THE CHOICE OF REVERSE LOGISTICS RECYCLING MODEL IN THE NONLINEAR DEMAND

Transcription

1 Journal of athematical Sciences: dvances and lications Volume ages 9-61 vailale at htt://scientificadvancescoin DOI: htt://dxdoiorg/101864/jmsaa_ THE CHOICE OF EVESE LOGISTICS ECYCLING ODEL IN THE NONLINE DEND LEING ZHOU and SHENGJI XUE Education College of Jinan University China @qqcom tlozy@jnueducn stract We discuss the choice of recycling mode of reverse logistics under nonlinear demand In determining ilateral monooly market and leading manufacturer of the closed-loo suly chain recycling mode ere analyzed ased on the game theory of Stackelerg equilirium constructing reverse logistics three asic recovery mode -- the manufacturer for recycling reclaiming retailers and third-arty recovery of nonlinear rogramming model through mathematical derivation ere decision analysis and lingo softare simulation verification It is ointed out that the retailer is resonsile for recycling is the est ay to recovery and finally the otimal model of otimization 1 Introduction With the exhaustion of the sources of the earth and the eole environmental rotection consciousness enhancement the aste roducts of recycling has ecome a kind of ne logistics mode-reverse logistics mode In the reverse logistics of three tyes (recycling re-use and re-manufacturing) ased on re-manufacturing reverse logistics has greater comlexity and higher economic value here in the collector 010 athematics Suject Classification: 9006 Keyords and hrases: nonlinear demand reverse logistics recycling mode Stackelerg equilirium nonlinear rogramming eceived ril Scientific dvances ulishers

2 40 LEING ZHOU and SHENGJI XUE ith the mode of selection is a research hotsot t resent the research of it has made aundant achievements [1-8] the most reresentative is Savaskan etc [1 ] of ork they are considered three kind of recyclings atterns namely the manufacturer direct recovery reclaiming the retailer entrusted y the manufacturer the manufacturer shall entrust a third arty comany for recovery use game-theoretic models to study ho the memers of the enterrise in different recovery mode of the otimal decision-making Hoever the research is all aout the market rice of the linear function is done under the assumtion of In real life of one-of-a-kind roduct needs more reresentation is that the rice of the non-linear function This aer is to ut forard the demand for recycling mode of decision-making Non-linear Demand Function Demand function in the reresentation of a certain market size in the actual commodity markets each roduct ill have a certain amount of market caacity In the form of linear demand function such as Savsakan s shon in Figure 1 Figure 1 Function diagram: D( )

3 THE CHOICE OF EVESE LOGISTICS ECYCLING 41 When α < 1 as shon in Figure hen the rice is relatively lo the demand for commodities as rices change raidly hen the commodity rice is too high the demand for commodities suject to rice influence is not ovious In a certain category of ricing reresents change rate of demand rice elasticity of demand is less than the lack of goods such as some necessities of life Figure Function diagram: D ( ) ( α < 1) α When α > 1 as shon in Figure hen the rice is relatively lo the demand for goods y the rice is not ovious in a certain range of rices ith the increase in rices the demand for goods decreased raidly The roortion of the change in demand is greater than the roortion of the rice change hich reresents the commodity is flexile Such as electronic roducts moile hones television and other household electrical aliances is the main tye of reverse logistics research

4 4 LEING ZHOU and SHENGJI XUE Figure Function diagram: D ( ) ( α > 1) α In vie of the comlexity of nonlinear comutation e make α the research in this aer to determine the market demand function D( ) in order to carry out relevant research and analysis ork rolem Descrition and Basic ssumtions In this aer the asic meanings of the symols: C m is ra materials for manufacturing a ne roduct cost C r is the cost of ne roducts for remanufacture of aste roducts into and ne roducts; C < C is sales rice; ϖ is the manufacturer s holesale rice r m; rovide to retailers; is the recycling rice of aste roducts is the ay rice of for manufacturers to recycle aste roducts oviously here C m C r ; is recovery rate 0 1; according to [1 6] I L L > 0 I is the fixed investment of recycling L is the

5 THE CHOICE OF EVESE LOGISTICS ECYCLING 4 scale arameter oviously the average cost of the manufacturer is C ( ) C C the average cost of the recycling aste m 1 r m ; roducts including the recycle rice aid to consumer and the average cost of transortation in order to ensure the manufacturers can make rofit from recycling and re-manufacturing; here e assume O is exogeneous does not affect the value of demand i j is the rofit function of i under j recycle model here i m t resectively hich indicated that manufacturers retailers and the third arty and the overall rofits; j ll variales and functions of suerscrit said otimal solution and otimal value ssumtions are as follos: roduct demand function D( ) ; market is leading manufacturer (in fact it is usually true) Thus the manufacturer can as a leader in Stackelerg game; all arameters and variales are greater than zero meet and the sales rice C m hich rice ound (the actual calculation desirale for larger ositive) Can e seen from the asic symols and assumtions: the numer of recyclale roducts can e recycled D ( ) α the total cost of recycling is L D( ) L α 4 Study on Three odels of ecovery Under the ilateral monooly market the reverse logistics of the closed loo suly chain can e divided into direct recycling of manufacturers manufacturers commissioned y the manufacturers to recycle manufacturers commissioned third arty recovery elated models are constructed as follos:

6 44 LEING ZHOU and SHENGJI XUE 41 anufacturers direct recovery-( ) nonlinear rogramming rolems In the model the manufacturer is resonsile for the roduction of ne roducts recycling of aste roducts and the roduction of ne roducts retailers are only resonsile for sales Stark and Berg game decision making rocess: the manufacturer is the leader in Stackelerg retailers as a folloer manufacturers determines a holesale rice and recovery according to the reaction function of retailers retailers selling rice is determined according to the manufacturer s decision We have: for a given holesale rice the retailer s otimal resonse to the solution: ax ( )( ) (41) such as seudo concave analysis shos that the rice is seudo concave so est resonse equivalent to the first-order conditions: 0 (4) manufacturers according to the retailer s reaction function determine the holesale rice and return rate the rofit function: ( )( c ( ) ) L m (4) Taking (4) as the ojective function considering (4) and other assumtions the range of recovery rate is eteen 1 and 0 as the corresonding constraints so as to construct the folloing nonlinear rogramming rolem: ax ( )( c ( ) ) L m

7 THE CHOICE OF EVESE LOGISTICS ECYCLING 45 0 ( ) cm ( ) s t The manufacturer is resonsile for the recovery of the total rofit function under the model T (44) Because it is a nonlinear rogramming cannot find out the variales of the analytical solution e can use the Lagrange multilier method to estalish the relationshi eteen variales Due to various constraints on the otimal solution of inequality is elaxation using Lagrange multilier λ constructing a ne function: [( )( c ( ) ) L ] λ( ) Γ λ m here (45) λ for ne function variales resectively on these variales for artial derivative (46) (48) (410) and (41) ordered the first order artial derivatives of 0 relation (47) (49) (411) (41) and (414) Γ ( c ( ) ) 6λ λ m (46) Γ Order 0 that is [ λ ( cm ( ) ) ] λ 0 (47) Γ λ) (48)

8 46 Γ Order 0 LEING ZHOU and SHENGJI XUE that is λ (49) Γ ( )( ) L (410) Γ Order 0 that is ( )( ) (411) L Γ λ (41) Γ Order 0 that is λ (41) (414) We ill get to the conversion could also e used to calculate values of each variale to otain (49) (411) (41) and (414) sustitution (47) tye equation e derive this equation: 5 4 [ ] ( ) ( L) 4 CmL ( ) ( L ) L 0 (415) There is no formula solution for one yuan and five equations We cannot find the analytic formula of each variale value But these relationshis can hel us to derive a comarison of the size of each target value in the folloing

9 THE CHOICE OF EVESE LOGISTICS ECYCLING 47 4 etailer resonsile for recycling-( ) nonlinear rogramming rolem The retailer is resonsile for recovering the model the retailer is resonsile for sales of ne roducts and recycling of aste roducts manufacturer is resonsile for the manufacture of ne roducts manufacturers need to through the ayment roducts recycling comensation rice from the hands of the retailers get recycled roducts remanufacturing The decision-making rocess is under Stackelerg game decision frameork manufacturers first comensation rice and the holesale rice of is determined according to the reaction function of retailers retailers according to the manufacturer s decision to determine the sales rice and recovery rate of the retailer s reaction for solving: ax ( )( ( ) ) L (416) s aove has a seudo concave on the rice of and and the otimal reaction is equivalent to the first order artial derivative of and : To 0 (417) 0 (418) the manufacturer determines the holesale rice and the recycling rice according to the retailer s resonse function W and its rofit function is ( )( c ( ) ) m (419)

10 LEING ZHOU and SHENGJI XUE 48 To (419) as the ojective function considering the tye (417) (418) and other assumtions as constraints and the range of the recovery rate in the range of 1 to 0 the construction of model nonlinear rogramming rolems are as follos: ( )( ( ) ) ax c m ( ) ( ( ) ) ( )( ) L s t In this model hen the ojective function tends to the maximum The total rofit function is resonsile for recovery in retailer model: T (40) We have 41 similar calculation: ( ) [ ] ( ) [ ] (41) ( )( ) L (4)

11 THE CHOICE OF EVESE LOGISTICS ECYCLING 49 4 Indirect recycling of third arty comanies-( ) nonlinear rogramming rolem Under the model of the third arty resonsile for recycling the manufacturer the third arty is resonsile for recycling the aste roducts and roviding it to the manufacturer The manufacturer is resonsile for manufacturing and re-creating ne roducts and the retailer is resonsile for the sales of ne roducts The decision-making rocess is under Stackelerg game decision frameork manufacturers according to third arty comanies and retailers reaction function to determine the holesale rice of and to third arty comensation rice retailers and third-arty comanies according to the manufacturer s decision resectively sales rice and recovery rate of and the retailer s otimal resonse to solve: ax ( )( ) (4) ccording to the analysis of the seudo concave has a seudo concave to the rice of so the otimal reaction is equivalent to the first order condition 0 (44) third arty comanies according to the manufacturer to determine the conversion rice of to determine the recovery rate of aste roducts the otimal resonse to the solution: ax ( )( ) L (45) From to has the seudo concave so the otimal reaction function is equivalent to the first order condition:

12 LEING ZHOU and SHENGJI XUE 50 0 (46) the manufacturer's rofit function is ( )( ( ) ) c m (7) The (47) as the ojective function considering (44) (46) and other assumtions recovery rate range set as the corresonding constraints is eteen 0 and 1 uilding a model nonlinear rogramming rolems are as follos: ( )( ( ) ) ax c m ( ) ( )( ) ( ) c L s t m ccording to the ojective function it can e concluded that the ojective function tends to the maximum value hen In the third arty commissioned y the comany recycling model the total rofit function T (48)

13 THE CHOICE OF EVESE LOGISTICS ECYCLING 51 Similar to the 41 calculation e have (49) ( )( ) (40) L 5 Comarison and Selection Strategies of Three ecovery odes 51 elational comarison When the manufacturer does not directly recycling commissioned y the retailer or third arty comanies recycling aste roducts ill ay the rice of from the last fe sections of the analysis e can see Tale 1 results: Tale 1 nalytic solution of odel odel ccording to the results of the last section the relationshi eteen the otimal rice and the otimal recovery rate of the three recovery models are listed as shon in Tale

14 Tale Otimal rice and otimal recovery of the three recovery models Variale odel odel odel ( ) ( ) [ ] ( ) [ ] ( ) ( )( ) L ( ) ( )( ) L ( ) ( )( ) L

15 THE CHOICE OF EVESE LOGISTICS ECYCLING 5 We assume that no matter hat kind of recycling mode the roduction of the roduct quality is the same hen the direct recovery of manufacturers retailers from the manufacturer to the holesale rice of roduct unit cost is er unit roduct of the retailers direct ayment is When the retailer s recovery his unit cost is no longer a holesale rice ut holesale rice minus the retailers due to the recycling of aste roducts and otained from the manufacturer of the comensation er unit of roduct the actual cost is W B hen retailers also must e considered for roduct recovery of fixed costs When the retailer is actually sending ( ) er unit the rational retailers consider hether to choose the recycling of aste roducts so e only need to discuss the case of ( ) > In revious studies this condition is satisfied under linear demand For the same reason hen a third arty comany for recycling ecause manufacturers ay in addition to the manufacturing cost also includes the urchase of aste roducts of the conversion rice manufacturers in order to ass on this art of the sending ill certainly holesale rice higher than their recovery is roosed in order to comensate for this hen < not the manufacturers commissioned third-arty comanies to recycle aste roducts t the same time e need to discuss the: That is > ( ) > 5 roositional analysis > On the asis of the comarative analysis e can get the folloing roosition:

16 54 LEING ZHOU and SHENGJI XUE roosition 1 In the leading maker of ilateral monooly market three kinds of reverse logistics recycling mode ( ) ( ) and ( ) commissioned y the manufacturer retailer is resonsile for recycling roducts and manufacturers directly recycled roducts than manufacturers commissioned third-arty comanies recovery hen is more rice advantage in the market esecially to the retailer recycling mode of rice advantage is more significant That is > s the recycling model ( ) has a strong rice advantage the choice of manufacturers to recycle is conducive to imroving the roduct demand there are: D < D < D roosition In three recycling models ( ) ( ) and ( ) the retailer recovery ( ) hen the recovery is the most and hen the manufacturer commissioned third arty recovery ( ) the recovery rate is the loest that is < > < roosition In the three modes of recycling the manufacturer s rofit in the retailer recovery model (m ) is the largest in manufacturer is resonsile for recycling (m m) is smaller in the manufacturer commissioned third-arty comanies resonsile for recycling (m ) minimum that is > > roosition 4 In the three recycling models the retailer s rofit is the largest in the direct recovery of manufacturers hen the retailer is recycling smaller the third side of the comany is resonsile for the recovery of the minimum That is > >

17 THE CHOICE OF EVESE LOGISTICS ECYCLING 55 6 Simulation ode In ilateral monooly market structure the leading maker a lant to roduce some electronic roducts imlementation of the closed-loo suly chain of reverse logistics and the demand function resents nonlinear morhology: D( ) and manufacturers resonsile for recycling of aste roducts the manufacturing cost of the ne roduct is C 90 (100 yuan) and the use of rocessing aste m roducts re-manufacturing cost of the ne roduct is: C 74 (100 yuan) the treatment y the manufacturer after the recovery of the aste roducts re utilization rate is: 01 and cannot e used for remanufacturing of used roducts also has a residual value: S 1 (100 yuan) Simulation calculation is carried out in order to facilitate the suort of more data our simulation resectively in the recovery of the scale arameter: L fixed cost of recycled roducts recycling resectively: 4 y LINGO11 softare to solve the three recycling mode of nonlinear rogramming gloal otimal solution relevant results as shon in Tale shos: r

18 Tale Comarison of simulation results of kinds of recycling channels arket scale ecovery cost ecovery model D T L

19 THE CHOICE OF EVESE LOGISTICS ECYCLING 57 From Tale and model results come to the same conclusion and in the recovery of fixed costs and scale arameters are consistent commissioned y the manufacturer retailer recycling ( ) consumers can gain the maximum rofit Because of the loest selling rice for manufacturers the U recovery mode otimal efficiency The imact of the social environment the highest recycling efficiency of retailers can ring greater ositive and effect to the environment This recycling model can take into account the comrehensive enefits of economy and environment not only conducive to the sustainale develoment of enterrises ut also contriute to the sustainale develoment of the society as a hole 7 odel esult nalysis Due to the comlexity of the calculation of the nonlinear rogramming rolem can not through mathematical derivation comletely descrie the three recycling model of closed-loo suly chain channel overall rofit e refer to the derived roosition and the simulation results comarison some conclusions are dran as follos: Conclusion 1 In the case of all arameters consistent > > The recovery rate of the size reflects the reverse logistics recycling mode of success or not in the leading maker of ilateral monooly market the retailer is resonsile for recycling ( ) recovery rate is highest manufacturers direct recovery ( ) times the recovery efficiency is the loest is the third arty comany resonsile for recycling ( ) mode Because the third arty comany for recycling he on the marginal rofit of ( ) and manufacturers to recycle the marginal rofit ( ) ecause ( ) ( ) so thirdarty comanies on the recovery of investment to gain than the direct

20 58 LEING ZHOU and SHENGJI XUE recovery of manufacturers is less rofitale In addition manufacturers can also e directly recovered through the retailer s ricing strategy to romote the recycling of aste roducts resulting in the recovery of the to level effect This henomenon is consistent ith the linear demand and nonlinear demand of the roduct Conclusion In the case of all arameters consistent < < D > D > D By retailer recycling ( ) hen the sales rice is the loest the second is the manufacturer directly recycling ( ) the highest is hen the third arty comanies to recover ( ) arket demand and sales rice is inversely roortional to the relationshi so the market demand for the relationshi eteen the erformance of: D > D > D In monooly market leading manufacturer and the third arty comany rofit only in the third arty comany for recycling is likely to emerge ut third-arty comanies cannot directly enjoy recovery reverse logistics or remanufacturing roduce excess rofits he otained the rofit is the manufacturer and the retailer enjoyed remanufacturing rofit margins For manufacturers he can get more rofit from the loer retail rice ecause the manufacturer ill e ale to roduce a loer holesale rice hich ill enefit from increasing market demand When recovery for retailers retail rices in all recycling channels loest retailers through his ricing strategy to influence market demand and direct rofit from so manufacturers authorized dealer for recycling the demand for a roduct This result is consistent ith the conclusions dran from the study of the recovery model ased on linear demand Conclusion In the case of all arameters consistent > > ; > > When the recovery rate and the loest rice the market demand of the roducts is maximum as the main ody of the closed-loo suly chain memer manufacturers to maximize rofits Through the analysis

21 THE CHOICE OF EVESE LOGISTICS ECYCLING 59 of the aove comarative simulation in the structure of the market leading manufacturers manufacturers commissioned recovery mode hen the est retailers But the retailer is not in ( ) in the most rofitale The rofits of the hole channel is not necessarily in direct recovery hen the largest manufacturer Conclusion 4 In the case of all the arameters ith the increase of the degree of difficulty of the recovery L and the recovery of the fixed cost of increased resulting in decrease and T are reduced increase D decrease Conclusion 5 In all cases ith the same arameters the leading manufacturer monooly market entrusted y the manufacturer retailer ( ) recovery is the otimal decision makers Because the high recovery rate means that the environmental contriution is large lo sales rices means that the majority of consumer grous enefit These variales are in the retailer recycling mode ( ) in the est erformance So model is the otimal decision Even though our simulation results sho that the overall rofit of the channel in this model does not have the highest ut the closed-loo suly chain in addition to the ursuit of economic rofits ut also take into account the imact on the environment from the imlementation of the closed-loo suly chain need to select ( ) is otimal decision 8 Coordination Otimization nalysis of ( ) odel From the aove research e dra the conclusion: the entrusted y the manufacturer retailer ( ) recovery mode is a leading manufacturer recycling mode under the monooly market of the most effective imlementation of the closed-loo suly chain

22 60 LEING ZHOU and SHENGJI XUE In ( ) recycling model: ecause there are > > so manufacturers are illing to choose to entrust the retailer for recycling ut there are > > and in some cases close to In order to successfully imlement the closed-loo suly chain as a market leader in the manufacturer in order to make a rofit he should e aid y the additional recycling susidies to encourage retailers to recycle The corresonding coordination mechanism is as follos: anufacturers and retailers have negotiated a distriution coefficient α of hich 0 < α < 100 When F >> In this model only take the rofit Fα% hen take Fα% ecause the F is relatively large so Fα% > When the manufacturer is assigned to the retailer from the rofit F α% distriution the retailer ill consider the commitment to recycling usiness t the same time as a government agency in order to environmental guidance should also e through government susidies and other means to encourage retailers to oen u recycling channels so that ( ) model to achieve coordinated and otimized ojectives

23 THE CHOICE OF EVESE LOGISTICS ECYCLING 61 eferences [1] C Savaskan S Bhattacharya and L Van Wassenhove Closed-loo suly chain models ith roduct remanufacturing anagement Science 50() (004) 9-5 [] C Savaskan and L Van Wassenhove everse channel design: The case of cometing retailers anagement Science 5(1) (006) 1-14 [] G U Qiaolun G O Tiegang and S H I Lianshuan rice decision analysis for reverse suly chain ased on game theory Systems Engineering Theory & ractice () (005) 0-5 (in Chinese) [4] Guohua Sun Qiushuang Chen Haitao Xu et al Coordination & decision-making eteen manufacturers and retailers in integrated remanufacturing/manufacturing system Comuter Integrated anufacturing System 1(1) (005) 17-1 (in Chinese) [5] Huang Zuqing and Qingli Da Study on efficiency of serial suly chains ith remanufacture Journal of anagement Sciences in China 9(4) (006) (in Chinese) [6] Xiaohua Han and Shengjia Xue Coordination mechanism analysis for closed-loo suly chain under asymmetric information Comuter Integrated anufacturing System 14(4) (008) (in Chinese) [7] Xiaohua Han and Shengjia Xue everse channel decision for cometing closed-loo suly chain ith dominant retailer Comuter Integrated anufacturing System 15(11) (009) 47-5 (in Chinese) [8] Shengjia Xue Some methods to constitute seudoconvex (seudoconcave) functions Journal of Guangxi University (Natural Science) (1) (1980) (in Chinese) [9] Shengjia Xue Notes on staility analysis in mathematical rogramming Journal of Jinan University (Natural Science) 19() (1998) 15-0 (in Chinese) g