Study on the Complexity of Closed-loop Supply Chain Based on Price Difference between New and Remanufactured Products

This study studied a more realistic closed-loop supply chain model, which is based on price difference between new and remanufactured products and it contains manufacturers, two recyclers and customers. In this model, assumed that new and remanufactured products have price difference, first we build a decision-making dynamic system model of the manufacturers, two recyclers and customers and then analyze the possibility of the existence of the system equilibrium points and their stability. Through numerical simulation, we use bifurcation diagram, Maximum Lyapunov index variation diagram and chaos attractor to estimate the complexity and chaos of the system comprehensively, observe the profit trends of the manufacturers and recyclers when system change from stability to chaos and analyze the system initial value sensitivity. The conclusion of the numerical simulation has a lot of guidance and reference value to the decision-makers in a closed-loop supply chain.


INTRODUCTION
The application of game theory and complexity theory in supply chain, using it to guide the decisionmakers, is intensively concerned by scholars home and abroad.The equilibrium selection problem in a nonlinear duopoly game with adaptive expectations is studied, proved that the iteration results was sensitive to the initial value (Bischi and Kopel, 2001).An output game model based on various decision rules and the stability of its equilibrium were discussed (Yali, 2012).A supply chain model which is constrained by forbidden returning and limited supply capacity was build and its bifurcation phenomenon was discussed (Jing and Xun, 2012).A duopoly game model with bounded rationality and time delayandits complexity were analyzed (Elsadany, 2010).Aduopoly advertising model and its chaos control based on heterogeneous expectations were studied (Juan et al., 2012).The price game model for four oligarchs with different decision rules and its chaos control were analyzed (Junling and Junhai, 2012).A supply chain output game model and its chaos and initial value sensitivity were discussed (Guanhui et al., 2011).
Meanwhile, scholars apply game and complexity theory to various economic environment successfully, such like pollution abatement (Dragone et al., 2010) and so on.In the field of closed-loop supply chain, a closed-loop supply chain model with uncertain demand was build and studied (Huiling, 2012); a duopoly manufacturers recycling game model was build and its chaos and initial value sensitivity when recovery price and adjustment rates vary were studied (Yuehong et al., 2011).
Based on the former work, to be more realistic, this study introduces the assumption that new and remanufactured products have price difference into the closed-loop supply chain model which is consist of manufacturers, recyclers and customers.After building the decision-making dynamic system model of the manufacturers and recyclers, this study analyzes the possibility of the existence of the system equilibrium points and their stability.Then through numerical simulation, we use bifurcation diagram, Maximum Lyapunov index variation diagram and chaos attractor to estimate the complexity and chaos of the system comprehensively, observe the profit trends of the manufacturers and recyclers when system change from stability to chaos and discuss the system initial value sensitivity.At last, we summarize and put forward the direction of further research.

MODEL ESTABLISHMENT
Model description: At present, the research about closed-loop supply chain is mostly focused on the analysis of Nash equilibrium point in price and output model and mostly only concerned the recycling market; seldom used complexity theory to study closed-loop supply chain and rarely combined the new product market and recycling market together.To be more realistic to the product market, this study concerns such a situation: new products and remanufactured products are the same in performance, but different in customer approval degree, then builds a three-tier closed-loop supply chain model, which contains manufacturers, recyclers and customers.In this model, the only manufacturer produces remanufactures and sales the products, two recyclers are in a duopoly state and recovery products.For this model, assumed that: • The manufacturer M and the two recyclers R1, R2 are independent decentralized decision-makers, at the discrete time periods t = 0, 1, 2, …, they all maximize their own profit (Fig. 1) The parameters in this model are denoted as follows: • Manufacturer M: market price of the new products n p , market price of the remanufactured products r p , manufacturer recovery price 0 p , new product cost per unit n c , recycling cost per unit r c • Recycler R1: Recovery price 1 1 0 p r p = , 1 r is the price transfer ratio of R1 • Recycler R2: Recovery price 2 2 0 p r p = , 2 r is the price transfer ratio of R2 Model and description: For manufacturer, the price function of new and remanufactured products is: are the replacement ratio between new and remanufactured products, 1 Q is the amount of new products, 2 Q is the amount of remanufactured products, For recyclers, the recovery amount function is: (2) ( ) is the environmental protection index, is the recovery quantity when recovery price is zero, . Since recyclers can't decide the environmental protection index, so here assumes 0 i k = , then the function between i q and i r , 0 p is: So, the profit function of manufacturer M and recyclers R1, R2 at period t is: The decision variables of manufacturer M are recovery price 0 p and new products amount 1 Q , the decision variables of recyclers R1, R2 are ( 1, 2)   i r i = , then we got the marginal profit function: As in real economic world, no participants can have complete information, so based on the assumption that decision-makers are limited rational, their decision way at period are the adjustment speed of 1 Q , 0 p , 1 r , 2 r respectively.Synthesize Eq. ( 1)-( 6), we got the discrete dynamic system model: Model analysis: Through the above analysis, we have built the discrete dynamic system model (7) of the manufacturer M and recyclers R1, R2, now we will solve the equilibrium of the system and analyze the stability of the equilibrium points.

e f e e E b A
A e e f f e e f f e e e f e f e e A e e f f Ẽ E are boundary equilibrium points, E 16 is the Nash equilibrium point.
The stability of the equilibrium point depends on the eigenvalues of the Jacobian matrix of system (7).The Jacobian matrix of system ( 7) is: ) ) 1 (( ( ) Calculate the Jacobian matrices of the 16 equilibrium points separately; judge their stability according to the value of the eigenvalues of their Jacobian matrix.Take equilibrium point E 1 as an example, its Jacobian matrix is: Solve the matrix and the Eigen values are:

NUMERICAL SIMULATIONS
In order to study the properties and characteristics of this system better, we will use certain value data to simulate the dynamic system.Assume the parameters values as follows: In the study, we'll use bifurcation diagram, Maximum Lyapunov index variation diagram and chaos attractor to study the dynamic properties of the system, observe the influence on all the participators' profits when the adjustment speed of decision variables change and analyze the initial value sensitivity of the decision variables.

Bifurcation and chaos numerical simulation:
In this part, as the situation of 2 v is like 1 v and 4 v is similar to 3 v , so we'll only research 1 v and 3 v .First, we study the bifurcation and chaos phenomenon of the system: when 2 v = 0.3, 3 v = 0.45, 4 v = 0.5, the trends of 1 0 1 2 , , , Q p r r when 1 v changes are showed in Fig. 2 and From Fig. 2 and 3, we can see that when increase, the decision variables of the system ( 3 1 0 1 2 : , , , v Q p r r ) will change from stability to the first Bifurcation and then to period-doubling, at last into chaos.If adjust decision variables too fast, the market will be chaotic and the decision will be very complicated.
Besides, in this model, without considering customers' environmental protection consciousness, the adjustment speed of new products amount recovery price 0 p by manufacturer will chaos of itself, it has no influence to the stability ) will change from stability to the first doubling, at last into chaos.If adjust decision variables too fast, the market will be chaotic and the decision will be very Besides, in this model, without considering customers' environmental protection consciousness, the adjustment speed of new products amount 1 Q and will only cause the the stability of the Next, we'll measure the property of the system through the Maximum Lyapunov index variation diagram, the Maximum Lyapunov index is showed in Fig. 4 and  v vary.When i v is small, the Maximum adjustments peed of cyclers will in fect both Next, we'll measure the property of the system through the Maximum Lyapunov index variation diagram, the Maximum Lyapunov index is showed in change.We can see that the variation status of the 4 and 5 fits well the situation of bifurcation and chaos in Fig. 2 and 3, when small, the Maximum Lyapunov index is less than zero; with the increase of v i , the system appears bifurcation phenomenon, the Maximum Lyapunov index equals zero; when big and the system becomes chaotic, the Maximum Lyapunov index is greater than zero.
Value that: in both values, the Maximum Lyapunov in system is greater than zero and the system is chaotic, as the chaos attractor confirmed in Fig. 6 and 7.The situation in Fig. 6 and 7 is accordant to the one in Fig. 2 to 5.
Influence of DVAS on profits: Profit factor of an enterprise, how would profits vary when Decision Variables Adjustment Speed (DVAS) increases?The trends of the profits are showed in when v 1 increases, in Fig. 9 when v before, the trend of v 2 is similar to v 1 the same as v 3 .In the picture, m π , 1 r π for the profits of the manufacturer, recycler R1, recycler R2 and the sum of them, respectively.
Attractor when system is in chaos caused by the Attractor when system is in chaos caused by the Lyapunov index is less than zero; with the increase of , the system appears bifurcation phenomenon, the Maximum Lyapunov index equals zero; when v i is too big and the system becomes chaotic, the Maximum 4 = 0.5 = 0.5 the Maximum Lyapunov index of the system is greater than zero and the system is chaotic, as .6 and 7. .6 and 7 is accordant to the one Profit S is the key factor of an enterprise, how would profits vary when Decision Variables Adjustment Speed (DVAS) v i increases?The trends of the profits are showed in Fig. 8 v 3 increases.As and v 4 is nearly 1 r , 2 r π , total π stand for the profits of the manufacturer, recycler R1, recycler v increases and reach a critical point, the profits of the manufacturer, recycler R1, recycler R2 and the total profit all begin to fall.The critical point value is also the bifurcation point in and the point when the Maximum Lyapunov index equals zero in Fig. 4.This means that, if the manufacturer adjusts its decision variable too fast, the system becomes chaos, the profits of all the ones will decrease and it's bad for the whole system.
In Fig. 9, when v 3 increases and reach a critical point, the profits of the manufacturer, recycler R1 and the total profit all fall, but the profit of recycler R2 increases.The critical point value is also the one where system changes from stability to chaos.This means recycler R2 can profit from market disorder, others' behalf will be harmed.This phenomenon claims the necessity of contract coordination and it is important.
Initial value sensitivity: Figure 10 initial value sensitivity of the system variables, in the iteration process, 1 v = 0.25, 2 v = 0.18, 0.5.
From Fig. 10 to 13 we can see that, the system variables are highly sensitive to the initial value.Set up two groups of initial values, each variable only differs increases and reach a critical point, the profits of the manufacturer, recycler R1, recycler R2 and the total profit all begin to fall.The critical point value is also the bifurcation point in Fig. 2 and the point when the Maximum Lyapunov index .4. This means that, if the decision variable too fast, the system becomes chaos, the profits of all the ones will decrease and it's bad for the whole system.
increases and reach a critical urer, recycler R1 and all fall, but the profit of recycler R2 increases.The critical point value is also the one where system changes from stability to chaos.This means recycler R2 can profit from market disorder, but the others' behalf will be harmed.This phenomenon claims the necessity of contract coordination and it is 10 to 13 showed the initial value sensitivity of the system variables, in the = 0.18, 3 v = 0.75 and 4 v = we can see that, the system variables are highly sensitive to the initial value.Set up initial values, each variable only differs

CONCLUSION
This study assumed that new and remanufactured products have price difference, discussed the market situation of a three-tier supply chain, which is consist of the manufacturer, two recyclers and customers.Through numerical simulation, the conclusions are: • When the decision-makers adjust their decision variables too fast, the market will become disordered; however, without considering the customers' environmental protection consciousness, the manufacturer and the recyclers have different influence • If the manufacturer adjusts its two decision variables too fast, all the members' profit of the supply chain will decrease; while any of the recyclers adjusts its decision variable too fast, for one of the recyclers, its profit increase, the profit of the rest in this supply chain would fall.This claims the necessity of contract coordination • All decision variables are sensitive to initial values The model and methods in this study would help the manufacturers and recyclers make more reasonable decisions, improve the stability of their profits and the market and have theoretical guidance and practical reference value.The price or output models of multiple manufacturers and products, with one period delay or even more, the retailers are involved and so on, are the direction of further research in future.

Fig
Fig. 4: v 1 -maximum Lyapunov index 5 when 1 v and 3 v change.We can see that the variation status of the Maximum Lyapunov index in Fig. situation of bifurcation and chaos in 1 v and 3

Fig. 6 :Fig. 7 :
Fig. 6: Attractor when system is in chaos caused by the increasing of v 3 Fig. 8: Profits when v 1 increases

Fig. 13 :
Fig.13: r 2 -initial value sensitivity difference of 477.5 times; the value of 2 r differs 0.2824, it means a difference of 282.4 times.This phenomenon is the characteristics of chaos movement, as time goes on; the adjacent orbital has a great deviation.This gives the enterprises a useful advice that chose your initial value carefully, or it may cause huge losses.