Managmnt Scinc and Enginring Vol 7 No 3 pp 3-34 DOI:3968/jms9335X374 ISSN 93-34 [Print] ISSN 93-35X [Onlin] wwwcscanadant wwwcscanadaorg Two Products Manufacturr s Production Dcisions with Carbon Constraint LU Li [a] ; CHEN Xu [a] [a] School of Managmnt and Economics Univrsity of Elctronic Scinc and Tchnology of China Chngdu China Corrsponding author Supportd by National Natural Scinc Foundation of China (No 778) Program for Nw Cntury Excllnt Talnts in Univrsity (NoNCET--87) Youth Foundation for Humanitis and Social Scincs of Ministry of Education of China (No YJC63) and Sichuan Provinc Ky Tchnology R&D Program (No FZ3) Rcivd 3 January 3; accptd 4 March 3 Abstract In this papr w considr a manufactur which producs both ordinary products and grn products in a monopoly markt and invstigat his production dcisions with carbon constraint Firstly w driv th manufacturr s optimal production and maximum profit without carbon constraint Thn w discuss th optimal production and maximum profit with carbon constraint in diffrnt situation Th rsults indicat that manufacturr s optimal production and maximum profit with carbon constraint ar lss than thm without carbon constraint optimal production is an incrasing function with carbon constraint Ky words: Production dcision; Carbon constraint; Complt monopoly LU Li CHEN Xu (3) Two Products Manufacturr s Production Dcisions with Carbon Constraint Managmnt Scinc and Enginring 7() 3-34 Availabl from: http://wwwcscanada nt/indxphp/ms/articl/viw/jms 9335X374 DOI: http://dxdoiorg/3968/jms9335x374 INTRODUCTION Govrnmnts of diffrnt nations hav paid high attntion to th svr situation of global warming Th intrnational convntion Kyoto protocol which has com into forc sinc 5 sts a systm calld Cap and Trad which is dsignd to st missions targts for ach country or rgion to rduc global grnhous gas mission ffctivly Du to this targt all countris and rgions hav to distribut th whol carbon mission targt to th micro ntrpris lvl as a rsult manufacturrs whos production ar dpndnt on carbon ought to fac th svr carbon constraint To ovrcom th carbon constraint manufacturrs can rduc production control pollution mission or produc grn product Among ths mans mor and mor manufacturrs hav chosn to produc grn product Comparing with th traditional ordinary product grn product is totally diffrnt in production cost carbon missions of unit product markt dmand and pric tc such as LED lamp and ordinary lamp Grn product has highr production cost and lowr carbon missions of unit product than ordinary product so it can b usd as an important man of saving carbon missions Although mor and mor manufacturs hav bgun to produc both ordinary products and grn product in th xisting litratur most of th rsarchs hav only focusd on th production dcisions of manufacturrs who producd only on singl product Dobos [] discussd th influnc of missions trading on manufactur s production and drivd th optimal production basd dynamic Arrow-Karlin modl Ltmath and Balakrishnan [] stablishd th production modl with mission cap using th mixd intgr programming mthod and solvd th problm to gt th optimal production Rong and Lahdlma [3] stablishd th optimal production modl of a hat and powr production company and solvd th problm using multi-priod stochastic optimization to driv th optimal production Du Shaofu tal [4] thought firms might obtain mission prmits in thr diffrnt ways govrnmnt uota and markt trad and purifying and thn stablishd an optimal production modl with mission prmits and 3
trading thn drivd th optimal production dcisions Tsai tal [5] took carbon tax as a production cost and invstigatd th grn product manufactur s production dcisions using mathmatical programming mthod and drivd th optimal productions in diffrnt situations In this papr w considr a manufactur which producs both ordinary products and grn products in a monopoly markt and invstigat its production dcisions with carbon constraint This papr solvs th following problms: i) what is th two products manufacturr s optimal production policy in th prsnc of carbon constrain? ii) How dos th carbon constrain influnc th manufacturr s dcision and maximum xpctd profit? MODEL DESCRIPTIONS AND ASSUMPTION W considr a manufactur which producs both ordinary products and grn products in a monopoly markt and invstigat his production dcisions on both two products with carbon constraint Th manufacturr s total amount of carbon mission of on priod allocatd by th govrnmnt is E units Production cost of ordinary products and grn product ar c and c rspctivly and c c Carbon mission of producing unit product of ordinary products and grn product ar and rspctivly and As th manufacturr is a complt monopoly company sals pric of unit product is ngativly rlatd to th production W suppos th rlationship is a linar ngativ dmand function As a rsult sals pric of unit product of two products ar p ( )a _ b and p ( )a _ b whr a a stands for th highst pric of ths two products and b b stands for th pric lasticity cofficint of ths two products and a b a b > Notably th sals pric of grn product is highr than ordinary on that isp ( ) > p ( ) Th manufacturr taks th maximizing th xpctd profit as th dcision goal and tak th production of ordinary product and grn product as dcision variabls and mak th optimal production dcision according to diffrnt carbon constraints MODEL FORMULATIONS AND SOLUTION Th Situation without Carbon Constrain Without th carbon constrain th manufacturr s profit is π ( ) p( ) c + p( ) c Th manufacturr s dcision is max ( ) π Proposition Without th carbon constrain th manufacturr s optimal production and maximum profit xist and ar uniu Th manufacturr s optimal production of ordinary product and grn product rspctivly ar ( a c) b and ( ) b and th manufacturr s maximum profit is π 4b + 4b ( ) Proof ( ) As π a b c ( ) b < π ( ) ; a b c ( ) b < and ( ) π ( ) π thn D D ( ) b < and ( ) π ( ) π π ( ) π ( ) ( ) π ( ) π ( ) π ( ) π 4bb > So ( ) π is concav of and that is whn two kind products ar producd th manufacturr s optimal production and maximum profit xist and ar uniu without carbon constrain π ( ) π ( ) Lt w hav ; lt b w hav Substituting b and in to π( ) w gt th manufacturr s maximum profit ( ) a c a c π + 4b 4b Th Situation with Carbon Constrain With th carbon constrain th manufacturr s profit is π ( ) p( ) c + p( ) c Th manufacturr s dcision is 3
maxπ( ) + E st ; Proposition Whn E + th manufacturr s optimal production of ordinary product b and grn product rspctivly ar ( ) and ( ) b ; Whn E < + th manufacturr s optimal production of ordinary product and grn product rspctivly ar ( a c) ( a c) + be and b + b ( a c) ( a c) + b E b + b Proof Whn E + th carbon constrain dosn t work and th manufacturr produc th products according th optimal policy without carbon constrain Thy ar a c b b ( ) and ( ) Whn E< + th carbon constrains work Th manufacturr cannot produc th two kinds of products according to th optimal policy simultanously At this tim w hav < Thn ( ) ( ) b > ( ) b a c ( a c) b b ( ) ( ) b > ( ) b ( a c) b b So π ( )is an incras function of and that is th manufacturr s profit will incras whn incras or This proprty shows that th manufacturr will us all th carbon uotas whn E< + Thn w hav E + At this point th manufacturr will compar th a b c marginal profits brought by unit carbon a b c and Finally th manufacturr will produc th two products whr th marginal profits of th two products ar ual a b c a b c So combin with E + w hav ( a c) ( a c) + be and b + b a c + b E b + b Put and max π w obtain th manufacturr s maximum profit with carbon constraint π ( ) Proposition indicats that whn th govrnmnt uotas is mor than th nd of manufacturr s optimal production th carbon constrain dosn t work In this situation th manufacturr will produc both two kinds of products in optimal production without carbon constrain Howvr whn th govrnmnt uotas is lss than th nd of manufacturr s optimal production th carbon constrain dosn t work In this situation th manufactur will compar th two products marginal profit brought by on unit of carbon and thn produc th two products whr th marginal profits of th two products ar ual 3 Impact Analysis of Carbon Constrain W discuss th impact of carbon constrain on th manufacturr s production dcision-making according to Proposition and Proposition W hav: Proposition 3 Whn E + th manufacturr s optimal production and maximum profit with and without carbon constrain ar ual Whn E< + th manufacturr s optimal production maximum profit ar smallr than th production without carbon constrain Proof Compar and and Thn a c Whn E + b And according to optimization b thory as into ( ) and both tak th valu at whr th function gt th gratst valu π ( ) ( π ) Whn E< + th carbon constrains work From Proposition w can driv ( a c) ( a c) + be < b + b ( a c) ( a c) + b E < b + b And according to optimization thory as and dos not tak th valu at whr th function gt th 33
gratst valu π ( ) ( π ) > Proposition 3 indicats that manufacturr s optimal production and maximum profit with carbon constraint ar lss than thm without carbon constraint Proposition 4 Whn E + and ar not affctd by E;Whn E< + both and ar th incrasing function with E Proof From Proposition Whn E< + b b Obviously d de and ar not affctd by E Whn E< + a c + be b + b a c + b E b + b d b Obviously > de b + b d de d b > So both and ar th de b + b incrasing function with E + Figur Rlationship btwn Optimal Production and Carbon Constraint Proposition 4 can b figurd out by figur It shows that whn th govrnmnt uotas is mor than th nd EE of manufacturr s optimal production manufacturr s optimal production ar not affctd by E howvr whn th govrnmnt uotas is lss than th nd of manufacturr s optimal production manufacturr s optimal production is an incrasing function with carbon constraint 3 CONCLUSIONS AND SUGGESTIONS FOR FURTHER RESEARCH In this papr w considr a manufactur which producs both ordinary products and grn proucts in a monopoly markt and invstigat his production dcisions with carbon constraint Firstly w driv th manufacturr s optimal production and maximum profit without carbon constraint Thn w discuss th optimal production with carbon constraint in diffrnt situation Th rsults indicat that manufacturr s optimal production and maximum profit with carbon constraint ar lss than thm without carbon constraint and optimal production is an incrasing function with carbon constraint Howvr this papr has only studid th situation with dmand crtainty W will do som furthr rsarch about th situation with random dmand and th situation of carbon mission trading to provid th manufactur with mor scintific production dcision-making guidanc REFERENCES [] Dobos I (5) Th ffcts of mission trading on production and invntoris in th Arrow-Karlin modl Intrnational Journal of Production Economics 93-94(8) 3-38 [] Ltmath P Balakrishnan N (5) Environmntal considration on th optimal product mix Europan Journal of Oprational Rsarch 67() 398-4 [3] Rong A Y Lahdlma R CO missions trading planning in combind hat and powr production via multi-priod stochastic optimization Europan Journal of Oprational Rsarch 76(3) 874-895 [4] Du SF Dong J F Liang L Zhang J J (9) Optimal production policy with mission prmits and trading Chins Journal of Managmnt Scinc 7(3) 8-86 [5] Tsai W H Lin W R Fan Y W L P L Lin S J Hsu J L () Applying a mathmatical programming approach for a grn product mix dcision Intrnational Journal of Production Rsarch DOI:8/75435 5549 34
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