research article joint optimal production planning for

Research ArticleJoint Optimal Production Planning for Complex Supply ChainsConstrained by Carbon Emission Abatement Policies

Longfei He Zhaoguang Xu and Zhanwen Niu

College of Management and Economics Tianjin University Tianjin 300072 China

Correspondence should be addressed to Zhaoguang Xu zhaoguangxu022163com

Received 4 April 2014 Accepted 2 June 2014 Published 16 July 2014

Academic Editor Xiaolin Xu

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

We focus on the joint production planning of complex supply chains facing stochastic demands and being constrained by carbonemission reduction policiesWe pick two typical carbon emission reduction policies to research how emission regulation influencesthe profit and carbon footprint of a typical supply chain We use the input-output model to capture the interrelated demand linkbetween an arbitrary pair of two nodes in scenarios without or with carbon emission constraintsWe design optimization algorithmto obtain joint optimal production quantities combination for maximizing overall profit under regulatory policies respectivelyFurthermore numerical studies by featuring exponentially distributed demand compare systemwide performances in variousscenarios We build the ldquocarbon emission elasticity of profit (CEEP)rdquo index as a metric to evaluate the impact of regulatory policieson both chainwide emissions and profit Our results manifest that by facilitating the mandatory emission cap in proper installationwithin the network one can balance well effective emission reduction and associated acceptable profit lossThe outcome that CEEPindexwhen implementingCarbon emission tax is elastic implies that the scale of profit loss is greater than that of emission reductionwhich shows that this policy is less effective than mandatory cap from industry standpoint at least

1 Introduction

People around the world have gradually realized that carbondioxide emission has become a global environmental andenergy issue as discussed in the World Climate Conferenceheld in 2009 in Copenhagen [1] To cut carbon emission intocertain levels governments in many countries put forwardemission reduction targets confronting pressure both frominternationally and domestically environmental concerns [2]Therefore governments have designed a series of carbonabatement policies to serve these carbon emission reductionpurposes say mandatory cap carbon and trade carbon taxand carbon offset policies in the report of Congress of theUnited States [3] Apparently a variety of manufacturingindustries as one of the main bodies emitting carbon dioxideare definitely subject to those emission reduction regulations

On the way to confront the carbon emission regulationsfirms over a great range of industries have to take a series ofactions on reducing their own carbon emission For firmsrsquoresponse to carbon emission regulations we can observe

from practice or reports that their common actions are oftenreflected in the replacement of energy-saving equipmentadoption of low-carbon technology for new product develop-ment substitution of fossil fuels and saving in consumptionof electricity power and so forth [4] For example manufac-turers can renew their old machine with high emission byturning to low-carbon emitted equipments

Different from those aforementioned means for meetingcarbon emission constraints a few of companies are awareof the possibility in reducing emission by operations adjust-ment This is an attractive and plausible solution in the low-carbon economic environment However this kind of actionsled only by single firm is not easy to achieve the anticipatedgoals There are two reasons a single firm is usually self-interest oriented even when choosing her emission reductionoperations strategies which will be often distorted by theinteractions with her supply chain partners the other is thecarbon footprint of final product which is accumulatedthrough thewhole supply chain process over installations oneby one as well as the pipeline However the carbon footprint

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 361923 14 pageshttpdxdoiorg1011552014361923

2 Discrete Dynamics in Nature and Society

reflected in the carbon label will affect the market demanddue to consumersrsquo low-carbon awareness and in turn influ-ence each firmrsquos profit eventually

Although the government has released a couple of carbonemission regulatory policies for changing all carbon emittersrsquobehaviors especially for firms in industries we should askwhether those policies can get their initial expected effect incarbon emission control Upon the intuition one can inferthat some carbon emission regulation may not work well if itis originated from controlling single firm emission behaviorbut not from supply chain The interactions among firms inthe supply chain usually incur behaviors deviating from thegoal of thewhole system In low carbon environment firms insupply network also have different appeals when confrontingvarious regulatory carbon emission policies The low-carbonconcerns will worsen the conflict of interests among nodefirms This is the reason why we focus on supply chainproduction planning under carbon emission constraints inthis study

In this paper our goal is to study the impact of variouscarbon emission regulations on the operations and then theperformance of a complex supply chain network that isassembly system like made up of a series of nodes with con-vergent material flows to the final product In this systemeach node firmrsquos production is constrained by her own emis-sion regulatory policy say eithermandatory cap or carbon taxas we exploit afterwards In contrast to single firm productionplanning problem under low-carbon scenario we turn tofocus on amore realistic and typical manufacturing system aswe show here which attracts us to investigate howwemanagea set of manufacturing units under low-carbon era comparedwith traditional setting without low-carbon thinking Thiskind of comparison can uncover the impact of carbon emis-sion regulations on the typical supply chain network Undersystemic centralized setting we formulate the problem asconstrained nonlinear programming with 119899 dimensionalvariables

On the other hand we conduct another kind of compar-ison of performances resulted from two distinctive emissionregulatory policies respectively We intend to measure andshow the difference of two policiesrsquo transmissionmechanismsthrough the whole supply network which is designed toinspire our interest and thinking on the link of supply chainoperations and public administrative policies To achieve thistarget we induce some metric index in terms of individuallevel and system level profits and emissions

There are several streams of research related to our workFor an extensive survey in general on the operationsmanage-ment in low-carbon environment see Benjaafar et al [4] andXia et al [1] Some researchers holding the macroperspectivehave studied the relationship between carbon emissions andthe national economy see Dhakal [5] Holtz-Eakin andSelden [6] and Zhang and Cheng [7] while many otherstudies consider the evaluation and influential factors ofcarbon emissions Similarly some researches pay attentionto the relationship between macroeconomic factors andcarbon emissions Leng [8] develops a theoretical analysis ofenvironmental economics for analyzing the impact of eco-nomic factors on carbon emissions In this direction further

analysis on how economic factors affect carbon emissionswasconducted by Zhu et al [9] and Sun and Chen [10] Theyalso propose that the adjustment of the industrial structureenhancement of energy efficiency and transformation of theenergy structure are among the emphases of energy conser-vation Whereas being obviously different from the above-mentioned literatures our study focuses on microfirm andespecially the network-level organization operations consid-ering emission constraints

Another relevant stream in this area focuses on the impactof policy instruments on the carbon emission reduction andthese instruments can be summarized asmandatory emissioncaps taxes on emissions cap-and-trade and cap-and-offsetsee Cropper and Oates [11] Hepburn [12] Webster et al [13]and Bureau [14] While these researches study the impact ofpublic policies on the carbon emission reduction they do notlink operations management and public policy making forreducing carbon emission and also do not compare the effec-tiveness of carbon emission control under different policies

Another relevant stream to our work are papers thatconsider the influence of carbon emissions policies on theindustrial firmsrsquo production and inventory decisions profitsand costs and so forth Benjaafar et al [4] and Benjaafar et al[15] are among the first who proposed how to achieve the tar-geted emission reduction effect only from the perspective ofmanufacturing operations when a single firm was subject tocarbon emissions regulations Pan [16] investigates the car-bon emission reduction and production planning of a singlecompany as well as duopoly companies both under thecarbon tax and cap-and-trade policies respectively But hisstudy does not take the excess inventory generated emissionsinto account As a contrast Kang and Yoon [17] study thecase considering the emissions incurred by inventory Forfurther exploring the relation between profit and emissionreductions Song and Leng [2] exploiting classic single-cyclenewsvendor model investigate the optimal quantity orderingunder three kinds of emission regulatory policies that ismandatory emission cap emissions tax and cap-and-tradeAfterwards Chen et al [18] research a similar problem butthey adopt the conventional EOQmodel by assuming that themarket demand rate is constant and shortage is not permit-ted Although considering the comparison of performancesincurred by different policies the above researches are limitedto a single firm and therefore do not study the impact ofcarbon emission regulations on supply chains On the con-trary this is just the gap our work attempts to fill

The most highly related research stream on low-carbonsupply chain mainly focus on supply chain design and pro-duction system decisions They argue that structure of thesupply chain plays an important role in the chainwide carbonemissions so it is important to optimize the supply chaindesign to achieve the emission control goal see Cachon [19]Ramudhin et al [20] andAbdallah et al [21] Ramudhin et al[20] established a mixed integer programming model for agreen supply chain redesign On its basis Ramudhin et al[22] consider the life cycle assessment in the supply chaindesign As a further study developed by Cachon [19] he con-siders the impact of two aspects supply chain structure andoperations on carbon emissions Another research direction

Discrete Dynamics in Nature and Society 3

on production decisions of supply chain under the low-carbon policies is primarily derived by Benjaafar et al [15]They study how to optimize operations (such as procurementproduction and inventory controlling) in supply chain underemission regulations Abdallah et al [21] establish a mix inte-ger programming (MIP) model to minimize carbon emis-sions Similar to our study Zhao and Lv [23] construct amodel for some supply chain operations and develop optimalpolicy to minimize the chainwide overall carbon emissionsThe supply chain structures considered in these literatures arerelatively simple and few studies above discuss the relationbetween system profit and carbon emission of the whole sup-ply chain Being compared with these literatures the supplychain structure studied in this paper is more likely to reflectthe reality of assembly-system-like supply chain the charac-teristics of which lead to more sophisticated operations tohandle under low-carbon oriented polies

In this study we concentrate on an assembly-system-likecomplex supply chain as described in He [24] but with exis-tence of various carbon emission regulations We take bothof production relevant and inventory-incurred carbon emis-sion into account with the supply chain firm sufferingrandom demands from the system outside Combining theinput-output technology we establish a system productionplanning model of the supply chain and get the optimalproduction planning for each firm as well as the whole supplychain What is more we demonstrate the impact of the emis-sions reduction on the production decision-making And weanalyze how to adjust emissions reduction policy for publicadministration to reach the goal of carbon emissions reduc-tion Moreover we introduce the concept of ldquocarbon emis-sions elasticity of profitrdquo index to evaluate the effect ofemissions reduction policy

The rest of the paper is organized as follows We begin byaddressing the problem this study focuses on and introducenotation and assumptions In Section 3 we discuss the com-plex supply chain production planning in the setting withoutcarbon emission regulations The results here are to becompared with those in other scenarios In Sections 4 and 5we investigate the system planning problem considering typ-ical carbon emission constraints of two low-carbon policiesnamely mandatory carbon emission cap and carbon emissiontax respectively We present in these two sections how tomake the optimal production planning by considering profitoptimization and emission requirements simultaneously InSection 6 we conduct numerical experiments and associatedanalysis to enrich and examine the solutions in afore sectionsSection 7 comes by summarizing the managerial insights andpossible extensions for research in the future

2 Notation Assumptions andProblem Formulation

In this paper we consider supply chains composed of a set ofnodes firms in which between a pair of nodes is the coupleddemand according to physical structure of the final productThe integrator of the assembly supply chain makes product

Firm 1

Firm 1

Firm 2 Firm 3

Supply chainborder

Independentdemand

DependentdemandFirm j

Firm m Firm n Firm p

Firm k

Figure 1 Schematic map of complex supply chain

with multiprocess and a complex structure For research sim-plicity without loss of generality we assume that the productconsists of a set of complementary component parts eachof which is uniquely produced by one node before the assem-bly Demand relationship existing between a pair of nodesis determined by corresponding material flows which areeventually installed by final product configuration In otherwords a node firm is the customer of her preceding node aswell as the supplier of her succeeding node meanwhile

There are two kinds of demands existing for each firm inthe supply chain network which are either from other nodefirms within or from organizations beyond the boundary ofthe network Sorted by the source of the demand the formeris called dependent demand and the latter independentdemand see Jacobs and Chase [25] There are quantity cor-relations between these two types of demands according toHe et al [26] which can bemodeled by physical input-outputanalysis including the direct consumption coefficients andthe total demand coefficients The structure of the supplychain and itsmaterial flows are shown in Figure 1 as describedin literature [27]

Embodied in the supply chain there are three descrip-tions featuring the system operations (1) to fabricate a finalproduct the production quantity for each company shouldmeet a specific relationship which is determined by theconfiguration of the components in the final product (2) eachnode firm should allocate her own capacity to satisfy relativelydeterministic internal dependent demands and stochasticexternal demand simultaneously (3)we assume the existenceof an integrator as system controller to make a centralizeddecision by assigning productions distributed among allnodes to maximize total supply chain profit under carbonemission constraints In this paper we call above systemrelated demand supply chain (RDSC) that is prevalent in theindustry

The complexity ofmanaging the RDSC ismainly reflectedin two aspects (1) we should consider multinode synchro-nously to seek a global optimizer pursuing themaximizationof individual own interests each node firm has hisherown interest appeal this typical self-interested decentralizeddecision-making modes usually results in suboptimal per-formance both for its upstream and downstream companiesdeviating from optimal decisions (2) stochastic demandsbeyond and within the network boundary are intertwinedwith each other The volatility of external random demand


is passed through other nodes and then nodes are coupledtogether across the entire supply chain network which raisesthe complexity of the production planning

Moreover driven by todayrsquos harsh climate change andchallenging environmental issues many countries haveenacted a series of laws and regulations to limit carbon emis-sions Two representative policies the mandatory emissioncap and carbon tax policies are selected Specific content ofmandatory emission caps policy is that government requirescarbon dioxide emissions of companies cannot exceed a givenvalue within a given period Carbon tax policy is to levyappropriate taxes according to the amount of carbon dioxideemitted during making business products or services for thepurpose of reducing carbon emissions

In this paper the related demand supply chain weresearched faces the governmentrsquos carbon reduction policymentioned above In this situation the RDSC should adjustits production operational decisions In the condition ofmandatory emission caps policy production of each com-pany cannot be excessive otherwise it will exceed the govern-mentrsquos mandatory carbon emission limits under carbon taxpolicy if the company producesmore it will emit muchmorecarbon emissions resulting in more carbon taxes but on theother hand company can sell more products and improve itsrevenue So it is important to formulate appropriate produc-tion volume for the company in the supply chain to balancethe relationship between carbon taxes and products revenuein order to obtain maximum total profit of the supply chain

We first figure out the optimal production decisions ofnode firms to maximize the supply chain profit in the condi-tion without considering the carbon emission policy And wegive out analytical solutions of optimal production planningof dependent demand in complex supply chains Then weconsider how to adjust the production strategies of thecompany to maximize the chainwide profit in the conditionof two kinds of carbon emissions policies as we mentionedWhat is more we investigate how to use these policies toachieve good effect of emission reduction while withoutsubstantial damages to the profit of the supply chain

Parameters Notation and Assumptions Parameters used inthis paper are shown below

Parameters

119873 = 1 2 119899 collection of all companies (prod-ucts) in the supply chain119883119894 stochastic market demand of independent

demand product its probability density function is119891119894(119909119871

119894) cumulative distribution function is 119865

119894(119909119871

119894)

and mean value is 120583119894

119864119894 the actual carbon emissions of each company

119862119894 carbon emission quotas of each company

119870119894 fixed cost of each product

V119894 variable cost of per unit of product

ℎ119894 inventory cost due to the presence of residual

product

119904119894 penalty cost due to shortage

119890119898

119894 carbon dioxide emissions generated by the pro-

duction of per unit of product119864119870

119894 fixed carbon emissions of each production

119890ℎ

119894 carbon emissions of per unit remaining inventory

119901119894 unit price of each product

120596 carbon tax of per unit of carbon emissions

Decision Variables

119902119894 stock after completion of the production of each

company119876119872

= [ 119902119872

119894 ] production planning for dependent

demand of each firm119876119871= [ 119902

119871

119894 ] production planning of independent

demand for each firm119876119878

= [ 119902119878

119894 ] total production planning for each

firm 119902119878119894= 119902119871

119894+ 119902119872

119894

To make our research clear and simple without loss ofgenerality and essence we need to give out some assumptionsas follows

Assumption 1 Each node firm only produces one kind ofcomponent in supply chain

Assumption 2 The initial inventory at each node firm is zero

Assumption 3 Demand relationships between a pair of nodesin the supply chain are determined by the structure of theproduct and do not change over a single cycle

Quantitative relationships between the inventory of com-panies upon completion of each production 119876 and produc-tion planning of independent demand 119876

119871can be connected

by the direct consumption coefficient 119886119894119895and matrix 119860 =

119886119894119895119899times119899

According to the input-output balanced equationssum119899

119895=1119886119894119895sdot 119902119895+ 119902119871

119894= 119902119894 so 119876 = 119860 sdot 119876 + 119876

119871 And we can get the

following formula from the relationship between 119876 and 119876119871

119876119871= (119868 minus 119860) sdot 119876 = 119861 sdot 119876 (1)

where 119861 = 119868 minus 119860 = 119887119894119895119899times119899

We can prove that (119868minus119860) is a full-rankmatrix and further

calculate its inverse matrix And another form of relationshipbetween 119876 and 119876

119871can be rewritten as

119876 = (119868 minus 119860)minus1

119876119871= 119867 sdot 119876

119871 (2)

where 119863 = (119868 minus 119860)minus1

= 119889119894119895119899times119899

and we call it absolutelynecessary coefficient matrix So it is obvious that the totalproduction planning amount of each company can be derivedfrom the production planning of independent demandsand the relationship between them is expressed as 119902

119894=

sum119899

119895=1119889119894119895119902119871

119895 119894 isin 119873 Based on this relationship we take the

production planning of independent demands as the decisionvariable


3 Production Planning Model of the SupplyChain without Carbon Emission Constraints

In this section we first consider production planning prob-lem in the same complex supply chain system without emis-sion constraints existing The results in this setting are to bescaleplate compared with that in other scenarios There existtwo kinds of costs to be considered in this study One is theinventory holding costThe company producesmore than themarket demand and the remaining products needed to bestoredThe parameter ℎ

119894is the stock cost per unit of product

Theother is the penalty costWhen the amount of the productcannot meet the demand of the market the company shouldbe punished with per unit product penalty cost 119904

119894 We have

assumed that there is no initial stock for each node firm Sothe ending stock level 119902

119894is equal to the total production

planning amount 119902119904119894which is the sum of production planning

of internal dependent demand and external independentdemand that is 119902

119894= 119902119904

119894= 119902119871

119894+ 119902119872

119894 And because supply and

procurement within the supply chain is determined by thebill of materials (BOM) table so the production planning ofdependent demand of a company is determined by the totalproduction planning amount of its downstream firmsThere-fore we can draw a conclusion that inventory costs andshortage costs are influenced by the relationship betweenproduction planning of independent demand and stochasticmarket demand

We establish a vector Δ119862(119876119871) = [ Δ119888

119894 ] considering

inventory underage and overage simultaneously We call itdifference cost function between the production planning ofinternal demand and that of external demand the expressionof which is as follows

Δ119888119894= ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

(3)

The practical significance of the formula above is that thedifferencesrsquo cost function contains the inventory costs andshortage costs And it can be transformed into piecewisefunction in the following

(119883119894minus 119902119871

119894)

minus

=

119902119871

119894minus 119883119894

where 119883119894lt 119902119871

119894

0 where 119883119894ge 119902119871

119894

(119883119894minus 119902119871

119894)

+

=

0 where 119883119894lt 119902119871

119894

119883119894minus 119902119871

119894where 119883

119894ge 119902119871

119894

(4)

In addition to these two kinds of costs we consideranother two kinds of costs that is fixed cost and variable costBecause we only consider the case of single-cycle the fixedcosts and variable costs per unit of product are constants andvariable costs are positively correlated with the total yields

In this part our target is to maximize the profit of thesupply chain and we construct the profit function according

to the relationship that profit is equal to the value of revenueminus cost The profit function is as follows

120587 (119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871

119894 119883119894)

minus[

[

ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895]

]

(5)

On the right-hand side of the above equation it is theprofit of each firm that located in the curly braces And thesumof individual company profit constitutes the supply chainprofit The first item in the curly braces is the revenue of thecompany and the company obtains it by selling the product tothe market We do not consider the revenue from sellingproduct to the other companies inside the supply chainbecause this part of revenue is purchasing cost for them andthis revenue and cost can be offset by the entire supply chainprofit And we can find that salesrsquo volumes of each node firmare the minimum of the production planning of independentdemand and the stochastic market demand If the companyproduces more than the market demand then the salesrsquo vol-umes are determined by the planned production of indepen-dent demand Otherwise the salesrsquo volumes are equal to themarket demandThe secondpartwithin the curly braces is thecost of the company which contains inventory costs shortagecosts fixed costs and variable costs

If we know the probability density function and the cum-ulative distribution function of the stochasticmarket demandfor node firm 119894 we can get the objective function tomaximizethe profit of the supply chain as follows

maxΠ = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894

minus 119870119894minus 119904119894120583119894minus V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119902119871

119894ge 0

(6)

It is obvious that the objective function is a second-ordercontinuous function If we want to get the optimal solution tomaximize the profit we should first prove that it is a concavefunction and thenwe calculate the first-order derivative of thefunction for the planned production of independent demandby equating it to zero By solving the equations system we canobtain the optimal value of production planning We prove


the concavity of the function in the appendix And the first-order derivative of the function is as follows by letting it bezero

minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896) 119865119896(119902119871

119896) +

119899

sum

119894=1

V119894119889119894119896

= 0 (7)

From this equation we can calculate the planning pro-duction of independent demand of each company Accordingto the relationship between planned production of indepen-dent demand and the total production of each companywe can calculate out some other kinds of production Andthe expression of the planned production of independentdemand is as follows

119902119871lowast

119896= 119865minus

119896[

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (8)

It is obvious that there is no relation between the optimalplanned productions of independent demand of a companywith three parameters of other companies in the supply chainthe price inventory holding cost and shortage penalty costper unit product It is only influenced by these three costfactors of its own but another finding is that optimal planningproduction of independent demand is related to the variablecost of per unit product of other companies in the supplychain To learn more details of the research content as wellas proofs shown in Section 3 please refer to literature [27]And we will do a further study in the section of numericalanalysis

4 Supply Chain Decision under theMandatory Emission Cap Policy

Chinarsquos carbon emission occupies nearly 14 of the worldrsquosemissions which is the highest in the world So China hasbeen facing the pressure to cut emissions from the interna-tional community in the past In contrast China has said itwould reduce its ldquocarbon intensityrdquo or the proportion ofemissions relative to economic output Related personnel ofNDRC Energy Research Institute said that the NDRC wasconsidering implementing an absolute upper limit of carbonemissions in the next five-year plan and now they werestudying what is the appropriate level

Mandatory emission cap policy is a good method toreduce carbon emissions but the problem is that companieswhich are facing pressure from government must haveflexibility on production decisions and also have an impact oncorporate profits This section is intended to study theimpact of mandatory emission cap policy on the productiondecisions of supply chain members

41 Joint Optimization Model under Mandatory Emission CapPolicy In the case of mandatory emission cap policy thecompanyrsquos total carbon emissions119864

119894must not exceed govern-

ment regulations value 119862119894

At this time the expected profit function is the same asthe scenarios under no carbon emissions Under the permit

that each company satisfies 119864119894le 119862119894 objective function of the

supply chain is

max119902119871

119894

prod

MC(119876119871) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus 119904

119894120583119894minus 119870119894

minusV119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119864119894le 119862119894

119902119871

119894ge 0

forall119894 isin 119873

(9)

where 119864119894= 119890119898

119894sdot sum119899

119895=1119889119894119895sdot 119902119871

119895+ 119864119870

119894+ 119890ℎ

119894sdot int

119902119871

119894

0119865119894(119909)119889119909

Companyrsquos total carbon emissions are equal to the sumof variable production relevant emissions fixed productionrelevant emissions and excess inventory relevant carbon emi-ssions Variable production relevant emissions are propor-tionally increasing in total number of products that is119890119898

119894sum119899

119895=1119889119894119895

sdot 119902119871

119895 fixed production of emissions is 119864

119870

119894 The

expected value of remaining inventory is int119902119871

119894

0119865119894(119909) 119889119909 so the

expected value of carbon emissions generated by the excessremaining inventory is 119890ℎ

119894sdot int

119902119871

119894

0119865119894(119909) 119889119909

42 Interior Point Method to Solve the Problem For generalunconstrained optimization problem there exists a variety ofefficient algorithms currently People usually transform theconstrained problem into an associated unconstrained prob-lem Specifically according to the constraint characteristicsthey construct some kind of ldquopunishmentrdquo function and thenadd it to the objective function then the constraint solvingproblem is transformed into a series of unconstrained prob-lem solving In this respect there are many algorithms suchas outside the penalty function method the penalty functionmethod and the multiplier method

As for the planning problem with inequality constraintsunder mandatory emission cap policy this paper uses thepenalty functionmethod to solve it In order tomake iterativepoints always possible the penalty function method builds ahigh ldquowallrdquo on the boundary of the feasible region When theiteration point is near the border the objective function valuesuddenly increases as a punishment to stop the iterationpoint across the border and thus the optimal solution isblocked in the feasible region

Before the use of penalty functionmethod we construct apenalty function and add it to the objective function Specificsteps are as follows


Step 1 Make the original problem into a minimizationproblem

min 1198691(119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119892119894(119902119871

119894) = 119862

119894minus 119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895

minus 119864119870

119894minus 119890ℎ

119894sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 ge 0

119902119871

119894ge 0


(10)

Step 2 Structure a newplanningwith the formmin119876119871isinR

119899120593(119876119871

119903(119896)

) = 1198691(119876119871) + 119903(119896)

sum119899

119894=1(1119892119894(119876119871)) where the penalty factor

119903(119896)

gt 0

Step 3 Given the initial point119876119871

(0)isin R119899 the allowable error

120576 gt 0 the penalty factor 120574(1) gt 0 and magnificence 119887 isin (0 1)set 119896 = 1

Step 4 Take 119876119871

(119896minus1) as the initial point then solve the fol-lowing planning problem min

119876119871isinR119899120593(119876119871 119903(119896)

) = 1198691(119876119871) +

119903(119896)

sum119899

119894=1(1119892119894(119876119871)) 119876119871

(119896) is the minimal point

Step 5 If 119903(119896)sum119899119894=1

(1119892119894(119876119871)) lt 120576 stop calculating the result

is 119876119871

(119896)

Step 6 Otherwise 120574(119896+1)

= 119887120574(119896) set 119896 = 119896 + 1 and go back to

step four

5 Supply Chain Decisions underCarbon Tax Policy

Carbon tax originated in the British economist Pigou ldquoPigoursquostheoryrdquo the theory is that in order to achieve effective controlof pollution and pollutant emissions purposes the govern-ment imposes tax on polluters according to the degree ofharm caused In order to achieve the reduction of carbondioxide emissions and fossil fuel consumption the govern-ment levies carbon tax on coal gasoline natural gas jet fueland other fossil fuel products according to the proportion oftheir carbon content Carbon tax can take many forms andthe simplest case is penalty with the linear growth of unitcarbon emissions

In this part we study the impact of carbon tax policy onsupply chain operations In order to obtain the optimal

supply chain profit it is important to balance the relationshipbetween the revenue and the tax brought by the production ofthe company If the companies can sell more products and therevenue may cover the carbon emissions tax it is clear thatcompanies will choose to produce more Conversely com-panies produce less We build the profit model of the supplychain under carbon tax policy in this part to study the impactof tax policy on production decisions in the complex supplychain

Company 119894 needs to pay taxes 120596 for releasing per unitof carbon emissions Companiesrsquo carbon emissions are119890119898

119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)

minus so companies need to paytaxes120596sdot[119890

119898

119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)

minus

]The carbon tax isa kind of cost in the profit function according to the profitfunction of no carbon emissions circumstances and the profitfunction can be expressed as

prod

CT(119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871

119894 119883119894) minus ℎ119894sdot (119883119894minus 119902119871

119894)

minus

minus 119904119894

sdot (119883119894minus 119902119871

119894)

+

minus 119870119894minus V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895minus 120596

sdot[

[

119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895+ 119864119870

119894

+ 119890ℎ

119894sdot (119883119894minus 119902119871

119894)

minus]

]

(11)

So the model of maximizing expected profit of entiresupply chain can be written as follows

max119864(prod

CT) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894+ 120596119890ℎ

119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus (V

119894+ 120596119890119898

119894)

sdot

119899

sum

119895=1

119889119894119895119902119871

119895minus 120596119864119870

119894minus 119904119894120583119894minus 119870119894

st 119902119871119894ge 0


(12)

First we assume the profits functionprodCT is second-ordercontinuous The presence of optimal solution for the aboveequation is that the first derivative of prodCT for a variety ofindependent demand 119902

119871

119896(119896 isin 119873) is zero and prodCT is a

concave functionAccording to the method of proving prod to be a concave

function it is easy to prove that the profit functionprodCT is alsoconcave function


Then we seek first-order partial derivative of prodCT for119902119871

119896(for all 119896 isin 119873) and set it to zero as follows

minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896) 119865119896(119902119871

119896)

+

119899

sum

119894=1

119889119894119896(V119894+ 120596119890119898

119894) = 0

(13)

So we can get the optimal production planning ofindependent demand

119902119871lowast

119896= 119865minus1

119896[

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] (14)

6 Numerical Experiments and Analysis

In this section we conduct numerical experiments to exam-ine some conclusions or findings in preceding sections Acomplex supply chain of one functional product is designedas in Figure 2

Among these products product A is the final productassembled which only faces random independent demandfrom external market while the products B C and D are allcomponent parts and they are confronted with dependentdemand from network nodes and independent demand ofexternal market According to the demand structure we canget the direct consumption coefficient matrix A = [0 0 0 0

2 0 0 0 1 1 0 0 1 3 1 0]Now independent demand 119909

119871

119894is assumed to subject to

exponential distribution with parameter 120582119894and its mean

value 120583119894 the probability density function 119891

119894(119909119871

119894) = 120582

119894119890minus120582119894119909119871

119894 and the cumulative distribution function is119865

119894(119909119871

119894) = 1minus119890

minus120582119894119909119871

119894 In seek of further analysis of joint optimization strategy of

production for the supply chain we set up several groups ofexperiments in this paper through numerical simulation tostudy the relationship between decision variables and somekey parameters under the carbon emissions policy

61 Analysis of Planned Production of Each Company underNo Carbon Emissions Constraints The optimal planned pro-duction of independent demand of each company is

119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (15)

As we can see from the formula the optimal planned pro-duction of independent demand of each company only hasthe correlation with factors of its own including the productmarket demand product prices the inventory cost penaltycost and the initial inventory and is not affected by thesefactors of other companies According to the absolutely nec-essary coefficientmatrix we get 119889

119894119896= 0 where 119896 gt 119894 It means

that when company 119894 is a customer to company 119896we get 119889119894119896

=

0 At this time companiesrsquo optimal planned production ofindependent demand will not be influenced by their cus-tomersrsquo variable cost per unit of product Andwe can get119889

119894119896gt

0 where 119896 lt 119894 It means that when company 119894 is a supplier of

Internal supply chain

AB

C

D

Outsidemarket

1 1 1 1

1 1

1 1

1 21 3

Figure 2 A manufacturing supply chain structure

company 119896 we get 119889119894119896

gt 0 Thus to customers their optimalplanned production of independent demand is influenced byvariable cost per unit of product of their suppliers Besidesplanned production of independent demand of customercompanies is on the decrease with the increasing of thesuppliersrsquo variable cost per unit of productThis phenomenonillustrates the importance of real powerful impact factormdashvariable cost per unit of product in the supply chain

(1) This section mainly analyzes optimal planned pro-duction of independent demand trends under the given120583 119875 119867 119878 and the changes in production costs therebygetting analysis of the influence of production cost vectoron the system performance Without loss of generality weset 120583 = [100 112 120 140]

119879 119875 = [10 6 4 1]119879 119867 =

[ℎ1 ℎ2 ℎ3 ℎ4] = [06 04 03 005]

119879 and 119878 = [1199041 1199042 1199043 1199044] =

[08 04 02 01]119879 The production planning of independent

demand and system profits are shown in Table 1From Table 1 it is not difficult to find that when the

assembly plant Arsquos production cost increases its productionplanning amount related to independent demand decreasesand that of its suppliers B C and D is unchanged At thispoint the system gross profit is decreasing When the factoryBrsquos production costs increases planned production of inde-pendent demandof the factoriesA andBdecreases while thatof C and D remains unchanged and the total profits of thesystem decrease When production cost of A B and suppliersC increases the planned production of independent demandof A B and C was decreased but that of raw materialssupplier D is unchanged And we can find that the totalprofit of the system has also gradually reduced in this caseIt is an interesting insight that when the production costs ofrawmaterial supplier D increased the planned production ofindependent demand of all the companies in the supply chainhas reduced and the total profit system has also graduallyreduced From the above analysis we can find that the totalprofit of the system is negatively correlated to the productioncosts of each company The planned production of indepen-dent demand of downstream company in the supply chain isaffected by production costs of its own and its upstream sup-pliers and there is a negative correlation between the plannedproduction of independent demand and the productioncosts But the planned production of independent demand ofthe upstream companies in the supply chain is only affectedby its own production costs


Table 1 Production planning and system profits sensitivity analysisto 119881119901

119881119901 Planned production of independent

demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617

(2) This section mainly analyzes rends of planned pro-duction of independent demand 119876

119871 in a given 120583 119881119901 119867

and 119878with the changes of product pricesThereby we analyzethe impact of the product price vector on system perform-ance Without loss of generality we order 120583 = [100 112

120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]

119879 Planned production of indepen-dent demand and system profits is shown in Table 2

It can be found in Table 2 that product price changes ofeach company will only affect the production planning ofindependent demand but will not affect other companiesWith prices gradually decreasing the production planningfor external random demand is to decline This indicates thatthe supply chain production operations decision is influencedby product prices vector Similarly the total profit of supplychain also goes down with decreasing prices

62TheAnalysis of Supply Chain Profits andCarbon Emissionsunder Mandatory Emission Caps Policy This part is mainlydevoted to the changing regularity of the total profit and thetotal carbon emission of the supply chain with the change of

Table 2 Planned production of independent demand and systemprofit sensitivity analysis to price 119875

Price 119875

Planned production ofindependent demand

Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168

carbon caps C under the given 120583 119875119867 119878119883 and119881119901 Now we

give the value of these factors as follows

119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)

We have gotten the optimal planning production of inde-pendent demand of each company under the condition of nocarbon policy Because of the existence of carbon discharge


during the process of production or stock in the supply chainwe can calculate each companyrsquos carbon discharge the totalprofit carbon emission of the supply chain system We call allthese values ldquodatum carbon emissionsrdquo ldquodatum systemprofitrdquoand ldquodatum total carbon emissionsrdquo This part majorly ana-lyzed how the government should formulate the emission capof each company when they develop the plans of reductionGenerally when the government develops a specific reductionpolicy they will consult annual carbon emission data whichwe have mentioned above that is ldquobaseline carbon emis-sionsrdquo We analyzed a case where the emission cap of eachcompany based on ldquobaseline carbon emissionsrdquo reduces anumber of percentage points respectively (one companyreduced emission cap while othersrsquo is constant)This is equiv-alent to the government stepped up restrictions on carbonemissions for each company It will inevitably lead to two con-sequences First is that the profit of systemwill be affected thesecond is that the systemrsquos total carbon emissions will bechanged And we obtain the change trends of system profitand carbon emission with the emission cap of each companyreducing from 1 percent to 80 percent respectively Theresults are shown in Figures 3 and 4

As can be seen from Figure 3 it is obvious that for eachcompany when the government tighten carbon emission capthe total profit of the supply chain system is reduced And thereduction volume is different when the governmentrsquos tighten-ingmeasures are applied in different company And it is obvi-ous that when the government strengthens restriction of thecarbon emissions of company A which is the assembly com-panies of the supply chain system profit is reduced less thanthe case that the emission caps of other companies reduce thesame percentage From this perspective it is better for supplychain when the government tighten carbon emission cap ofcompany A But from the perspective of the government itwould not be the best decision because government alsoneeds to take the pressure of reduce emissions into accountFor further study we use Figure 4 to explore the changes ofthe system carbon emissions when the emission cap of eachcompany is reduced by percentage

We can find a remarkable phenomenon from Figure 4when the emission cap of company A is reduced by a certainpercentage from the ldquodatum carbon emissionsrdquo (the emissioncap of other companies are unchanged at this time) thesupply chain carbon emissions is higher than the case that theemission cap of other company is reduced by the same pro-portion From the government perspective it could not be aperfect consequence The government hopes that emissionsreduction volume reduces more the better So we can drawa conclusion that when the government can only tighten theemission cap of one company it will tighten the emission capof company D by a certain proportion to achieve a higheremission reduction effect compared with the same reductionpercentage of emission caps of other companies It was totallyopposed to the results of Figure 3 in which the supply chainwants the tightening policy to be applied on company A Sothe effect of the carbon reduction with tightening emissioncap of a company in the supply chain is ambivalent from theperspective of the government and supply chain We tryto introduce a kind of evaluation index to balance the

200

300

400

500

600

700

800

Supp

ly ch

ain

profi

t

B

C

D

A

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80

The proportion of carbon emission cap reduction of each firm ()

Figure 3 System profit when each company reduces carbonemissions by 1ndash80

468

10121416182022

Syste

m to

tal c

arbo

n em

issio

n

0 8 16 24 32 40 48 56 64 72 80

times103

A

BCD


Figure 4 System emissions when each company reduces carbonemissions by 1ndash80

contradiction between the government and the supply chainWe define the indicators for ldquocarbon emission elasticity ofprofitrdquo Similar to the price elasticity of demand the value ofcarbon emission elasticity of profit is the quotient from divid-ing change rate of system profits relative to the datum supplychain profit by change rate of supply chain carbon emissionrelative to datum total carbon emissions If the value isbetween 0 and 1 it means that the change rate of supply chainprofit is less than the change rate of supply chain carbon emis-sion In this condition the government will be very satisfiedwith the high proportion of carbon emissions reduction andthe supply chain is also happy that its profit is not reduced bya high proportion Therefore it is a very good result that thevalue of carbon emission elasticity of profit is between 0 and 1In addition the carbon emission elasticity of profit is differentwhen the government tightens the emission cap on differentcompanies Figure 5 analyzes the change of carbon emissionelasticity of profit with the respectively reduction of emissioncap of each company by a certain proportion

As can be seen from Figure 5 carbon emission elasticityof profit is all between 0 and 1 with respective reduction of


0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

ion

elas

ticity

of

profi

t of t

he su

pply

chai

n


AB

CD

Figure 5 Carbon emission elasticity of profit with the respectivereduction of emission cap of each company by a certain proportion

emission cap of each company by a certain proportion andthe value increases with the increase of the reduction propor-tion What is more we can find that for A company whenits emission cap reduces a certain proportion the carbonemission elasticity of profit is lower compared with thesituation that other companiesrsquo emission cap reduce the samepercentage For example when the reduction proportion ofemission cap of company A is 2 the carbon emission elas-ticity of profit is 0012 At this time the carbon emission elas-ticity of profit is lowest comparedwith the situation that othercompaniesrsquo emission cap reduce 2 In other words in thissituation if the change rate of carbon emissions was 1 thenthe change rate of supply chain profit is only 0012The effect isvery significant for supply chain and the government So bothsides will be more inclined to reduce a certain proportion ofemission cap of company A

63TheAnalysis of Supply Chain Profits andCarbon EmissionsunderCarbonTaxPolicy At this point all companiesrsquo optimalplanned production of independent demand

119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)

This part mainly analyzes trends of the optimal plannedproduction of independent demand119876 total profit and actualcarbon emissions of supply chain with changes in the carbontax 120596 per unit of carbon emissions under the given 120583 119875119867 119878V 119890119898 119864119870 and 119890

ℎ The values of these factors are as follows

120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)

Table 3 Sensitivity analysis of production planning of independentdemand and the supply chain profit to carbon tax 120596 of unit carbonemission

120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466

At this point we can also get the planned productionof independent demand and the supply chain profit underdifferent carbon tax The results are shown in Table 3


The Effect of Imposing Carbon Tax Per Unit Carbon Emissionon Production Planning for Independent Demand Figure 6shows that with the gradual increase of carbon tax ofper unit carbon emissions planned production of inde-pendent demand is decreased approximately linearly Andunder a given carbon tax the planned production of inde-pendent demand of the company A is the lowest

The Effect of Imposing Carbon Tax Per Unit Carbon Emissionson Overall Supply Chain Carbon Emission As can be seenfrom Figure 7 within a certain range the system profit andtotal carbon emission reduce with the increasing of carbontax And this phenomenon illustrates that it is indeed an effi-cient and not complicated mechanism to decline the systemrsquostotal carbon emissions through increasing the carbon taxFrom the governmentrsquos perspective the carbon tax policy is agood policy because emissions can be greatly reduced butfrom the point of view of supply chain the carbon taxreduces systemprofit greatly Similar to the case ofmandatoryemission cap policy this section also build a similar system ofcarbon emission elasticity of profit andwe compare the resultwith the case of mandatory emission cap policy With otherparameters remaining unchanged we take the optimal solu-tion under no carbon emissions limits as the reference point(in this case the carbon tax is 0) and we obtained the profitand carbon emissions of the supply chain in the referencepoint Similarly we obtain the carbon emission elasticity ofprofit under different carbon tax in Figure 8

As can be seen from Figure 8 with the gradual increaseof carbon tax rate the carbon emission elasticity of profitincreased and then decreased which are all greater than 1This shows that the change rate of profit is greater than thechange rate of the carbon emissions It is clear that the supplychain and the government are more difficult to accept theresults Because the profit loss in this situation can not beneglected and unfortunately the carbon emissions reductionis not comparatively significant In comparison a mandatoryemissions cap policy is more excellent because regardless ofcarbon emissions austerity policies imposed on which com-pany the carbon emission elasticity of profit is less than onewhich is an optimum result for the government and supplychain Therefore for the supply chain we studied the carbontax policy is not prior to mandatory emission cap policy to beused in practice

7 Conclusion

In this paper we focus on how to get the optimal productionplanning confronting various carbon emission regulations fora complex supply chain comprising nodes firms with cou-pled internal dependent demand flows and random marketdemands We solve the joint production optimization prob-lem of the supply chain under mandatory emission cap andemission tax respectively and uncover the impact of theseregulatory policies on the profit and total emission of thesupply chainWe compare the optimal production quantitiesprofits and overall emissions arising respectively from threescenarios that is no emission policy mandatory emissioncap and emission tax One of contributions of this study is

0

50

100

150

200

250

300

350

Tax rate of carbon emission

A

B

C

D

Plan

ned

prod

uctio

n of

inde

pend

ent d

eman

d

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6

Figure 6 Effect of carbon tax of per unit carbon emissions onplanned production of independent demand

0

100

200

300

400

500

600

700

8000

008

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


Syste

m p

rofit

0

5

10

15

20

25

Syste

m ca

rbon

emiss

ions

System profitSystem carbon emissions

times103

Figure 7 Effect of carbon tax of per unit carbon emissions on supplychain total carbon emission

that we introduce the ldquocarbon emission elasticity of profit(CEEP)rdquo index to evaluate the influence of carbon emissionregulatory policies on profit and total emissions of a supplychain and her node firms Taking advantage of the CEEPindex we find that under the mandatory emission cap policyif we only decrease mandatory emission cap of the assemblycompany by a certain proportion the CEEP index is lowestcompared with situations where we instead decrease the capof other node firms by same scale Meanwhile the value ofthe index is between 0 and 1 namely nonelastic whichimplies we can reach the emission control target at a relativelylow price of profit loss That is obviously acceptable both forthe supply chain and public administration As for thescenario of carbon tax policy we find that the CEEP is elasticwhich in turn indicates it should not be prior to mandatory


138

1385

139

1395

14

1405

141

1415

142

1425


Carbon emission elasticity of profit

Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6

Figure 8 Carbon emission elasticity of profit under different carbontax

emission cap policy to be adopted and preferred by industryand government

There may be some extension in the future For exampleone can consider the multiple-horizon production planningproblem in the same supply chain structure under low-carbon constraints One can also add the pricing decision intothe consideration

Appendix

Proof The objective function of the supply chain

max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894

minus119870119894minus 119904119894120583119894minus V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)

In order to prove that the target functionprod is the concavefunction this needs to prove that function 119869(119876

119871) is convex

functionConstruct a function 120601(119909) = int

119909

0119865(119905)119889119905 its second deriva-

tive 12060110158401015840(119909) = 119891(119909) gt 0 thus 120601(119909) is a convex function to its

domain for all 1199091 1199092gt 0 we have 120601(120582119909

1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905

where 0 le 120582 le 1 120582 = 1 minus 120582∘So we get int120582119902

119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)

In this formula (119901119894+ ℎ119894+ 119904119894) gt 0 and we can calculate

from the formula and get int120582119902119894

1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0

that means 119869(119876119871) is a convex function Therefore prod is a

concave function then there is a global optimum point


Conflict of Interests

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

Acknowledgments

The authors would like to express their sincere thanks to theanonymous referees and editors for their time and patiencedevoted to the review of this paper This work is partiallysupported by NSFC Grant (no 71202086)

References

[1] L J Xia D Zhao and B Yuan ldquoCarbon efficient supply chainmanagement literature review with extensionsrdquo Applied Mech-anics and Materials vol 291ndash294 pp 1407ndash1412 2013

[2] J Song and M Leng ldquoAnalysis of the single-period problemunder carbon emissions policiesrdquo in Handbook of NewsvendorProblems vol 176 of International Series in Operations Researchand Management Science pp 297ndash313 Springer New York NYUSA 2012

[3] Congress of the United States Policy options for reducing Co2emissions a CBO study This study was provided by the Con-gressional Budget Office (CBO) 2008

[4] S Benjaafar Y Li and M Daskin ldquoCarbon footprint and themanagement of supply chains Insights from simple modelsrdquoIEEE Transactions on Automation Science and Engineering vol10 no 1 pp 99ndash116 2013

[5] S Dhakal ldquoUrban energy use and carbon emissions from citiesin China and policy implicationsrdquo Energy Policy vol 37 no 11pp 4208ndash4219 2009

[6] D Holtz-Eakin and T M Selden ldquoStoking the fires CO2emis-

sions and economic growthrdquo Journal of Public Economics vol57 no 1 pp 85ndash101 1995

[7] X Zhang and X Cheng ldquoEnergy consumption carbon emis-sions and economic growth in Chinardquo Ecological Economicsvol 68 no 10 pp 2706ndash2712 2009

[8] X Leng Research on the relationship between carbon emissionand Chinarsquos economics [Thesis] Fudan University 2012

[9] Q Zhu X Z Peng Z M Lu and K Y Wu ldquoFactor decompo-sition and empirical analysis of changes in carbon emissions inChinarsquos energy consumptionrdquo Resources Science vol 31 no 12pp 2072ndash2079 2009

[10] J W Sun and Z G Chen ldquoResearch on Chinarsquos carbon emis-sions footprint based on input-output analysisrdquo China Popula-tion Resources and Environment vol 20 no 5 pp 28ndash34 2010

[11] M L Cropper and E Oates ldquoEnvironmental economics asurveyrdquo Journal of Economic Literature vol 30 no 2 pp 675ndash740 1992

[12] C Hepburn ldquoRegulation by prices quantities or both a reviewof instrument choicerdquoOxford Review of Economic Policy vol 22no 2 pp 226ndash247 2006

[13] M Webster I S Wing and L Jakobovits ldquoSecond-best instru-ments for near-term climate policy intensity targets vs thesafety valverdquo Journal of Environmental Economics and Manage-ment vol 59 no 3 pp 250ndash259 2010

[14] B Bureau ldquoDistributional effects of a carbon tax on car fuels inFrancerdquo Energy Economics vol 33 no 1 pp 121ndash130 2011

[15] S Benjaafar Y LiMDaskin L Qi and S KennedyTheCarbonFootprint of UHT Milk University of Minnesota MinneapolisMN USA 2010

[16] D PanOptimal decision of the companies production under low-carbon economy Yanshan University 2012

[17] B S Kang and J S Yoon ldquoSheet forming apparatus with flexiblerollersrdquo PCT Patent pending 2013

[18] X Chen S Benjaafar and A Elomri ldquoThe carbon-constrainedEOQrdquo Operations Research Letters vol 41 no 2 pp 172ndash1792013

[19] G P Cachon ldquoCarbon footprint and the management of supplychainsrdquo in Proceedings of the INFORMS Annual Meeting SanDiego Calif USA 2009

[20] A Ramudhin A Chaabane M Kharoune and M PaquetldquoCarbon market sensitive green supply chain network designrdquoin Proceedings of the IEEE International Conference on IndustrialEngineering and EngineeringManagement (IEEM rsquo08) pp 1093ndash1097 December 2008

[21] T Abdallah A Diabat and D Simchi-Levi ldquoA carbon sensitivesupply chain network problemwith green procurementrdquo inPro-ceedings of the 40th International Conference on Computers andIndustrial Engineering (CIE40 10) Awaji Japan July 2010

[22] A Ramudhin A Chaabane M Kharoune and M PaquetldquoDesign of sustainable supply chains under the emission tradingschemerdquo International Journal of Production Economics vol 135no 1 pp 37ndash49 2012

[23] D Zhao and J Lv ldquoCarbon-efficient strategy for integratedsupply chain considering carbon emission rights and tradingrdquoIndustrial Engineering andManagement vol 17 no 5 pp 65ndash712012

[24] L He Supply network optimization based on coordination mech-anism design considering diverse chain structures [Dissertation]Tianjin University 2010

[25] F R Jacobs and R B ChaseOperations Management Mechan-ical Industry Press Beijing China 2011

[26] L He H Lu andD Zhao ldquoQuick response based joint dynamicproduction optimization for a complicated supply chain underinternally interwined demand dependencerdquo Journal of SystemsScience and Mathematical Sciences vol 34 no 1 pp 20ndash422014

[27] L He ldquoOptimal joint output analysis of complex supplynetworks with stochastic demandsrdquo Working Paper TianjinUniversity 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


reflected in the carbon label will affect the market demanddue to consumersrsquo low-carbon awareness and in turn influ-ence each firmrsquos profit eventually

Although the government has released a couple of carbonemission regulatory policies for changing all carbon emittersrsquobehaviors especially for firms in industries we should askwhether those policies can get their initial expected effect incarbon emission control Upon the intuition one can inferthat some carbon emission regulation may not work well if itis originated from controlling single firm emission behaviorbut not from supply chain The interactions among firms inthe supply chain usually incur behaviors deviating from thegoal of thewhole system In low carbon environment firms insupply network also have different appeals when confrontingvarious regulatory carbon emission policies The low-carbonconcerns will worsen the conflict of interests among nodefirms This is the reason why we focus on supply chainproduction planning under carbon emission constraints inthis study

In this paper our goal is to study the impact of variouscarbon emission regulations on the operations and then theperformance of a complex supply chain network that isassembly system like made up of a series of nodes with con-vergent material flows to the final product In this systemeach node firmrsquos production is constrained by her own emis-sion regulatory policy say eithermandatory cap or carbon taxas we exploit afterwards In contrast to single firm productionplanning problem under low-carbon scenario we turn tofocus on amore realistic and typical manufacturing system aswe show here which attracts us to investigate howwemanagea set of manufacturing units under low-carbon era comparedwith traditional setting without low-carbon thinking Thiskind of comparison can uncover the impact of carbon emis-sion regulations on the typical supply chain network Undersystemic centralized setting we formulate the problem asconstrained nonlinear programming with 119899 dimensionalvariables

On the other hand we conduct another kind of compar-ison of performances resulted from two distinctive emissionregulatory policies respectively We intend to measure andshow the difference of two policiesrsquo transmissionmechanismsthrough the whole supply network which is designed toinspire our interest and thinking on the link of supply chainoperations and public administrative policies To achieve thistarget we induce some metric index in terms of individuallevel and system level profits and emissions

There are several streams of research related to our workFor an extensive survey in general on the operationsmanage-ment in low-carbon environment see Benjaafar et al [4] andXia et al [1] Some researchers holding the macroperspectivehave studied the relationship between carbon emissions andthe national economy see Dhakal [5] Holtz-Eakin andSelden [6] and Zhang and Cheng [7] while many otherstudies consider the evaluation and influential factors ofcarbon emissions Similarly some researches pay attentionto the relationship between macroeconomic factors andcarbon emissions Leng [8] develops a theoretical analysis ofenvironmental economics for analyzing the impact of eco-nomic factors on carbon emissions In this direction further

analysis on how economic factors affect carbon emissionswasconducted by Zhu et al [9] and Sun and Chen [10] Theyalso propose that the adjustment of the industrial structureenhancement of energy efficiency and transformation of theenergy structure are among the emphases of energy conser-vation Whereas being obviously different from the above-mentioned literatures our study focuses on microfirm andespecially the network-level organization operations consid-ering emission constraints

Another relevant stream in this area focuses on the impactof policy instruments on the carbon emission reduction andthese instruments can be summarized asmandatory emissioncaps taxes on emissions cap-and-trade and cap-and-offsetsee Cropper and Oates [11] Hepburn [12] Webster et al [13]and Bureau [14] While these researches study the impact ofpublic policies on the carbon emission reduction they do notlink operations management and public policy making forreducing carbon emission and also do not compare the effec-tiveness of carbon emission control under different policies

Another relevant stream to our work are papers thatconsider the influence of carbon emissions policies on theindustrial firmsrsquo production and inventory decisions profitsand costs and so forth Benjaafar et al [4] and Benjaafar et al[15] are among the first who proposed how to achieve the tar-geted emission reduction effect only from the perspective ofmanufacturing operations when a single firm was subject tocarbon emissions regulations Pan [16] investigates the car-bon emission reduction and production planning of a singlecompany as well as duopoly companies both under thecarbon tax and cap-and-trade policies respectively But hisstudy does not take the excess inventory generated emissionsinto account As a contrast Kang and Yoon [17] study thecase considering the emissions incurred by inventory Forfurther exploring the relation between profit and emissionreductions Song and Leng [2] exploiting classic single-cyclenewsvendor model investigate the optimal quantity orderingunder three kinds of emission regulatory policies that ismandatory emission cap emissions tax and cap-and-tradeAfterwards Chen et al [18] research a similar problem butthey adopt the conventional EOQmodel by assuming that themarket demand rate is constant and shortage is not permit-ted Although considering the comparison of performancesincurred by different policies the above researches are limitedto a single firm and therefore do not study the impact ofcarbon emission regulations on supply chains On the con-trary this is just the gap our work attempts to fill

The most highly related research stream on low-carbonsupply chain mainly focus on supply chain design and pro-duction system decisions They argue that structure of thesupply chain plays an important role in the chainwide carbonemissions so it is important to optimize the supply chaindesign to achieve the emission control goal see Cachon [19]Ramudhin et al [20] andAbdallah et al [21] Ramudhin et al[20] established a mixed integer programming model for agreen supply chain redesign On its basis Ramudhin et al[22] consider the life cycle assessment in the supply chaindesign As a further study developed by Cachon [19] he con-siders the impact of two aspects supply chain structure andoperations on carbon emissions Another research direction







Firm 1

Firm 1

Firm 2 Firm 3

Supply chainborder

Independentdemand



Firm k












Parameters




119894(119909119871

119894)







product


119890119898




119890ℎ




Decision Variables


company119876119872

= [ 119902119872



119871



= [ 119902119878


firm 119902119878119894= 119902119871

119894+ 119902119872

119894








119886119894119895119899times119899


119895=1119886119894119895sdot 119902119895+ 119902119871

119894= 119902119894 so 119876 = 119860 sdot 119876 + 119876



119876119871= (119868 minus 119860) sdot 119876 = 119861 sdot 119876 (1)





119876 = (119868 minus 119860)minus1

119876119871= 119867 sdot 119876

119871 (2)


= 119889119894119895119899times119899


119894=

sum119899

119895=1119889119894119895119902119871








119894 We have





119894= 119902119904

119894= 119902119871

119894+ 119902119872






Δ119888119894= ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

(3)


(119883119894minus 119902119871

119894)

minus

=

119902119871

119894minus 119883119894

where 119883119894lt 119902119871

119894

0 where 119883119894ge 119902119871

119894

(119883119894minus 119902119871

119894)

+

=

0 where 119883119894lt 119902119871

119894

119883119894minus 119902119871

119894where 119883

119894ge 119902119871

119894

(4)




120587 (119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871

119894 119883119894)

minus[

[

ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895]

]

(5)



maxΠ = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

st 119902119871

119894ge 0

(6)




minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896) 119865119896(119902119871

119896) +

119899

sum

119894=1

V119894119889119894119896

= 0 (7)


119902119871lowast

119896= 119865minus

119896[

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (8)










supply chain is

max119902119871

119894

prod

MC(119876119871) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus 119904

119894120583119894minus 119870119894

minusV119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119864119894le 119862119894

119902119871

119894ge 0


(9)

where 119864119894= 119890119898

119894sdot sum119899

119895=1119889119894119895sdot 119902119871

119895+ 119864119870

119894+ 119890ℎ

119894sdot int

119902119871

119894

0119865119894(119909)119889119909


119894sum119899

119895=1119889119894119895

sdot 119902119871


119870

119894 The


119894

0119865119894(119909) 119889119909 so the


119894sdot int

119902119871

119894

0119865119894(119909) 119889119909






min 1198691(119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119892119894(119902119871

119894) = 119862

119894minus 119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895

minus 119864119870

119894minus 119890ℎ

119894sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 ge 0

119902119871

119894ge 0


(10)


119899120593(119876119871

119903(119896)

) = 1198691(119876119871) + 119903(119896)

sum119899


119903(119896)

gt 0




Step 4 Take 119876119871


119876119871isinR119899120593(119876119871 119903(119896)

) = 1198691(119876119871) +

119903(119896)

sum119899

119894=1(1119892119894(119876119871)) 119876119871


Step 5 If 119903(119896)sum119899119894=1


is 119876119871

(119896)

Step 6 Otherwise 120574(119896+1)

= 119887120574(119896) set 119896 = 119896 + 1 and go back to

step four






119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)


119898

119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)

minus


prod

CT(119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871


119894)

minus

minus 119904119894

sdot (119883119894minus 119902119871

119894)

+


119899

sum

119895=1

119889119894119895119902119871

119895minus 120596

sdot[

[

119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895+ 119864119870

119894

+ 119890ℎ

119894sdot (119883119894minus 119902119871

119894)

minus]

]

(11)


max119864(prod

CT) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894+ 120596119890ℎ

119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus (V

119894+ 120596119890119898

119894)

sdot

119899

sum

119895=1

119889119894119895119902119871

119895minus 120596119864119870

119894minus 119904119894120583119894minus 119870119894

st 119902119871119894ge 0


(12)


119871







minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896) 119865119896(119902119871

119896)

+

119899

sum

119894=1

119889119894119896(V119894+ 120596119890119898

119894) = 0

(13)


119902119871lowast

119896= 119865minus1

119896[

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] (14)





119871




119894(119909119871

119894) = 120582

119894119890minus120582119894119909119871


119894(119909119871

119894) = 1minus119890

minus120582119894119909119871




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (15)


119894119896= 0 where 119896 gt 119894 It means


=


119894119896gt



AB

C

D

Outsidemarket

1 1 1 1

1 1

1 1

1 21 3


company 119896 we get 119889119894119896



119879 119875 = [10 6 4 1]119879 119867 =

[ℎ1 ℎ2 ℎ3 ℎ4] = [06 04 03 005]

119879 and 119878 = [1199041 1199042 1199043 1199044] =







demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617


119871 in a given 120583 119881119901 119867


120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]





Price 119875


Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168



119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)






200

300

400

500

600

700

800

Supp

ly ch

ain

profi

t

B

C

D

A

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80



468

10121416182022

Syste

m to

tal c

arbo

n em

issio

n

0 8 16 24 32 40 48 56 64 72 80

times103

A

BCD






0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

ion

elas

ticity

of

profi

t of t

he su

pply

chai

n


AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


0

50

100

150

200

250

300

350


A

B

C

D

Plan

ned

prod

uctio

n of

inde

pend

ent d

eman

d

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


0

100

200

300

400

500

600

700

8000

008

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


Syste

m p

rofit

0

5

10

15

20

25

Syste

m ca

rbon

emiss

ions


times103




138

1385

139

1395

14

1405

141

1415

142

1425



Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Firm 1

Firm 1

Firm 2 Firm 3

Supply chainborder

Independentdemand



Firm k












Parameters




119894(119909119871

119894)







product


119890119898




119890ℎ




Decision Variables


company119876119872

= [ 119902119872



119871



= [ 119902119878


firm 119902119878119894= 119902119871

119894+ 119902119872

119894








119886119894119895119899times119899


119895=1119886119894119895sdot 119902119895+ 119902119871

119894= 119902119894 so 119876 = 119860 sdot 119876 + 119876



119876119871= (119868 minus 119860) sdot 119876 = 119861 sdot 119876 (1)





119876 = (119868 minus 119860)minus1

119876119871= 119867 sdot 119876

119871 (2)


= 119889119894119895119899times119899


119894=

sum119899

119895=1119889119894119895119902119871








119894 We have





119894= 119902119904

119894= 119902119871

119894+ 119902119872






Δ119888119894= ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

(3)


(119883119894minus 119902119871

119894)

minus

=

119902119871

119894minus 119883119894

where 119883119894lt 119902119871

119894

0 where 119883119894ge 119902119871

119894

(119883119894minus 119902119871

119894)

+

=

0 where 119883119894lt 119902119871

119894

119883119894minus 119902119871

119894where 119883

119894ge 119902119871

119894

(4)




120587 (119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871

119894 119883119894)

minus[

[

ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895]

]

(5)



maxΠ = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

st 119902119871

119894ge 0

(6)




minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896) 119865119896(119902119871

119896) +

119899

sum

119894=1

V119894119889119894119896

= 0 (7)


119902119871lowast

119896= 119865minus

119896[

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (8)










supply chain is

max119902119871

119894

prod

MC(119876119871) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus 119904

119894120583119894minus 119870119894

minusV119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119864119894le 119862119894

119902119871

119894ge 0


(9)

where 119864119894= 119890119898

119894sdot sum119899

119895=1119889119894119895sdot 119902119871

119895+ 119864119870

119894+ 119890ℎ

119894sdot int

119902119871

119894

0119865119894(119909)119889119909


119894sum119899

119895=1119889119894119895

sdot 119902119871


119870

119894 The


119894

0119865119894(119909) 119889119909 so the


119894sdot int

119902119871

119894

0119865119894(119909) 119889119909






min 1198691(119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119892119894(119902119871

119894) = 119862

119894minus 119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895

minus 119864119870

119894minus 119890ℎ

119894sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 ge 0

119902119871

119894ge 0


(10)


119899120593(119876119871

119903(119896)

) = 1198691(119876119871) + 119903(119896)

sum119899


119903(119896)

gt 0




Step 4 Take 119876119871


119876119871isinR119899120593(119876119871 119903(119896)

) = 1198691(119876119871) +

119903(119896)

sum119899

119894=1(1119892119894(119876119871)) 119876119871


Step 5 If 119903(119896)sum119899119894=1


is 119876119871

(119896)

Step 6 Otherwise 120574(119896+1)

= 119887120574(119896) set 119896 = 119896 + 1 and go back to

step four






119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)


119898

119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)

minus


prod

CT(119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871


119894)

minus

minus 119904119894

sdot (119883119894minus 119902119871

119894)

+


119899

sum

119895=1

119889119894119895119902119871

119895minus 120596

sdot[

[

119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895+ 119864119870

119894

+ 119890ℎ

119894sdot (119883119894minus 119902119871

119894)

minus]

]

(11)


max119864(prod

CT) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894+ 120596119890ℎ

119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus (V

119894+ 120596119890119898

119894)

sdot

119899

sum

119895=1

119889119894119895119902119871

119895minus 120596119864119870

119894minus 119904119894120583119894minus 119870119894

st 119902119871119894ge 0


(12)


119871







minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896) 119865119896(119902119871

119896)

+

119899

sum

119894=1

119889119894119896(V119894+ 120596119890119898

119894) = 0

(13)


119902119871lowast

119896= 119865minus1

119896[

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] (14)





119871




119894(119909119871

119894) = 120582

119894119890minus120582119894119909119871


119894(119909119871

119894) = 1minus119890

minus120582119894119909119871




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (15)


119894119896= 0 where 119896 gt 119894 It means


=


119894119896gt



AB

C

D

Outsidemarket

1 1 1 1

1 1

1 1

1 21 3


company 119896 we get 119889119894119896



119879 119875 = [10 6 4 1]119879 119867 =

[ℎ1 ℎ2 ℎ3 ℎ4] = [06 04 03 005]

119879 and 119878 = [1199041 1199042 1199043 1199044] =







demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617


119871 in a given 120583 119881119901 119867


120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]





Price 119875


Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168



119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)






200

300

400

500

600

700

800

Supp

ly ch

ain

profi

t

B

C

D

A

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80



468

10121416182022

Syste

m to

tal c

arbo

n em

issio

n

0 8 16 24 32 40 48 56 64 72 80

times103

A

BCD






0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

ion

elas

ticity

of

profi

t of t

he su

pply

chai

n


AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


0

50

100

150

200

250

300

350


A

B

C

D

Plan

ned

prod

uctio

n of

inde

pend

ent d

eman

d

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


0

100

200

300

400

500

600

700

8000

008

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


Syste

m p

rofit

0

5

10

15

20

25

Syste

m ca

rbon

emiss

ions


times103




138

1385

139

1395

14

1405

141

1415

142

1425



Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Parameters




119894(119909119871

119894)







product


119890119898




119890ℎ




Decision Variables


company119876119872

= [ 119902119872



119871



= [ 119902119878


firm 119902119878119894= 119902119871

119894+ 119902119872

119894








119886119894119895119899times119899


119895=1119886119894119895sdot 119902119895+ 119902119871

119894= 119902119894 so 119876 = 119860 sdot 119876 + 119876



119876119871= (119868 minus 119860) sdot 119876 = 119861 sdot 119876 (1)





119876 = (119868 minus 119860)minus1

119876119871= 119867 sdot 119876

119871 (2)


= 119889119894119895119899times119899


119894=

sum119899

119895=1119889119894119895119902119871








119894 We have





119894= 119902119904

119894= 119902119871

119894+ 119902119872






Δ119888119894= ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

(3)


(119883119894minus 119902119871

119894)

minus

=

119902119871

119894minus 119883119894

where 119883119894lt 119902119871

119894

0 where 119883119894ge 119902119871

119894

(119883119894minus 119902119871

119894)

+

=

0 where 119883119894lt 119902119871

119894

119883119894minus 119902119871

119894where 119883

119894ge 119902119871

119894

(4)




120587 (119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871

119894 119883119894)

minus[

[

ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895]

]

(5)



maxΠ = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

st 119902119871

119894ge 0

(6)




minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896) 119865119896(119902119871

119896) +

119899

sum

119894=1

V119894119889119894119896

= 0 (7)


119902119871lowast

119896= 119865minus

119896[

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (8)










supply chain is

max119902119871

119894

prod

MC(119876119871) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus 119904

119894120583119894minus 119870119894

minusV119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119864119894le 119862119894

119902119871

119894ge 0


(9)

where 119864119894= 119890119898

119894sdot sum119899

119895=1119889119894119895sdot 119902119871

119895+ 119864119870

119894+ 119890ℎ

119894sdot int

119902119871

119894

0119865119894(119909)119889119909


119894sum119899

119895=1119889119894119895

sdot 119902119871


119870

119894 The


119894

0119865119894(119909) 119889119909 so the


119894sdot int

119902119871

119894

0119865119894(119909) 119889119909






min 1198691(119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119892119894(119902119871

119894) = 119862

119894minus 119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895

minus 119864119870

119894minus 119890ℎ

119894sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 ge 0

119902119871

119894ge 0


(10)


119899120593(119876119871

119903(119896)

) = 1198691(119876119871) + 119903(119896)

sum119899


119903(119896)

gt 0




Step 4 Take 119876119871


119876119871isinR119899120593(119876119871 119903(119896)

) = 1198691(119876119871) +

119903(119896)

sum119899

119894=1(1119892119894(119876119871)) 119876119871


Step 5 If 119903(119896)sum119899119894=1


is 119876119871

(119896)

Step 6 Otherwise 120574(119896+1)

= 119887120574(119896) set 119896 = 119896 + 1 and go back to

step four






119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)


119898

119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)

minus


prod

CT(119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871


119894)

minus

minus 119904119894

sdot (119883119894minus 119902119871

119894)

+


119899

sum

119895=1

119889119894119895119902119871

119895minus 120596

sdot[

[

119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895+ 119864119870

119894

+ 119890ℎ

119894sdot (119883119894minus 119902119871

119894)

minus]

]

(11)


max119864(prod

CT) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894+ 120596119890ℎ

119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus (V

119894+ 120596119890119898

119894)

sdot

119899

sum

119895=1

119889119894119895119902119871

119895minus 120596119864119870

119894minus 119904119894120583119894minus 119870119894

st 119902119871119894ge 0


(12)


119871







minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896) 119865119896(119902119871

119896)

+

119899

sum

119894=1

119889119894119896(V119894+ 120596119890119898

119894) = 0

(13)


119902119871lowast

119896= 119865minus1

119896[

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] (14)





119871




119894(119909119871

119894) = 120582

119894119890minus120582119894119909119871


119894(119909119871

119894) = 1minus119890

minus120582119894119909119871




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (15)


119894119896= 0 where 119896 gt 119894 It means


=


119894119896gt



AB

C

D

Outsidemarket

1 1 1 1

1 1

1 1

1 21 3


company 119896 we get 119889119894119896



119879 119875 = [10 6 4 1]119879 119867 =

[ℎ1 ℎ2 ℎ3 ℎ4] = [06 04 03 005]

119879 and 119878 = [1199041 1199042 1199043 1199044] =







demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617


119871 in a given 120583 119881119901 119867


120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]





Price 119875


Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168



119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)






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119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


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Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894 We have





119894= 119902119904

119894= 119902119871

119894+ 119902119872






Δ119888119894= ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

(3)


(119883119894minus 119902119871

119894)

minus

=

119902119871

119894minus 119883119894

where 119883119894lt 119902119871

119894

0 where 119883119894ge 119902119871

119894

(119883119894minus 119902119871

119894)

+

=

0 where 119883119894lt 119902119871

119894

119883119894minus 119902119871

119894where 119883

119894ge 119902119871

119894

(4)




120587 (119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871

119894 119883119894)

minus[

[

ℎ119894sdot (119883119894minus 119902119871

119894)

minus

+ 119904119894sdot (119883119894minus 119902119871

119894)

+

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895]

]

(5)



maxΠ = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

st 119902119871

119894ge 0

(6)




minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896) 119865119896(119902119871

119896) +

119899

sum

119894=1

V119894119889119894119896

= 0 (7)


119902119871lowast

119896= 119865minus

119896[

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (8)










supply chain is

max119902119871

119894

prod

MC(119876119871) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus 119904

119894120583119894minus 119870119894

minusV119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119864119894le 119862119894

119902119871

119894ge 0


(9)

where 119864119894= 119890119898

119894sdot sum119899

119895=1119889119894119895sdot 119902119871

119895+ 119864119870

119894+ 119890ℎ

119894sdot int

119902119871

119894

0119865119894(119909)119889119909


119894sum119899

119895=1119889119894119895

sdot 119902119871


119870

119894 The


119894

0119865119894(119909) 119889119909 so the


119894sdot int

119902119871

119894

0119865119894(119909) 119889119909






min 1198691(119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119892119894(119902119871

119894) = 119862

119894minus 119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895

minus 119864119870

119894minus 119890ℎ

119894sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 ge 0

119902119871

119894ge 0


(10)


119899120593(119876119871

119903(119896)

) = 1198691(119876119871) + 119903(119896)

sum119899


119903(119896)

gt 0




Step 4 Take 119876119871


119876119871isinR119899120593(119876119871 119903(119896)

) = 1198691(119876119871) +

119903(119896)

sum119899

119894=1(1119892119894(119876119871)) 119876119871


Step 5 If 119903(119896)sum119899119894=1


is 119876119871

(119896)

Step 6 Otherwise 120574(119896+1)

= 119887120574(119896) set 119896 = 119896 + 1 and go back to

step four






119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)


119898

119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)

minus


prod

CT(119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871


119894)

minus

minus 119904119894

sdot (119883119894minus 119902119871

119894)

+


119899

sum

119895=1

119889119894119895119902119871

119895minus 120596

sdot[

[

119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895+ 119864119870

119894

+ 119890ℎ

119894sdot (119883119894minus 119902119871

119894)

minus]

]

(11)


max119864(prod

CT) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894+ 120596119890ℎ

119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus (V

119894+ 120596119890119898

119894)

sdot

119899

sum

119895=1

119889119894119895119902119871

119895minus 120596119864119870

119894minus 119904119894120583119894minus 119870119894

st 119902119871119894ge 0


(12)


119871







minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896) 119865119896(119902119871

119896)

+

119899

sum

119894=1

119889119894119896(V119894+ 120596119890119898

119894) = 0

(13)


119902119871lowast

119896= 119865minus1

119896[

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] (14)





119871




119894(119909119871

119894) = 120582

119894119890minus120582119894119909119871


119894(119909119871

119894) = 1minus119890

minus120582119894119909119871




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (15)


119894119896= 0 where 119896 gt 119894 It means


=


119894119896gt



AB

C

D

Outsidemarket

1 1 1 1

1 1

1 1

1 21 3


company 119896 we get 119889119894119896



119879 119875 = [10 6 4 1]119879 119867 =

[ℎ1 ℎ2 ℎ3 ℎ4] = [06 04 03 005]

119879 and 119878 = [1199041 1199042 1199043 1199044] =







demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617


119871 in a given 120583 119881119901 119867


120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]





Price 119875


Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168



119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)






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times103

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0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

ion

elas

ticity

of

profi

t of t

he su

pply

chai

n


AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


0

50

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Plan

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Syste

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Syste

m ca

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times103




138

1385

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14

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1425



Carb

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023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896) 119865119896(119902119871

119896) +

119899

sum

119894=1

V119894119889119894119896

= 0 (7)


119902119871lowast

119896= 119865minus

119896[

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (8)










supply chain is

max119902119871

119894

prod

MC(119876119871) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus 119904

119894120583119894minus 119870119894

minusV119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119864119894le 119862119894

119902119871

119894ge 0


(9)

where 119864119894= 119890119898

119894sdot sum119899

119895=1119889119894119895sdot 119902119871

119895+ 119864119870

119894+ 119890ℎ

119894sdot int

119902119871

119894

0119865119894(119909)119889119909


119894sum119899

119895=1119889119894119895

sdot 119902119871


119870

119894 The


119894

0119865119894(119909) 119889119909 so the


119894sdot int

119902119871

119894

0119865119894(119909) 119889119909






min 1198691(119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119892119894(119902119871

119894) = 119862

119894minus 119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895

minus 119864119870

119894minus 119890ℎ

119894sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 ge 0

119902119871

119894ge 0


(10)


119899120593(119876119871

119903(119896)

) = 1198691(119876119871) + 119903(119896)

sum119899


119903(119896)

gt 0




Step 4 Take 119876119871


119876119871isinR119899120593(119876119871 119903(119896)

) = 1198691(119876119871) +

119903(119896)

sum119899

119894=1(1119892119894(119876119871)) 119876119871


Step 5 If 119903(119896)sum119899119894=1


is 119876119871

(119896)

Step 6 Otherwise 120574(119896+1)

= 119887120574(119896) set 119896 = 119896 + 1 and go back to

step four






119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)


119898

119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)

minus


prod

CT(119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871


119894)

minus

minus 119904119894

sdot (119883119894minus 119902119871

119894)

+


119899

sum

119895=1

119889119894119895119902119871

119895minus 120596

sdot[

[

119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895+ 119864119870

119894

+ 119890ℎ

119894sdot (119883119894minus 119902119871

119894)

minus]

]

(11)


max119864(prod

CT) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894+ 120596119890ℎ

119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus (V

119894+ 120596119890119898

119894)

sdot

119899

sum

119895=1

119889119894119895119902119871

119895minus 120596119864119870

119894minus 119904119894120583119894minus 119870119894

st 119902119871119894ge 0


(12)


119871







minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896) 119865119896(119902119871

119896)

+

119899

sum

119894=1

119889119894119896(V119894+ 120596119890119898

119894) = 0

(13)


119902119871lowast

119896= 119865minus1

119896[

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] (14)





119871




119894(119909119871

119894) = 120582

119894119890minus120582119894119909119871


119894(119909119871

119894) = 1minus119890

minus120582119894119909119871




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (15)


119894119896= 0 where 119896 gt 119894 It means


=


119894119896gt



AB

C

D

Outsidemarket

1 1 1 1

1 1

1 1

1 21 3


company 119896 we get 119889119894119896



119879 119875 = [10 6 4 1]119879 119867 =

[ℎ1 ℎ2 ℎ3 ℎ4] = [06 04 03 005]

119879 and 119878 = [1199041 1199042 1199043 1199044] =







demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617


119871 in a given 120583 119881119901 119867


120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]





Price 119875


Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168



119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)






200

300

400

500

600

700

800

Supp

ly ch

ain

profi

t

B

C

D

A

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80



468

10121416182022

Syste

m to

tal c

arbo

n em

issio

n

0 8 16 24 32 40 48 56 64 72 80

times103

A

BCD






0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

ion

elas

ticity

of

profi

t of t

he su

pply

chai

n


AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


0

50

100

150

200

250

300

350


A

B

C

D

Plan

ned

prod

uctio

n of

inde

pend

ent d

eman

d

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


0

100

200

300

400

500

600

700

8000

008

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


Syste

m p

rofit

0

5

10

15

20

25

Syste

m ca

rbon

emiss

ions


times103




138

1385

139

1395

14

1405

141

1415

142

1425



Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











min 1198691(119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

st 119892119894(119902119871

119894) = 119862

119894minus 119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895

minus 119864119870

119894minus 119890ℎ

119894sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 ge 0

119902119871

119894ge 0


(10)


119899120593(119876119871

119903(119896)

) = 1198691(119876119871) + 119903(119896)

sum119899


119903(119896)

gt 0




Step 4 Take 119876119871


119876119871isinR119899120593(119876119871 119903(119896)

) = 1198691(119876119871) +

119903(119896)

sum119899

119894=1(1119892119894(119876119871)) 119876119871


Step 5 If 119903(119896)sum119899119894=1


is 119876119871

(119896)

Step 6 Otherwise 120574(119896+1)

= 119887120574(119896) set 119896 = 119896 + 1 and go back to

step four






119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)


119898

119894sum119899

119895=1119889119894119895sdot119902119871

119895+119864119870

119894+119890ℎ

119894sdot(119883119894minus 119902119871

119894)

minus


prod

CT(119876119871) =

119899

sum

119894=1

119901119894sdotmin (119902

119871


119894)

minus

minus 119904119894

sdot (119883119894minus 119902119871

119894)

+


119899

sum

119895=1

119889119894119895119902119871

119895minus 120596

sdot[

[

119890119898

119894

119899

sum

119895=1

119889119894119895sdot 119902119871

119895+ 119864119870

119894

+ 119890ℎ

119894sdot (119883119894minus 119902119871

119894)

minus]

]

(11)


max119864(prod

CT) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894minus (119901119894+ ℎ119894+ 119904119894+ 120596119890ℎ

119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 minus (V

119894+ 120596119890119898

119894)

sdot

119899

sum

119895=1

119889119894119895119902119871

119895minus 120596119864119870

119894minus 119904119894120583119894minus 119870119894

st 119902119871119894ge 0


(12)


119871







minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896) 119865119896(119902119871

119896)

+

119899

sum

119894=1

119889119894119896(V119894+ 120596119890119898

119894) = 0

(13)


119902119871lowast

119896= 119865minus1

119896[

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] (14)





119871




119894(119909119871

119894) = 120582

119894119890minus120582119894119909119871


119894(119909119871

119894) = 1minus119890

minus120582119894119909119871




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (15)


119894119896= 0 where 119896 gt 119894 It means


=


119894119896gt



AB

C

D

Outsidemarket

1 1 1 1

1 1

1 1

1 21 3


company 119896 we get 119889119894119896



119879 119875 = [10 6 4 1]119879 119867 =

[ℎ1 ℎ2 ℎ3 ℎ4] = [06 04 03 005]

119879 and 119878 = [1199041 1199042 1199043 1199044] =







demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617


119871 in a given 120583 119881119901 119867


120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]





Price 119875


Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168



119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)






200

300

400

500

600

700

800

Supp

ly ch

ain

profi

t

B

C

D

A

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80



468

10121416182022

Syste

m to

tal c

arbo

n em

issio

n

0 8 16 24 32 40 48 56 64 72 80

times103

A

BCD






0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

ion

elas

ticity

of

profi

t of t

he su

pply

chai

n


AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


0

50

100

150

200

250

300

350


A

B

C

D

Plan

ned

prod

uctio

n of

inde

pend

ent d

eman

d

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


0

100

200

300

400

500

600

700

8000

008

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


Syste

m p

rofit

0

5

10

15

20

25

Syste

m ca

rbon

emiss

ions


times103




138

1385

139

1395

14

1405

141

1415

142

1425



Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus119901119896minus 119904119896+ (119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896) 119865119896(119902119871

119896)

+

119899

sum

119894=1

119889119894119896(V119894+ 120596119890119898

119894) = 0

(13)


119902119871lowast

119896= 119865minus1

119896[

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] (14)





119871




119894(119909119871

119894) = 120582

119894119890minus120582119894119909119871


119894(119909119871

119894) = 1minus119890

minus120582119894119909119871




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1V119894119889119894119896

(119901119896+ ℎ119896+ 119904119896)

] (15)


119894119896= 0 where 119896 gt 119894 It means


=


119894119896gt



AB

C

D

Outsidemarket

1 1 1 1

1 1

1 1

1 21 3


company 119896 we get 119889119894119896



119879 119875 = [10 6 4 1]119879 119867 =

[ℎ1 ℎ2 ℎ3 ℎ4] = [06 04 03 005]

119879 and 119878 = [1199041 1199042 1199043 1199044] =







demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617


119871 in a given 120583 119881119901 119867


120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]





Price 119875


Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168



119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)






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468

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0 8 16 24 32 40 48 56 64 72 80

times103

A

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0010203040506070809

1

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Car

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AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


0

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138

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6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












demandSystemprofit

[06 04 04 008] [10473 16780 21030 30520] 80306[08 04 04 008] [9985 16780 21030 30520] 78261[10 04 04 008] [9520 16780 21030 30520] 76311[12 04 04 008] [9076 16780 21030 30520] 74451[14 04 04 008] [8650 16780 21030 30520] 72679[16 04 04 008] [8242 16780 21030 30520] 70990[16 06 04 008] [7472 15396 21030 30520] 64635[16 08 04 008] [6758 14163 21030 30520] 58837[16 10 04 008] [6091 13054 21030 30520] 53549[16 12 04 008] [5465 12044 21030 30520] 48731[16 14 04 008] [4877 11118 21030 30520] 44349[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 06 008] [7108 15396 18291 30520] 59255[16 04 08 008] [6091 14163 16063 30520] 48919[16 04 10 008] [5167 13054 14184 30520] 39806[16 04 12 008] [4321 12044 12560 30520] 31784[16 04 14 008] [3542 11118 11130 30520] 24747[16 04 04 008] [8242 16780 21030 30520] 70990[16 04 04 010] [7850 16205 20727 28516] 67055[16 04 04 012] [7472 15659 20430 26764] 63284[16 04 04 014] [7108 15138 20141 25207] 59670[16 04 04 016] [6758 14640 19859 23806] 56203[16 04 04 018] [6419 14163 19583 22532] 52876[16 04 04 020] [6091 13706 19313 21365] 49682[16 04 04 022] [5773 13267 19050 20287] 46617


119871 in a given 120583 119881119901 119867


120 140]119879 V = [16 04 04 008] 119867 = [ℎ

1 ℎ2 ℎ3 ℎ4] =

[06 04 03 005]119879 119878 = [119904

1 1199042 1199043 1199044] = [08 04 02 01]

119879and 119870 = [30 21 16 10]





Price 119875


Systemprofit

[100 6 4 1] [8242 16780 21030 30520] 70990[95 6 4 1] [7793 16780 21030 30520] 68233[90 6 4 1] [7324 16780 21030 30520] 65581[85 6 4 1] [6831 16780 21030 30520] 63044[80 6 4 1] [6313 16780 21030 30520] 60636[75 6 4 1] [5766 16780 21030 30520] 58368[70 6 4 1] [5188 16780 21030 30520] 56259[65 6 4 1] [4574 16780 21030 30520] 54328[10 60 4 1] [8242 16780 21030 30520] 70990[10 55 4 1] [8242 15925 21030 30520] 66691[10 50 4 1] [8242 14998 21030 30520] 62498[10 45 4 1] [8242 13989 21030 30520] 58433[10 40 4 1] [8242 12879 21030 30520] 54520[10 35 4 1] [8242 11647 21030 30520] 50793[10 30 4 1] [8242 10262 21030 30520] 47297[10 25 4 1] [8242 8682 21030 30520] 44099[10 6 40 1] [8242 16780 21030 30520] 70990[10 6 38 1] [8242 16780 20485 30520] 69016[10 6 36 1] [8242 16780 19913 30520] 67062[10 6 34 1] [8242 16780 19313 30520] 65130[10 6 32 1] [8242 16780 18682 30520] 63223[10 6 30 1] [8242 16780 18015 30520] 61343[10 6 28 1] [8242 16780 17309 30520] 59493[10 6 26 1] [8242 16780 16558 30520] 57679[10 6 4 20] [8242 16780 21030 39280] 83852[10 6 4 18] [8242 16780 21030 37913] 81229[10 6 4 16] [8242 16780 21030 36398] 78626[10 6 4 14] [8242 16780 21030 34699] 76047[10 6 4 12] [8242 16780 21030 32765] 73499[10 6 4 10] [8242 16780 21030 30520] 70990[10 6 4 08] [8242 16780 21030 27845] 68538[10 6 4 06] [8242 16780 21030 24536] 66168



119875 = [10 6 4 1]119879 120583 = [100 112 120 140]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(16)






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0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80



468

10121416182022

Syste

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arbo

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issio

n

0 8 16 24 32 40 48 56 64 72 80

times103

A

BCD






0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

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elas

ticity

of

profi

t of t

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AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


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004

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4

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6


Syste

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0

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times103




138

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1395

14

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Carb

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6

016

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02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













200

300

400

500

600

700

800

Supp

ly ch

ain

profi

t

B

C

D

A

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80



468

10121416182022

Syste

m to

tal c

arbo

n em

issio

n

0 8 16 24 32 40 48 56 64 72 80

times103

A

BCD






0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

ion

elas

ticity

of

profi

t of t

he su

pply

chai

n


AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


0

50

100

150

200

250

300

350


A

B

C

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Plan

ned

prod

uctio

n of

inde

pend

ent d

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d

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


0

100

200

300

400

500

600

700

8000

008

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


Syste

m p

rofit

0

5

10

15

20

25

Syste

m ca

rbon

emiss

ions


times103




138

1385

139

1395

14

1405

141

1415

142

1425



Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0010203040506070809

1

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76

Car

bon

emiss

ion

elas

ticity

of

profi

t of t

he su

pply

chai

n


AB

CD




119902119871lowast

119896= minus120583119896ln[1 minus

119901119896+ 119904119896minus sum119899

119894=1119889119894119896(V119894+ 120596119890119898

119894)

(119901119896+ ℎ119896+ 119904119896+ 120596119890ℎ

119896)

] minus 1199090

119896

(17)



120583 = [100 112 120 140]119879 119875 = [10 6 4 1]

119879

119867 = [06 04 03 005]119879 119878 = [08 04 02 01]

119879

119870 = [30 21 16 10]119879 V = [16 04 04 008]

119879

119890119898

= [2 3 5 8]119879 119864

119870= [20 14 12 10]

119879

119890ℎ= [05 04 03 01]

119879

(18)


120596


Systemprofits

Systemcarbon

emissions0 [8242 16780 21030 30520] 70990 20526000008 [8078 16545 20869 29840] 69362 20188940016 [7917 16315 20709 29191] 67760 19856000024 [7758 16089 20551 28572] 66184 19530210032 [7602 15868 20395 27978] 64635 1921121004 [7448 15651 20242 27409] 63111 18898680048 [7297 15438 20090 26862] 61611 18592300056 [7147 15230 19941 26336] 60136 18291810064 [7000 15025 19793 25829] 58684 17996940072 [6856 14824 19647 25340] 57256 1770747008 [6713 14626 19502 24867] 55851 17423160088 [6572 14432 19360 24410] 54468 17143820096 [6433 14241 19219 23968] 53108 16869260104 [6297 14053 19080 23539] 51769 16599290112 [6162 13869 18942 23122] 50452 1633376012 [6028 13687 18806 22718] 49155 16072490128 [5897 13509 18672 22325] 47880 15815360136 [5767 13333 18539 21944] 46625 15562200144 [5639 13160 18407 21572] 45390 15312900152 [5513 12990 18277 21210] 44175 1506733016 [5388 12822 18148 20857] 42979 14825370168 [5265 12656 18021 20513] 41803 14586910176 [5143 12493 17895 20177] 40645 14351850184 [5023 12333 17771 19849] 39506 14120070192 [4904 12175 17647 19528] 38386 138914902 [4786 12019 17525 19215] 37283 13666010208 [4670 11865 17404 18909] 36199 13443550216 [4555 11713 17285 18609] 35132 13224020224 [4442 11563 17166 18316] 34083 13007340232 [4330 11415 17049 18028] 33051 1279343024 [4219 11270 16933 17747] 32036 12582230248 [4109 11126 16818 17471] 31038 12373660256 [4001 10984 16704 17200] 30056 12167650264 [3894 10843 16592 16935] 29091 11964150272 [3787 10705 16480 16674] 28142 1176308028 [3682 10568 16369 16419] 27209 11564390288 [3578 10433 16259 16168] 26292 11368020296 [3476 10299 16151 15921] 25390 11173910304 [3374 10167 16043 15679] 24504 10982010312 [3273 10037 15937 15441] 23633 1079228032 [3173 9908 15831 15207] 22777 1060466






7 Conclusion


0

50

100

150

200

250

300

350


A

B

C

D

Plan

ned

prod

uctio

n of

inde

pend

ent d

eman

d

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


0

100

200

300

400

500

600

700

8000

008

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


Syste

m p

rofit

0

5

10

15

20

25

Syste

m ca

rbon

emiss

ions


times103




138

1385

139

1395

14

1405

141

1415

142

1425



Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













7 Conclusion


0

50

100

150

200

250

300

350


A

B

C

D

Plan

ned

prod

uctio

n of

inde

pend

ent d

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d

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


0

100

200

300

400

500

600

700

8000

008

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6


Syste

m p

rofit

0

5

10

15

20

25

Syste

m ca

rbon

emiss

ions


times103




138

1385

139

1395

14

1405

141

1415

142

1425



Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










138

1385

139

1395

14

1405

141

1415

142

1425



Carb

on em

issio

n el

astic

ity o

f pro

fit

000

8

004

007

2

010

4

013

6

016

8

02

023

2

026

4

029

6




Appendix


max119902119871

119894ge0

prod = 119864 (120587) =

119899

sum

119894=1

(119901119894+ 119904119894) sdot 119902119871

119894

minus (119901119894+ ℎ119894+ 119904119894) int

119902119871

119894

0

119865 (119883119894) 119889119883119894


119899

sum

119895=1

119889119894119895119902119871

119895

(A1)

can be written as

min119902119871

119894ge0119894isin119873

119869 (119876119871) =

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119871

119894+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119871

119894

0

119865119894 (119909) 119889119909 + 119904

119894120583119894

+119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119871

119895

(A2)


119871) is convex


119909




1+ 1205821199092) le 120582120601(119909

1) +

120582120601(1199092) so int

1205821199091+1205821199092

0119865(119905)119889119905 le 120582 int

1199091

0119865(119905)119889119905 + 120582 int

1199092

0119865(119905)119889119905


119894

1+120582119902119894

2

0119865119894(119909)119889119909 le

120582int

119902119894

1

0119865119894(119909)119889119909 + 120582int

119902119894

2

0119865119894(119909)119889119909

∘

Thus

119869 (120582119876119871

1+ 120582119876119871

2) minus [120582119869 (119876

119871

1) + 120582119869 (119876

119871

2)]

=

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot (120582119902

119894

1+ 120582119902119894

2) + (119901

119894+ ℎ119894+ 119904119894)

sdot int

(120582119902119894

1+120582119902119894

2)

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894

+ V119894sdot

119899

sum

119895=1

119889119894119895(120582119902119895

1+ 120582119902119895

2)

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

1+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

1

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+V119894sdot

119899

sum

119895=1

119889119894119895119902119895

1

minus 120582

119899

sum

119894=1

minus (119901119894+ 119904119894) sdot 119902119894

2+ (119901119894+ ℎ119894+ 119904119894)

sdot int

119902119894

2

0

119865119894 (119909) 119889119909 + 119904

119894120583119894+ 119870119894+ V119894sdot

119899

sum

119895=1

119889119894119895119902119895

2

=

119899

sum

119894=1

(119901119894+ ℎ119894+ 119904119894) [int

120582119902119894

1+120582119902119894

2

0

119865119894 (119909) 119889119909 minus 120582int

119902119894

1

0

119865119894 (119909) 119889119909

minus120582int

119902119894

2

0

119865119894 (119909) 119889119909]

(A3)



1+120582119902119894

2

0119865119894(119909)119889119909 minus 120582int

119902119894

1

0119865119894(119909)119889119909 minus

120582int

119902119894

2

0119865119894(119909)119889119909 le 0 So 119869(120582119876

119871

1+120582119876119871

2) minus [120582119869(119876

119871

1) + 120582119869(119876

119871

2)] lt 0






Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article joint optimal production planning for

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