d1(c + ρ)1113858 11138592


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2


θ1 + θ


2td1 minus d23

1113888 1113889

2

minustd

22 ρd2( 1113857

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

θ1 + θ

middotd1ρ( 1113857

2

2td1 minus d23

2(1 + δ) + δ2θ

1 + θ1113890 1113891

(30)








2

2+

ks2

21113888 1113889 (31)



pclowast


22 + kd

231113872 1113873


231113872 1113873

(32)


hclowast


221113872 1113873


231113872 1113873

(33)


sclowast



231113872 1113873

(34)


πclowast


+ T2 2kd1 minus d

221113872 1113873 + 2kTd3 D minus cd1( 1113857


231113872 1113873

(35)








8 Complexity















Complexity 9


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19








Complexity 3










Cm(h) th

2

2 (1)






Ce(s) ks

2

2 (3)





4 Complexity





2

2 (4)


πe ρq minusks

2

2 (5)



2

2minus

ks2

2 (6)



pblowast


2td1 minus d23

1113888 1113889

2

middotρk


22

k 2td1 minus d231113872 1113873

middot t

(7)


hblowast


middotd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(8)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (9)


πblowastm

t

2middot

D minus d1(c + ρ)1113858 11138592

2td1 minus d23

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(10)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(11)


πblowast t middot


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(12)




2

2 (13)


πe ρq minusks

2

2 (14)



Complexity 5



2

2minus

ks2

2 (15)



pslowast

(c + ρ) +ρd1

kmiddot

td2

2td1 minus d23

1113888 1113889

2


22

k 2td1 minus d231113872 1113873

(16)


hslowast

ρk

middotd1d3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3

k


22

2td1 minus d23

(17)


sslowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (18)


πslowastm


+ 2d3T D minus d1(c + ρ)1113858 1113859 + 2d1T2

2 2td1 minus d231113872 1113873

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(19)



tρ D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(20)


πslowast


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(21)



m πblowast







6 Complexity







2

21113888 1113889 + θδ

ks2

2minus ρq1113888 1113889

(23)




psflowast


ρk

middotd2

2td1 minus d23

1113888 1113889

2

+t D minus d1(c + ρ)1113858 1113859 + d3T

2td1 minus d23

minus δρ middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

(24)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(25)


ssflowast

d2 middotρk

middotd23 minus td1

2tβ minus d23 (26)


Usflowastm T middot


2td1 minus d23

+t

2(1 + θ)middot


2td1 minus d23

+ ρd2( 11138572

middotδθ2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2td1 minus d23

middot1 + θ2k


2td1 minus d23

1113888 1113889

2

(27)


πsflowaste

ρd2( 11138572

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minusθ

1 + θmiddott(1 + δ) ρd1( 1113857

2

2td1 minus d23

(28)


πsflowastm


2d1

2td1 minus d23

+ ρ2d22 middot

δk

middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minusθ

1 + θ1113888 1113889

2

middottδ2ρ2d2

1

2 2td1 minus d231113872 1113873

(29)


Complexity 7

πsflowast

t D minus d1(c + ρ)1113858 11138592


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2


θ1 + θ


2td1 minus d23

1113888 1113889

2

minustd

22 ρd2( 1113857

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

θ1 + θ

middotd1ρ( 1113857

2

2td1 minus d23

2(1 + δ) + δ2θ

1 + θ1113890 1113891

(30)








2

2+

ks2

21113888 1113889 (31)



pclowast


22 + kd

231113872 1113873


231113872 1113873

(32)


hclowast


221113872 1113873


231113872 1113873

(33)


sclowast



231113872 1113873

(34)


πclowast


+ T2 2kd1 minus d

221113872 1113873 + 2kTd3 D minus cd1( 1113857


231113872 1113873

(35)








8 Complexity















Complexity 9


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19










Cm(h) th

2

2 (1)






Ce(s) ks

2

2 (3)





4 Complexity





2

2 (4)


πe ρq minusks

2

2 (5)



2

2minus

ks2

2 (6)



pblowast


2td1 minus d23

1113888 1113889

2

middotρk


22

k 2td1 minus d231113872 1113873

middot t

(7)


hblowast


middotd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(8)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (9)


πblowastm

t

2middot

D minus d1(c + ρ)1113858 11138592

2td1 minus d23

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(10)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(11)


πblowast t middot


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(12)




2

2 (13)


πe ρq minusks

2

2 (14)



Complexity 5



2

2minus

ks2

2 (15)



pslowast

(c + ρ) +ρd1

kmiddot

td2

2td1 minus d23

1113888 1113889

2


22

k 2td1 minus d231113872 1113873

(16)


hslowast

ρk

middotd1d3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3

k


22

2td1 minus d23

(17)


sslowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (18)


πslowastm


+ 2d3T D minus d1(c + ρ)1113858 1113859 + 2d1T2

2 2td1 minus d231113872 1113873

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(19)



tρ D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(20)


πslowast


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(21)



m πblowast







6 Complexity







2

21113888 1113889 + θδ

ks2

2minus ρq1113888 1113889

(23)




psflowast


ρk

middotd2

2td1 minus d23

1113888 1113889

2

+t D minus d1(c + ρ)1113858 1113859 + d3T

2td1 minus d23

minus δρ middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

(24)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(25)


ssflowast

d2 middotρk

middotd23 minus td1

2tβ minus d23 (26)


Usflowastm T middot


2td1 minus d23

+t

2(1 + θ)middot


2td1 minus d23

+ ρd2( 11138572

middotδθ2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2td1 minus d23

middot1 + θ2k


2td1 minus d23

1113888 1113889

2

(27)


πsflowaste

ρd2( 11138572

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minusθ

1 + θmiddott(1 + δ) ρd1( 1113857

2

2td1 minus d23

(28)


πsflowastm


2d1

2td1 minus d23

+ ρ2d22 middot

δk

middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minusθ

1 + θ1113888 1113889

2

middottδ2ρ2d2

1

2 2td1 minus d231113872 1113873

(29)


Complexity 7

πsflowast

t D minus d1(c + ρ)1113858 11138592


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2


θ1 + θ


2td1 minus d23

1113888 1113889

2

minustd

22 ρd2( 1113857

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

θ1 + θ

middotd1ρ( 1113857

2

2td1 minus d23

2(1 + δ) + δ2θ

1 + θ1113890 1113891

(30)








2

2+

ks2

21113888 1113889 (31)



pclowast


22 + kd

231113872 1113873


231113872 1113873

(32)


hclowast


221113872 1113873


231113872 1113873

(33)


sclowast



231113872 1113873

(34)


πclowast


+ T2 2kd1 minus d

221113872 1113873 + 2kTd3 D minus cd1( 1113857


231113872 1113873

(35)








8 Complexity















Complexity 9


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19





2

2 (4)


πe ρq minusks

2

2 (5)



2

2minus

ks2

2 (6)



pblowast


2td1 minus d23

1113888 1113889

2

middotρk


22

k 2td1 minus d231113872 1113873

middot t

(7)


hblowast


middotd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(8)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (9)


πblowastm

t

2middot

D minus d1(c + ρ)1113858 11138592

2td1 minus d23

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(10)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(11)


πblowast t middot


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(12)




2

2 (13)


πe ρq minusks

2

2 (14)



Complexity 5



2

2minus

ks2

2 (15)



pslowast

(c + ρ) +ρd1

kmiddot

td2

2td1 minus d23

1113888 1113889

2


22

k 2td1 minus d231113872 1113873

(16)


hslowast

ρk

middotd1d3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3

k


22

2td1 minus d23

(17)


sslowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (18)


πslowastm


+ 2d3T D minus d1(c + ρ)1113858 1113859 + 2d1T2

2 2td1 minus d231113872 1113873

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(19)



tρ D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(20)


πslowast


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(21)



m πblowast







6 Complexity







2

21113888 1113889 + θδ

ks2

2minus ρq1113888 1113889

(23)




psflowast


ρk

middotd2

2td1 minus d23

1113888 1113889

2

+t D minus d1(c + ρ)1113858 1113859 + d3T

2td1 minus d23

minus δρ middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

(24)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(25)


ssflowast

d2 middotρk

middotd23 minus td1

2tβ minus d23 (26)


Usflowastm T middot


2td1 minus d23

+t

2(1 + θ)middot


2td1 minus d23

+ ρd2( 11138572

middotδθ2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2td1 minus d23

middot1 + θ2k


2td1 minus d23

1113888 1113889

2

(27)


πsflowaste

ρd2( 11138572

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minusθ

1 + θmiddott(1 + δ) ρd1( 1113857

2

2td1 minus d23

(28)


πsflowastm


2d1

2td1 minus d23

+ ρ2d22 middot

δk

middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minusθ

1 + θ1113888 1113889

2

middottδ2ρ2d2

1

2 2td1 minus d231113872 1113873

(29)


Complexity 7

πsflowast

t D minus d1(c + ρ)1113858 11138592


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2


θ1 + θ


2td1 minus d23

1113888 1113889

2

minustd

22 ρd2( 1113857

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

θ1 + θ

middotd1ρ( 1113857

2

2td1 minus d23

2(1 + δ) + δ2θ

1 + θ1113890 1113891

(30)








2

2+

ks2

21113888 1113889 (31)



pclowast


22 + kd

231113872 1113873


231113872 1113873

(32)


hclowast


221113872 1113873


231113872 1113873

(33)


sclowast



231113872 1113873

(34)


πclowast


+ T2 2kd1 minus d

221113872 1113873 + 2kTd3 D minus cd1( 1113857


231113872 1113873

(35)








8 Complexity















Complexity 9


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19



2

2minus

ks2

2 (15)



pslowast

(c + ρ) +ρd1

kmiddot

td2

2td1 minus d23

1113888 1113889

2


22

k 2td1 minus d231113872 1113873

(16)


hslowast

ρk

middotd1d3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3

k


22

2td1 minus d23

(17)


sslowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (18)


πslowastm


+ 2d3T D minus d1(c + ρ)1113858 1113859 + 2d1T2

2 2td1 minus d231113872 1113873

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(19)



tρ D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(20)


πslowast


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

d42

2td1 minus d23

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(21)



m πblowast







6 Complexity







2

21113888 1113889 + θδ

ks2

2minus ρq1113888 1113889

(23)




psflowast


ρk

middotd2

2td1 minus d23

1113888 1113889

2

+t D minus d1(c + ρ)1113858 1113859 + d3T

2td1 minus d23

minus δρ middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

(24)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(25)


ssflowast

d2 middotρk

middotd23 minus td1

2tβ minus d23 (26)


Usflowastm T middot


2td1 minus d23

+t

2(1 + θ)middot


2td1 minus d23

+ ρd2( 11138572

middotδθ2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2td1 minus d23

middot1 + θ2k


2td1 minus d23

1113888 1113889

2

(27)


πsflowaste

ρd2( 11138572

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minusθ

1 + θmiddott(1 + δ) ρd1( 1113857

2

2td1 minus d23

(28)


πsflowastm


2d1

2td1 minus d23

+ ρ2d22 middot

δk

middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minusθ

1 + θ1113888 1113889

2

middottδ2ρ2d2

1

2 2td1 minus d231113872 1113873

(29)


Complexity 7

πsflowast

t D minus d1(c + ρ)1113858 11138592


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2


θ1 + θ


2td1 minus d23

1113888 1113889

2

minustd

22 ρd2( 1113857

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

θ1 + θ

middotd1ρ( 1113857

2

2td1 minus d23

2(1 + δ) + δ2θ

1 + θ1113890 1113891

(30)








2

2+

ks2

21113888 1113889 (31)



pclowast


22 + kd

231113872 1113873


231113872 1113873

(32)


hclowast


221113872 1113873


231113872 1113873

(33)


sclowast



231113872 1113873

(34)


πclowast


+ T2 2kd1 minus d

221113872 1113873 + 2kTd3 D minus cd1( 1113857


231113872 1113873

(35)








8 Complexity















Complexity 9


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19







2

21113888 1113889 + θδ

ks2

2minus ρq1113888 1113889

(23)




psflowast


ρk

middotd2

2td1 minus d23

1113888 1113889

2

+t D minus d1(c + ρ)1113858 1113859 + d3T

2td1 minus d23

minus δρ middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

(24)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(25)


ssflowast

d2 middotρk

middotd23 minus td1

2tβ minus d23 (26)


Usflowastm T middot


2td1 minus d23

+t

2(1 + θ)middot


2td1 minus d23

+ ρd2( 11138572

middotδθ2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2td1 minus d23

middot1 + θ2k


2td1 minus d23

1113888 1113889

2

(27)


πsflowaste

ρd2( 11138572

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minusθ

1 + θmiddott(1 + δ) ρd1( 1113857

2

2td1 minus d23

(28)


πsflowastm


2d1

2td1 minus d23

+ ρ2d22 middot

δk

middotθ

1 + θmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minusθ

1 + θ1113888 1113889

2

middottδ2ρ2d2

1

2 2td1 minus d231113872 1113873

(29)


Complexity 7

πsflowast

t D minus d1(c + ρ)1113858 11138592


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2


θ1 + θ


2td1 minus d23

1113888 1113889

2

minustd

22 ρd2( 1113857

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

θ1 + θ

middotd1ρ( 1113857

2

2td1 minus d23

2(1 + δ) + δ2θ

1 + θ1113890 1113891

(30)








2

2+

ks2

21113888 1113889 (31)



pclowast


22 + kd

231113872 1113873


231113872 1113873

(32)


hclowast


221113872 1113873


231113872 1113873

(33)


sclowast



231113872 1113873

(34)


πclowast


+ T2 2kd1 minus d

221113872 1113873 + 2kTd3 D minus cd1( 1113857


231113872 1113873

(35)








8 Complexity















Complexity 9


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19

πsflowast

t D minus d1(c + ρ)1113858 11138592


2 2td1 minus d231113872 1113873

+ρd2( 1113857

2


θ1 + θ


2td1 minus d23

1113888 1113889

2

minustd

22 ρd2( 1113857

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minust

2middot

θ1 + θ

middotd1ρ( 1113857

2

2td1 minus d23

2(1 + δ) + δ2θ

1 + θ1113890 1113891

(30)








2

2+

ks2

21113888 1113889 (31)



pclowast


22 + kd

231113872 1113873


231113872 1113873

(32)


hclowast


221113872 1113873


231113872 1113873

(33)


sclowast



231113872 1113873

(34)


πclowast


+ T2 2kd1 minus d

221113872 1113873 + 2kTd3 D minus cd1( 1113857


231113872 1113873

(35)








8 Complexity















Complexity 9


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19















Complexity 9


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19


th2

2minus (1 minus y)

ks2

2 (36)



th2

2minus

yks2

2 (37)


satisfies


x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(38)




πcm ge π

slowastm

πce ge π

slowaste

⎧⎨

⎩ (39)


πcm ϕπc




⎧⎨

⎩ (40)


πslowastm


πslowaste

πclowast (41)


πcm ge π

slowastm

πce ge π

slowaste

Ucm geU

sflowastm

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)


πcm ϕπc



sflowaste

Ucm πc




minus ϕπclowast



⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(43)


πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast leϕle

πclowast

minus πsflowaste

πclowast (44)


πclowast

minus πsflowaste

πclowast minus

πsflowastm

πclowast +

θδθ(δ + 1) + 1

middotπclowast

minus πsflowast

πclowast1113890 1113891

θ + 1


minus πsflowast

πclowast

(45)







10 Complexity



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19



satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(46)


satisfies

ρ 560356(1 minus ϕ)

x ϕ

y 1 minus ϕ

z ϕ

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(47)














Complexity 11






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19






25

20

15

10

05

Profi

t (times1

06 )

088 090 092 094φ

πeslowast

πeclowast

πmslowast

πmclowast

(a)

092 093 094 095φ

25

20

15

10

05

Prof

it (times

106 )

πesflowast

πeclowast

πmsflowast

πmclowast

(b)


090

088

092

094

096

098

φ

5 10 15 20


ρ

(a)

098

096

094

092

φ

5 10 15 20


ρ

(b)


12 Complexity

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19

600 800 1000 1200 1400 1600 180000612

00614

00616

00618

00620

00622

00624

00626

Span

of φ

T

(a)

00285

00280

00275

00270

00265

00260

00255

Span

of φ

600 800 1000 1200 1400 1600 1800T

(b)


006 008 010 012 014 016 018 020

093

094

095

096

097


φ

θ

(a)

120 140 160 180 200

0945

0950

0955

0960

0965

0970

0975

φ

δ


(b)


600 800 1000 1200 1400 1600 1800090

092

094

096

098

φ

T


(a)

600 800 1000 1200 1400 1600 1800

0970

0965

0960

0955

0950

0945

0940

φ

T


(b)


Complexity 13











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19











Appendix




Hm

z2πm

zp2

z2πm

zp zh

z2πm

zh zp

z2πm

zh2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

minus 2d1 d3

d3 minus t

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (A1)



pblowast


23

2td1 minus d23

hblowast


2td1 minus d23

(A2)


sblowast

d2 middotρk

middotd23 minus td1

2td1 minus d23 (A3)


14 Complexity

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19

pblowast


middottd2

2td1 minus d23

1113888 1113889

2


22ρ

k 2td1 minus d231113872 1113873

(A4)


hblowast

d1 middotρk

middotd3

tmiddot

td2

2td1 minus d23

1113888 1113889

2

+d3


22ρ

2td1 minus d23

(A5)


πblowastm t middot

D minus d1(c + ρ)1113858 11138592

2 2td1 minus d231113872 1113873

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A6)



D minus d1(c + ρ)

2td1 minus d23

+ρd2( 1113857

2

2kmiddot

d23 minus td1

2td1 minus d23

1113888 1113889

2

(A7)


πblowast t middot


2 2td1 minus d231113872 1113873

+d22ρ2

middotρk

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustd

422

middotρk

1113874 11138752

middot1

2td1 minus d23

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

(A8)



pslowast

minus pblowast

Td3

2td1 minus d23

hslowast

minus hblowast

2Td1

2td1 minus d23

sslowast

minus sblowast

ρd2

kmiddot

d23 minus td1

2td1 minus d23

minusρd2

kmiddot

d23 minus td1

2td1 minus d23

0


m Td3 D minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23


e Td1d3

2td1 minus d23




e1113872 1113873

Td3 D + d1 minus d1(c + ρ)1113858 1113859 + T

2d1

2td1 minus d23

(B1)










psflowast


ρt

kmiddot

d2

2td1 minus d23

1113888 1113889

2

+D minus d1(c + ρ)1113858 1113859t + d3T

2td1 minus d23

minusθ

1 + θmiddot

d23 minus td1

2td1 minus d23

middot δρ

(C1)


hsflowast

d3 middotρk


d2

2td1 minus d23

1113888 1113889

2

+d3 D minus d1(c + ρ)1113858 1113859 + 2Td1

2td1 minus d23


23

middotθ

1 + θ

(C2)



πsflowastm minus

td21ρ

2δ2

2 2td1 minus d231113872 1113873

f2

+δρ2d2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

f + A

(C3)



Complexity 15




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19




πsflowaste

ρ2d22

2kmiddot

d23 minus td1

2td1 minus d223

1113888 1113889

2

+ ρd1t D minus d1(c + ρ)1113858 1113859 + Td3

2td1 minus d23

minust(1 + δ) d1ρ( 1113857

2

2td1 minus d23

middotθ

1 + θ

(C4)



πsflowast



sslowast

ssflowast

d2 middotρk

middotd23 minus td1

2td1 minus d23

d2

kmiddot

d23 minus td1

2td1 minus d23

middot ρ (D1)





2 + kd23) de-





Since

Usflowastm


2td1 minus d23


2

2(1 + θ) 2td1 minus d231113872 1113873


2

2k

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minus(1 + θ)tρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

dUsflowastm

dθ


2td1 minus d23

middot T



231113872 1113873

middot t +δ2k


middotd23 minus td1

2td1 minus d23

1113888 1113889

2

minustρ2d4

2

2k2 2td1 minus d

231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


231113872 1113873 minus td

22

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2


2td1 minus d23



231113872 1113873

+ρd2( 1113857

2


23 minus td

221113872 1113873

k 2td1 minus d231113872 1113873

middotd23 minus td1

2td1 minus d23

1113888 1113889

2

gt 0

(E1)

16 Complexity





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19





x ϕ

y 1 minus ϕ

z ϕ (0lt ϕlt 1)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(F1)



2

2minus ϕ

ks2

2+ ϕTh


2

2t minus

ks2

2+ Th1113890 1113891 ϕπ

(F2)



2

2minus (1 minus ϕ)

ks2

2


2

2minus

ks2

2+ Th1113890 1113891 (1 minus ϕ)π

(F3)


Data Availability




Acknowledgments


References


















Complexity 17
































18 Complexity







Complexity 19
































18 Complexity







Complexity 19







Complexity 19

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