Advance Jounal of Food Science and Technology

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1 Advance Jounal of Food Science and Technology 5(): , 03 ISSN: ; e-issn: Maxwell Scienific Oganizaion, 03 Subied: July 9, 03 Acceped: Augus 03, 03 Published: Decebe 05, 03 Sudy on he Coplexiy of Closed-loop Supply Chain Based on Pice Diffeence beween New and Reanufacued Poducs Bin Chen and Junhai Ma College of Manageen and conoics, Tianjin Univesiy, Tianjin 30007, China Absac: This sudy sudied a oe ealisic closed-loop supply chain odel, which is based on pice diffeence beween new and eanufacued poducs and i conains anufacues, wo ecycles and cusoes. In his odel, assued ha new and eanufacued poducs have pice diffeence, fis we build a decision-aking dynaic syse odel of he anufacues, wo ecycles and cusoes and hen analyze he possibiliy of he exisence of he syse equilibiu poins and hei sabiliy. Though nueical siulaion, we use bifucaion diaga, Maxiu Lyapunov index vaiaion diaga and chaos aaco o esiae he coplexiy and chaos of he syse copehensively, obseve he pofi ends of he anufacues and ecycles when syse change fo sabiliy o chaos and analyze he syse iniial value sensiiviy. The conclusion of he nueical siulaion has a lo of guidance and efeence value o he decision-akes in a closed-loop supply chain. Keywods: Coplexiy, closed-loop supply chain, pice diffeence, eanufacued poducs INTRODUCTION The applicaion of gae heoy and coplexiy heoy in supply chain, using i o guide he decisionakes, is inensively concened by scholas hoe and aboad. The equilibiu selecion poble in a nonlinea duopoly gae wih adapive expecaions is sudied, poved ha he ieaion esuls was sensiive o he iniial value (Bischi and Kopel, 00). An oupu gae odel based on vaious decision ules and he sabiliy of is equilibiu wee discussed (Yali, 0). A supply chain odel which is consained by fobidden euning and liied supply capaciy was build and is bifucaion phenoenon was discussed (Jing and Xun, 0). A duopoly gae odel wih bounded aionaliy and ie delayandis coplexiy wee analyzed (lsadany, 00). Aduopoly adveising odel and is chaos conol based on heeogeneous expecaions wee sudied (Juan e al., 0). The pice gae odel fo fou oligachs wih diffeen decision ules and is chaos conol wee analyzed (Junling and Junhai, 0). A supply chain oupu gae odel and is chaos and iniial value sensiiviy wee discussed (Guanhui e al., 0). Meanwhile, scholas apply gae and coplexiy heoy o vaious econoic envionen successfully, such like polluion abaeen (Dagone e al., 00) and so on. In he field of closed-loop supply chain, a closed-loop supply chain odel wih unceain deand was build and sudied (Huiling, 0); a duopoly anufacues ecycling gae odel was build and is chaos and iniial value sensiiviy when ecovey pice and adjusen aes vay wee sudied (Yuehong e al., 0). Based on he foe wok, o be oe ealisic, his sudy inoduces he assupion ha new and eanufacued poducs have pice diffeence ino he closed-loop supply chain odel which is consis of anufacues, ecycles and cusoes. Afe building he decision-aking dynaic syse odel of he anufacues and ecycles, his sudy analyzes he possibiliy of he exisence of he syse equilibiu poins and hei sabiliy. Then hough nueical siulaion, we use bifucaion diaga, Maxiu Lyapunov index vaiaion diaga and chaos aaco o esiae he coplexiy and chaos of he syse copehensively, obseve he pofi ends of he anufacues and ecycles when syse change fo sabiliy o chaos and discuss he syse iniial value sensiiviy. A las, we suaize and pu fowad he diecion of fuhe eseach. MODL STABLISHMNT Model descipion: A pesen, he eseach abou closed-loop supply chain is osly focused on he analysis of Nash equilibiu poin in pice and oupu odel and osly only concened he ecycling ake; seldo used coplexiy heoy o sudy closed-loop supply chain and aely cobined he new poduc ake and ecycling ake ogehe. To be oe ealisic o he poduc ake, his sudy concens such a siuaion: new poducs and eanufacued poducs ae he sae in pefoance, bu diffeen in cusoe Coesponding Auho: Bin Chen, College of Manageen and conoics, Tianjin Univesiy, Tianjin 30007, China 566

2 appoval degee, hen builds a hee-ie closed-loop supply chain odel, which conains anufacues, ecycles and cusoes. In his odel, he only anufacue poduces eanufacues and sales he poducs, wo ecycles ae in a duopoly sae and ecovey poducs. Fo his odel, assued ha: The anufacue M and he wo ecycles R, R ae independen decenalized decision-akes, a he discee ie peiods = 0,,,, hey all axiize hei own pofi (Fig. ) The ecovey quaniy is only elevan o he ecovey pice, anufacue us ecovey he all fo he ecycles and i can ecycle all of he, no wase New and eanufacued poducs have pice diffeence Pice educes when oupu ises The aoun of eanufacued poducs is less han deand, so i can sell ou The paaees in his odel ae denoed as follows: Manufacue M: ake pice of he new poducs p n, ake pice of he eanufacued poducs p, anufacue ecovey pice p 0, new poduc cos pe uni c n, ecycling cos pe uni c Recycle R: Recovey pice p p 0, is he pice ansfe aio of R Recycle R: Recovey pice p p 0, is he pice ansfe aio of R Adv. J. Food Sci. Technol., 5(): , 03 Model and descipion: Fo anufacue, he pice funcion of new and eanufacued poducs is: pn a bq dq p a bq dq () a i 0, bi, di 0( i,) ae he eplaceen aio beween new and eanufacued poducs, Q is he aoun of new poducs, Q is he aoun of eanufacued poducs, Q q q. Fo ecycles, he ecovey aoun funcion is: q k e p f p q k e p f p () k i 0 i, is he envionenal poecion index, is he ecovey quaniy when ecovey pice is zeo, e i 0( i,) is he cusoes ecovey pice sensiive coefficien, f i 0( i,) is he ecycles copeiion coefficien, ei fi i,. Since ecycles can decide he envionenal poecion index, so hee assues ki 0, hen he funcion beween q i and i, p 0 is: 567 q( e f ) p0 q ( e f ) p 0 (3) So, he pofi funcion of anufacue M and ecycles R, R a peiod is: p c Q p p0 cq p p q p p q n n 0 0 (4) The decision vaiables of anufacue M ae ecovey pice p 0 and new poducs aounq, he decision vaiables of ecycles R, R ae i i(,), hen we go he aginal pofi funcion: a bq ( ) ( dd)(( e f) ( ) Q ( e f) ( )) p0( ) cn ( a ( dd) Q( ) b(( e f) ( ) p0 ( e f) ( )) p0( ) p0( ) c ) (( e f) ( ) ( e f) ( )) p0( e f e ) p0( e f e) (5) As in eal econoic wold, no paicipans can have coplee infoaion, so based on he assupion ha decision-akes ae liied aional, hei decision way a peiod is: QQ vq Q p0 p0 vp0 p0 v4 (6) v i 0 i,,3,4 ae he adjusen speed of Q, p 0,, especively. Synhesize q. ()-(6), we go he discee dynaic syse odel: Q ( ) Q ( ) vq ( )( a bq ( ) ( d d ) (( ) ( ) ( ) ( )) ( ) ) ( ) )(( ) ( ) ( ) ( ))) e f e f p0 cn p0( ) p0( ) vp0( )(( a ( dd) Q( ) b(( e f) ( ) ( e f) ( )) p0( ) p0 c e f e f ( ) () ()( p0( e f e )) ( ) ( ) v4( )( p0( e f e)) (7)

3 Adv. J. Food Sci. Technol., 5(): , 03 Model analysis: Though he above analysis, we have buil he discee dynaic syse odel (7) of he anufacue M and ecycles R, R, now we will solve he equilibiu of he syse and analyze he sabiliy of he equilibiu poins. Solve he discee dynaic syse odel (7) and he equilibiu poins ae: (0,0,0,0), (,0,0,0), b (0, R,0,0), (0,0, R,0) 3 4 a c n a c R R n 5 (0,0,0, ), 6 (,,0,0), b acn acn 7 8 b b (,0, R,0), (,0,0, R) a c 9 (0,,,0), b( e f) a c 0 b( e f) (0,,0, ), (0,0, R, R) A ( a c ) A( d d )( e f ) A n (,,,0) 4bA A A ( a c ) A( d d )( e f ) A 3 n (,,0, ), 4bA 3 A3 a cn (,0, R, R) b a c e f ee e f ee (0,,, ) ( ba4 ) 4ee ff 4ee ff A ( a c ) AA ( d d ) A e f ee e f ee 5 n 4 (,,, ) ba5 A5 4ee ff 4ee ff A ( a c )( d d ) b( a c ) n A ( d d ) ( e f ) b ( b ( e f ) ) JQ (, p,, ) 0 v( a4bq ( dd)(( e v( dd)(( ef) v( dd) v( dd) f) ( ef) ( ef) ) Q ( ef) Qp 0 ( ef) Qp 0 ) p0 cn ) v(( a( dd) v( ef)( a( d v( ef)( a( d v( dd) Q4 b(( ef) d) Q4 b(( ef) d) Q4 b(( ef) (( ef) ( ef) ) p04p0 ( ef) ) p0 p0 ( ef) ) p0p0 ( e f) ) p0 c )(( ef) ( e c) p0 c) p0 f) )) 0 vp 3 0( e f vp 3 0( e f vfp 3 0 e ) 4 e ) 0 vp 4 0( e f vp 4 0( e f vfp 4 0 e ) 4 e ) Calculae he Jacobian aices of he 6 equilibiu poins sepaaely; judge hei sabiliy accoding o he value of he eigenvalues of hei Jacobian aix. Take equilibiu poin as an exaple, is Jacobian aix is: v( ac n ) J Solve he aix and he igen values ae: v( a c n ), 3 4, accoding o he assupion befoe, v 0, a c n 0, so, hen is unsable. As he sae,, 3,, 5 ae all unsable equilibiu poins, 6 is a local sable Nash equilibiu poin. NUMRICAL SIMULATIONS In ode o sudy he popeies and chaaceisics of his syse bee, we will use ceain value daa o siulae he dynaic syse. Assue he paaees values as follows: a 9, a 6, b.3, b 0.8, d 0.5, d 0.6, c 5, c, e.3, e.9, f.4, f.. n In he sudy, we ll use bifucaion diaga, A3 ( dd ) ( e f ) b ( b ( e f ) ) Maxiu Lyapunov index vaiaion diaga and chaos aaco o sudy he dynaic popeies of he syse, ( e f)( ef ee ) ( e f)( ef ee ) obseve he influence on all he paicipaos pofis A4 when he adjusen speed of decision vaiables change 4ee ff and analyze he iniial value sensiiviy of he decision vaiables. A5 A4( dd) 4 b( ba4 ) Bifucaion and chaos nueical siulaion: In his pa, as he siuaion of v is like v ~ 5 ae bounday equilibiu poins, 6 is he and v 4 is siila o v Nash equilibiu poin. 3, so we ll only eseach v and v 3. Fis, we sudy he The sabiliy of he equilibiu poin depends on bifucaion and chaos phenoenon of he syse: when he eigenvalues of he Jacobian aix of syse (7). v = 0.3, v 3 = 0.45, v 4 = 0.5, he ends of Q, p0,, The Jacobian aix of syse (7) is: when v changes ae showed in Fig. and 568

4 Adv. J. Food Sci. Technol., 5(): , 03 Fig. : The CLSC sucue v v v v Fig. : Bifucaion when v inceases Fig. 4: v -axiu Lyapunov index Fig. 3: Bifucaion when v 3 inceases 3 shows he ends of Q, p0,, when v 3 changes and v = 0.5, v = 0.8, v 4 = 0.5. Fo Fig. and 3, we can see ha when v, v 3 incease, he decision vaiables of he syse ( v : Q, p 0; : Q, p0,, ) will change fo sabiliy o he fis Bifucaion and hen o peiod-doubling, a las ino chaos. If adjus decision vaiables oo fas, he ake will be chaoic and he decision will be vey coplicaed. Besides, in his odel, wihou consideing cusoes envionenal poecion consciousness, he adjusen speed of new poducs aoun Q and ecovey pice p 0 by anufacue will only cause he chaos of iself, i has no influence o he sabiliy of he Fig. 5: v 3 -axiu Lyapunov index ecycles decision; howeve, he adjusens peed of pice ansfeaio i by hee cycles will in fec boh of he. Nex, we ll easue he popey of he syse houghh he Maxiu Lyapunov index vaiaion diaga, he Maxiu Lyapunov index is showed in Fig. 4 and 5 when v and v 3 change. We can see ha he vaiaion saus of he Maxiu Lyapunov index in Fig. 4 and 5 fis well he siuaion of bifucaion and chaos in Fig. and 3, when v and v vay. When v is sall, he Maxiu 3 n i 569

5 Adv. J. Food Sci. Technol., 5(): , 03 v v Fig. 6: Aaco when syse is in chaos caused by he inceasing of v 3 v Fig. 8: Pofis when v inceases v Fig. 7: Aaco when syse is in chaos caused by he inceasing of v 3 Lyapunov index is less han zeo; wih he incease of v i, he syse appeas bifucaion phenoenon, he Maxiu Lyapunov index equals zeo; when v i is oo big and he syse becoes chaoic, he Maxiu Lyapunov index is geae han zeo. Value ha: v = 0..85, v = 0.3, v 3 = 0.45, v 4 = 0.5 v = 0..5, v = 0.8, v 3 = 0.7, v 4 = 0.5 in boh values, he Maxiu Lyapunov index of he syse is geae han zeo and he syse is chaoic, as he chaos aaco confied in Fig. 6 and 7. The siuaion in Fig. 6 and 7 is accodan o he one in Fig. o 5. Influence of DVAS on pofis: Pofi S is he key faco of an enepise, how would pofis vay when Decision Vaiables Adjusen Speed (DVAS) v i inceases? The ends of he pofis ae showed in Fig. 8 when v inceases, in Fig. 9 when v 3 inceases. As befoe, he end of v is siila o v and v 4 is nealy he sae as v 3. In he picue,,,, oal sand fo he pofis of he anufacue, ecycle R, ecycle R and he su of he, especively. 570 Fig. 9: Pofis when v 3 inceases In Fig. 8, when v inceases and each a ciical poin, he pofis of he anufacue, ecycle R, ecycle R and he oal pofi all begin o fall. The ciical poin value is also he bifucaion poin in Fig. and he poin when he Maxiu Lyapunov index equals zeo in Fig. 4. This eans ha, if he anufacue adjuss is decision vaiable oo fas, he syse becoes chaos, he pofis of all he ones will decease and i s bad fo he whole syse. In Fig. 9, when v 3 inceases and each a ciical poin, he pofis of he anufacue, ecycle R and he oal pofi all fall, bu he pofi of ecycle R inceases. The ciical poin value is also he one whee syse changes fo sabiliy o chaos. This eans ecycle R can pofi fo ake disode, bu he ohes behalf will be haed. This phenoenon clais he necessiy of conac coodinaion and i is ipoan. Iniial value sensiiviy: Figue 0 o 3 showed he iniial value sensiiviy of he syse vaiables, in he ieaion pocess, v = 0.5, v = 0.8, v 3 = 0.75 and v 4 = 0.5. Fo Fig. 0 o 3 we can see ha, he syse vaiables ae highly sensiive o he iniial value. Se up wo goups of iniial values, each vaiable only diffes

6 Adv. J. Food Sci. Technol., 5(): , 03 Fig. 0: Q -iniial value sensiiviy Fig. 3: -iniial value sensiiviy diffeence of ies; he value of diffes 0.84, i eans a diffeence of 8.4 ies. This phenoenon is he chaaceisics of chaos oveen, as ie goes on; he adjacen obial has a gea deviaion. This gives he enepises a useful advice ha chose you iniial value caefully, o i ay cause huge losses. CONCLUSION Fig. : p 0 -iniial value sensiiviy This sudy assued ha new and eanufacued poducs have pice diffeence, discussed he ake siuaion of a hee-ie supply chain, which is consis of he anufacue, wo ecycles and cusoes. Though nueical siulaion, he conclusions ae: When he decision-akes adjus hei decision vaiables oo fas, he ake will becoe disodeed; howeve, wihou consideing he cusoes envionenal poecion consciousness, he anufacue and he ecycles have diffeen influence If he anufacue adjuss is wo decision vaiables oo fas, all he ebes pofi of he supply chain will decease; while any of he ecycles adjuss is decision vaiable oo fas, fo one of he ecycles, is pofi incease, he pofi of he es in his supply chain would fall. This clais he necessiy of conac coodinaion All decision vaiables ae sensiive o iniial values Fig. : -iniial value sensiiviy The odel and ehods in his sudy would help Afe abou weny ies ieaions, all he values he anufacues and ecycles ake oe easonable of he vaiables had significan diffeences. Afe abou decisions, ipove he sabiliy of hei pofis and he fify ies ieaive opeaion, he value of Q ake and have heoeical guidance and pacical diffes0.98, i eans a diffeence of 9.8 ies; he efeence value. The pice o oupu odels of uliple anufacues and poducs, wih one peiod delay o value of p 0 diffes 0.46, i eans a diffeence of even oe, he eailes ae involved and so on, ae he 4.6 ies; he value of diffes , i eans a diecion of fuhe eseach in fuue. 57

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