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Research on game model and complexity of retailer collecting and selling in closed-loop supply chain
Applied Mathematical Modelling 37 (2013) 5047–5058
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Research on game model and complexity of retailer collecting and selling in closed-loop supply chain Yuehong Guo a,b, Junhai Ma a,⇑ a b
College of Management Economic, Tianjin University, Tianjin 300072, China School of Economics and Management, Tianjin University of Technology and Education, Tianjin 300222, China
a r t i c l e
i n f o
Article history: Received 2 March 2012 Received in revised form 7 August 2012 Accepted 11 September 2012 Available online 21 September 2012 Keywords: Repeated games Closed-loop supply chain Complexity Chaos
a b s t r a c t A closed-loop supply chain is a complex system in which node enterprises play important roles and exert great influence. Firstly, this paper established a collecting price game model for a close-loop supply chain system with a manufacturer and a retailer who have different rationalities. It assumed that the node enterprises took the marginal utility maximization as the basis of decision-making. Secondly, through numerical simulation, we analyzed complex dynamic phenomena such as the bifurcation, chaos and continuous power spectrum and so on. Thirdly, we analyzed the influences of the system parameters; this further explained the complex nonlinear dynamics behavior from the perspective of economics. The results have significant theoretical and practical application value. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The importance of the environmental performance of products and processes for sustainable manufacturing and service operations increasingly is being recognized in practice. While legislation introduced in Europe, North America, and Japan encourages this awareness, many corporations have recognized reducing production costs and improving profits from used products. China has required that electrical products manufacturer must be responsible for collecting waste products since 2003. Remanufactured products are typically upgraded to quality standards of new products, so that they can be sold in new product markets. (see Fig. 1). This paper draws on and contributes to several streams of literature, each of which we review below. A growing literature in operations management addresses reverse logistics management issues for remanufactured products. We refer the reader to Fleischmann et al. [1] and Guide et al. [2] for complete literature reviews. They defined as the supply chain to reverse supply chain which formed a complete closed-loop system (closed-loop supply chain, hereinafter referred to as CLSC). McGuire and Staelin [3] studied the pricing methods under three collecting channels which manufacturer, retailer and third-party collector collected products respectively. Guide and Wassenhove [4] to Xerox Corporation and Kodak Corporation studied that the manufacturers are responsible for the management of the collecting. Savaskan and Bhattacharya [5] studied three kinds of collecting channels of manufacturers collecting, retailer collecting and the third-party collecting in closed-loop supply chain. They found that the collecting distance was closer from consumers, the collecting efforts was more effective. In the literature describing reverse logistics management mode of the third-party. In this collecting mode, manufacturers entrust a third-party to perform extended producer responsibility and manage used-products.
⇑ Corresponding author. E-mail address: [email protected] (J. Ma). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.034
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Manufacturer
Retailer
Customer
Fig. 1. A manufacturer subcontract the sale and collection activity to an retailer.
Gu et al. [6] and Wang et al. [7] studied pricing analysis of reverse supply chain using the game theory method under retailers collecting mode and concluded the optimal strategy of single stage. Ge [8] established the demand model of new products and remanufacturing products from social environmental protection consciousness, consumers’ utility function to new products and remanufacturing products, and drew the conclusions. The first is the optimal pricing strategy in different collecting channels in decentralized decision-making. The second is the influence of social environmental protection consciousness. Xiao-min Zhao et al. [9] studied closed-loop supply chains management-managerial innovation on meeting weee eu directive in Chinese electronic industries. Weixin Yao [10] established atomic models for closed-loop supply chain in e-commerce. V.D.R. Guide Jr et al. [11] had a research on contingence planning for closed-loop supply chains with product recovery, and got meaningful conclusion. Sjur. Didrik. Flhm et al. [12] analyzed price expectations and cobwebs under uncertainty. 2. Model assumptions and notation 2.1. Assumption This model is based on these following assumptions. (1) This paper considers remanufactured products and new products don’t form a competitive market. When Remanufactured products do not meet the market demand, it supplemented by new products. New products and remanufactured products are no difference. The manufacturer manufactures new products and remanufacturing products. (2) The closed-loop supply chain consists of a manufacturer and a retailer, the retailer collects waste products from consumers and returns the manufacturer. The manufacturer transfers payments to the retailer, the retailer sales products to consumers. The manufacturers and the retailer are independent decision makers, and their strategic space is to choose the best collecting price. Their goal is to maximize returns in discrete time period as t = 0, 1, 2 . . . . (3) The number of the collecting is increasing function of collecting price. The collecting capability and manufacturing capability are unlimited. In order to simplify the problem and emphasize main parameters influence of the system, all the collecting products can be manufactured. 2.2. Notation description bm(t)—It is the transfer collecting price which the manufacturer paying to the retailer, the manufacturer’s decision variable. br(t), r2(t) — br(t) is the unit collecting price of the retailer paying to consumer markets. r2(t) is the profitability of the retailer, and which is the retailer’s decision variable. The formula is as follows.
br ðtÞ ¼ bm ðtÞð1 r2 ðtÞÞ;
ð1Þ
cm — It is the marginal cost of manufacturing a new product with raw materials, to be a constant. cr — It is the marginal cost of remanufacturing a used-product into a new one, to be a constant. pm(t), rm(t) — pm(t) is the unit wholesale price in t period which is the manufacturer’s decision variable. rm(t) is the profitability of the manufacturer, and which is the manufacturer’s decision variable. The formula is as follows.
pm ðtÞ ¼ ðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ;
ð2Þ
pr(t), r1(t) — pr(t) is the unit selling price in t period which is the retailer’s decision variable. r1(t) is the selling profitability of the retailer which is the retailer’s decision variable. The formula is as follow:
pr ðtÞ ¼ pm ðtÞð1 þ r1 ðtÞÞ:
ð3Þ
Ai — It is the collecting cost of retailer which including logistics cost etc., a constant, where i = 1, 2, A1 are selling costs, A2 are collecting costs. D–D is the demand for products in the market as a function of product price, i.e., D = a bpr(t) where a is market scale, b is price sensitivity of consumers. The value of b depends on many factors, including economic factors, social factors and psychological factors. It reflects in three aspects: firstly, it is the quantity and similar degree of products, in general, the higher the replacement of goods, the higher the value of b, on the other hand, the lower the replacement of goods, the lower the
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value of b. Secondly, it is the importance of goods, the value of b is lower of basic necessities of life, while the value of b is higher of luxury goods that is non-essential. Thirdly, it is adjustment time of consumers to demand, in general, the shorter the adjustment time, the lower of b; the longer the adjustment time, the higher of b. G — It is the market supply for the used-product in the market as a function of collection price. We assume that supply associated with collection price only and it is linear increasing function of the collection cost, i.e., G = k + bbr(t)k, b > 0. k is the quantity of volunteer to return used-products from consumers, and on another level as a measure of social awareness; b is the price sensitivity of consumers for collection price. Furthermore, D is greater than G. Pm — It is the profit function of the manufacturer. The function is as follows.
Pm ¼ ½pm ðtÞ bm ðtÞ cr ½k þ bbr ðtÞ þ ½pm ðtÞ cm ½a bpr ðtÞ k bbr ðtÞ;
ð4Þ
Pr —It is the profit function of the retailer. The function is as follows.
Pr ¼ ½pr ðtÞ pm ðtÞ A1 ½a bpr ðtÞ þ ½bm ðtÞ br ðtÞ A2 ½k þ bbr ðtÞ;
ð5Þ
P— It is the profit function of the closed-loop supply chain. The function is as follows.
P ¼ Pm þ Pr :
ð6Þ
Will be substituted into Eqs. (4)–(6), Pm(t), Pr(t) and P(t) are as follows.
8 Pm ðtÞ ¼ ½cm cr bm ðtÞ½k þ bbm ðtÞð1 r 2 ðtÞÞ þ ½ð1 þ r m ðtÞÞðcr þ bm ðtÞÞ cm > > > > > > ½a bð1 þ r m ðtÞÞðcr þ bm ðtÞÞð1 þ r 1 ðtÞÞ; > > > > > > > Pr ðtÞ ¼ ½r 1 ðtÞðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ A1 ½a bðbm ðtÞ þ cr Þð1 þ r m ðtÞÞð1 þ r1 ðtÞÞ > < þ½bm ðtÞr 2 ðtÞ A2 ½k þ bbm ðtÞð1 r 2 ðtÞÞ; > > > > PðtÞ ¼ Pm ðtÞ þ Pr ðtÞ > > > > > > ¼ ð1 þ rm ðtÞÞðcr þ bm ðtÞÞð1 þ r1 ðtÞÞða þ bðcm þ A1 ÞÞ bð1 þ r m ðtÞÞ2 ðcr þ bm ðtÞÞ2 ð1 þ r 1 ðtÞÞ2 > > > > : aðcm þ A1 Þ þ bm ðtÞð1 r 2 ðtÞÞðbðcm cr A2 Þ kÞ bbm ðtÞ2 ð1 r 2 ðtÞÞ2 þ kðcm cr A2 Þ:
ð7Þ
3. Stackelberg model 3.1. Model and analysis Suppose that the manufacturer and the retailer are principal and subordinate relationship, and the manufacturer is Stacklelberg leader, the retailer is follower, then they process sequential dynamic game, the game equilibrium is Stacklelberg equilibrium. In this game, the manufacturer make the decision of sales price and collection price according to the market information, then the retailer make decision according to the decision-making of the manufacture. Using backward induction, first find the second stage of the game model response function, to get Stackelberg balanced. First of all requirements is as follows.
max Pr ¼ ða bðbm ðtÞ þ cr Þð1 þ r m ðtÞÞð1 þ r1 ðtÞÞÞðr1 ðtÞðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ A1 Þ þ ðk þ bbm ðtÞð1 r 2 ðtÞÞÞðbm ðtÞr 2 ðtÞ A2 Þ ð8Þ The optimal marginal profit of the retailer can be concluded by the first-order conditions of formula (8). The result is as follows.
( @ Pr @r 1
¼ ðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ½a bð2r1 ðtÞ þ 1Þð1 þ r m ðtÞÞðbm ðtÞ þ cr Þ þ bA1 ;
@ Pr @r 2
¼ bm ðtÞ½k þ bbm ðtÞ 2br2 ðtÞbm ðtÞ þ bA2 :
ð9Þ
The retailer’s reaction functions are as follows by solving formula (9).
8 < r 1 ¼ 2b
aþbA1
ðbm þcr Þð1þrm Þ
: r ¼ bbm þbA 2 þk : 2 2bb
12 :
ð10Þ
m
Formula (10) is the optimal decision-making of the retailer on the premise of the manufacturer’s selling price and collecting price are certain, the retailer obtains the decision after she observed the manufacturer behavior. It is the retailer’s reaction function. Substitution formula (10) into formula (7), by @ Pm/@rm = 0, @ Pm/@bm = 0, the optimal result of the manufacturer can be obtained. Assume that the parameters values for a = 100, cm = 60, cr = 30, b = 1.2, b = 1, k = 0.1, A1 = 0.5, A2 = 0.5. Therefore, the Stac kelberg equilibrium is r m ; bm ; r 1 ; r 2 ¼ ð0:36; 12:25; 0:11; 0:59Þ. When making price decision, restricted by the objective conditions such as the decision-making ability and so on, each decision maker cannot get the whole market information and his price decision is also not completely rational. Also, because
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of the differences such as in the input on information collections and in the experiences of decision maker and so on, the information grasped by each node enterprise is not necessarily same exactly in the condition of incomplete market information, and they cannot have the exact same rational levels, so the final price decision and its setting rule of each maker are not exactly the same. Therefore, the manufacturer makes decision based on limited rational expectations. She adjusts the game process on the basis of marginal profits. If the marginal profits in period t are positive, then they will continue their output adjustment strategy in period t + 1. The model can be constructed as follows.
8 < r m ðt þ 1Þ ¼ r m ðtÞ þ a1 rm ðtÞ @@rPm ; m
: b ðt þ 1Þ ¼ b ðtÞ þ a b ðtÞ @ Pm ; m m 2 m @bm
ð11Þ
where a1 is the adjustment coefficient of rm. a2 is adjustment coefficient of bm. Then, the dynamic adjustment of the closed-loop supply chain system can be written as follows.
8 > r m ðt þ 1Þ ¼ r m ðtÞ þ a1 rm ðtÞ ðbm ðtÞ þ cr Þ abA12þbcm bðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ ; > > > > > > > > b ðt þ 1Þ ¼ b ðtÞ þ a b ðtÞ ð1þrm ðtÞÞðaþbcm bA1 Þ bðb ðtÞ þ c Þð1 þ r ðtÞÞ2 > m m 2 m m r m > 2 > > < r ÞþbA2 ; bbm ðtÞ kbðcm c 2 > > > > > > aþbA1 > 12 ; r 1 ðtÞ ¼ 2bðbm ðtÞþc > > r Þð1þr m ðtÞÞ > > > > : r ðtÞ ¼ bbm ðtÞþbA2 þk 2 2bbm ðtÞ
ð12Þ
From system (12) can be concluded that the manufacture firstly makes decision, and the decision variables directly relate to a1 and a2. But the retailer’s decision variables directly relate to rm and bm. The following simulates complex dynamics characteristics of system (12) through numerical simulation. 3.2. Numerical simulation Using MATLAB, influence of the parameters on the system (12) can be analyzed through numerical simulation. (1) The influence of a to the supply chain Assume that a1 is at [0, 0.002] and a2 is fixed, the initial value rm and bm are 0.4 and 20, rm, bm, r1 and r2 are as shown in Fig. 2 (a)–(d). The Figure shows that when 0 < a1 < 0.0014, the system is in stable state, after multi game, rm, bm, r1 and r2 is stable at the point of (0.6085, 14.67, 0.0884, 0.5517). When a1 = 0.0014, the system (12) occurs the first bifurcation, then after cycle 2, 4 cycle, etc., the system is gradually into the chaotic state. The influence of a to profits of the supply chain shows as Fig. 3(a)–(c). In stable state, the most profits of the manufacturer, the retailer and the supply chain are respectively 194.4, 96.6 and 291(Due to the numerical simulation and values, there is a normal error.). Any participant in the closed-loop supply chain, not to break this pattern, the stability of the market can be maintained. But the ultimate goal of enterprises is to win the best interests. Fig. 3(b) shows that there are two situations after the first bifurcation. The first situation is that the retailer profits increase significantly, the second situation is less profitable. If the retailer choose to adjust along the direction of increased profits from the supply chain point of view, to improve the system’s total profits, but at the manufacturer’s point of view allows manufacturers a substantial decline in profits, apparently the manufacturer does not allow this happens, then the best choice for manufacturers to maintain the same steady state, while the manufacturer is a leader in the supply chain, the ability to maintain the steady state of the system. But with the changes of products life cycle, the manufacturer as the leader becomes the follower, resulting in loss of balance, confusion, and ultimately into the chaotic state. Therefore, in the supply chain system, the game parties must not only consider their own profits, taking into account also the game the opponent’s response to achieve the optimal state of the system. (2) System variables power spectrum According to the numerical simulation, not only system variables graph in the phase space along with time can be drew, but also power spectrum graph of system variables can be estimated by period chart method. When a1 ¼ 0:002; a2 ¼ 0:001; rm ; bm e and en+1 = xn+1 dn+1 are shown as Fig. 4 (a)–(d). According to the numerical results, no matter how large the collection price adjustment parameters, the collection price traverse the whole value area over time, but r m ; bm e and en+1 = xn+1 dn+1 pm in the system (12) always limit in a certain range, which also verifies boundedness and ergodicity of chaos. (3) The influence of initial setup The butterfly effect is an important symbol of chaotic motions, which is sensitive to initial state, and small changes of the initial conditions may cause the adjacent orbital evolution to index form separate after multiple episodes. For the above conditions, when a1 = 0.002, a2 = 0.001, sensitive dependence on initial value of rm ; bm e and en+1 = xn+1 dn+1 are as shown in Fig. 5(a)–(d). rm values are respectively taken 0.4 and 0.4001, after 59 cycles iteration, rm difference
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Fig. 2. (a) a1 is at [0, 0.002], a2 = 0.001, rm variation diagram. (b) a1 is at [0, 0.002], a2 = 0.001, bm variation diagram. (c) a1 is at [0, 0.002], a2 = 0.001, r1 variation diagram. (d) a1 is at [0, 0.002], a2 = 0.001, r2 variation diagram.
values is 0.48 which is 4800 times than the initial difference value. bm values are respectively taken 20 and 20. 001, after 92 cycles iteration, bm difference values is 0.7904 which is 790.4 times than the initial difference value. Derived from Fig. 5, we can conclude that system variables value are approximate in the first period, with increase of iterations the difference increase obviously. The system variables at the beginning of the period of time relatively close, with the increase in the number of iterations, the difference became clear intuitively that the system sensitive dependence on initial; manufacturer is the leader, retailers to follow, manufacturers decision-making, rational decision-making behavior retailers in the observed behavior of the manufacturer, the retailer’s strategy is superior to the manufacturer’s strategy, leading manufacturer of the initial impact is far much larger than the retailer. (4) The influence of a to the system Other parameters constant, when a = 120, a1 is at [0, 0.002], a2 is 0.001,the initial value of rm and bm are respectively 0.4 and 20, rm, bm, e and en+1 = xn+1 dn+1 are as shown in Fig. 6(a)–(d). The Figure shows that when 0 < a1 < 0.0014, the system is in stable state. after multi game, rm, bm, r1 and r2 is stable at the point of (0.7978, 14.69, 0.1301, 0.5516). When a1 = 0.0014, the system (12) occurs the first bifurcation, then after cycle 2, 4 cycle, etc., the system is gradually into the chaotic state. The influence of a to profits of the supply chain shows as Fig. 7(a)–(c). In stable state, the most profits of the manufacturer, the retailer and the supply chain are respectively 350.2, 179.1 and 530. The comparison analysis of Figs. 6 and 2 shows that with the increase of market demand, rm and e significantly increase, but bm and en+1 = xn+1 dn+1 have no significant change. The comparison analysis of Figs. 8 and 3 shows that profits of the manufacturer and the retailer increase significantly with the increase of a. The total system profit is shown in Fig. 8 when a is gradually from 100 to 120. The conclusion is drawn that to changes in market demand did not have a direct impact in the collecting market, and this reflects the current level of collecting is relatively low in China, and a good collecting system have not be formed; In addition, it reflects the enterprises meet market demand through new materials.
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Fig. 3. (a) a1 is at [0, 0.002], a2 = 0.001 Pm variation diagram. (b) a1 is at [0, 0.002], a2 = 0.001 Pr variation diagram. (c) a1 is at [0, 0.002], a2 = 0.001 P variation diagram.
4. Centralized control decision-making model 4.1. Model and analysis Centralized control is that the manufacturers and the retailer codetermine to realize profits maximization of the supply chain system.
max P ¼ Pm þ Pr
ð13Þ
We get formula (4) and (5) into the formula (13), and attain the profits function of supply chain which as follows
2 2 2 max P ¼ bm þ cr 1 þ r m 1 þ r 1 ða þ bðcm þ A1 ÞÞ b 1 þ r 1 1 þ r m bm þ cr aðcm þ A1 Þ 2 2 þ bm 1 r 2 ðbðcm cr A2 Þ kÞ bbm 1 r2 þ kðcm cr A2 Þ;
ð14Þ
PðtÞ ¼ Pm ðtÞ þ Pr ðtÞ ¼ ð1 þ rm ðtÞÞðcr þ bm ðtÞÞð1 þ r1 ðtÞÞða þ bðcm þ A1 ÞÞ bð1 þ r m ðtÞÞ2 ðcr þ bm ðtÞÞ2 ð1 þ r 1 ðtÞÞ2 aðcm þ A1 Þ þ bm ðtÞ ð1 r 2 ðtÞÞðbðcm cr A2 Þ kÞ bbm ðtÞ2 ð1 r 2 ðtÞÞ2 þ kðcm cr A2 Þ: The second-order derivatives about rm ; bm e and en+1 = xn+1 dn+1 are as follows in formula (15).
8 2 @ PðtÞ=@r m ðtÞ2 ¼ 2bð1 þ r 1 ðtÞÞ2 ðbm ðtÞ þ cr Þ2 < 0; > > > > < @ 2 PðtÞ=@b ðtÞ2 ¼ 2bð1 þ r ðtÞÞ2 ð1 þ r ðtÞÞ2 2bð1 r ðtÞÞ2 < 0; m 1 m 2 > @ 2 PðtÞ=@r 1 ðtÞ2 ¼ 2bð1 þ r m ðtÞÞ2 ðbm ðtÞ þ cr Þ2 < 0; > > > : 2 @ PðtÞ=@r 2 ðtÞ2 ¼ 2bbm ðtÞ2 < 0:
ð15Þ
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Fig. 4. (a) a1 = 0.002, a2 = 0.001, power spectrum of rm. (b) a1 = 0.002, a2 = 0.001, power spectrum of bm. (c) a1 = 0.002, a2 = 0.001, power spectrum of r1. (d) a1 = 0.002, a2 = 0.001, power spectrum of r2.
P is strictly concave function about rm ; bm e and en+1 = xn+1 dn+1, therefore, the optimal solutions can be obtained by @ P=@r m ðtÞ ¼ 0; @ P=@bm ðtÞ ¼ 0; @ P=@r 1 ¼ 0; @ P=@r 2 ¼ 0. The process is as follows. 8 ða þ bðA1 þ cm ÞÞ 2bð1 þ r 1 ðtÞÞð1 þ r m ðtÞÞðbm ðtÞ þ cr Þ ¼ 0; > > > > < ð1 þ r ðtÞÞð1 þ r ðtÞÞða þ bðA þ c ÞÞ 2bð1 þ r ðtÞÞ2 ð1 þ r ðtÞÞ2 ðb ðtÞ þ c ÞÞ; m 1 1 m 1 m m r 2 > > þð1 r ðtÞÞðbðc c A Þ kÞ 2bb ðtÞð1 r ðtÞÞ ¼ 0; 2 m r 2 m 2 > > : bðcm cr A2 Þ þ k þ 2bbm ðtÞð1 r 2 ðtÞÞ ¼ 0:
ð16Þ
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Fig. 5. (a) a1 = 0.002, a2 = 0.001, the initial value sensitivity of rm. (b) a1 = 0.002, a2 = 0.001, the initial value sensitivity of bm. (c) a1 = 0.002, a2 = 0.001, the initial value sensitivity of r1. (d) a1 = 0.002, a2 = 0.001, the initial value sensitivity of r2.
8 < r 1 ¼ 2b
aþbðA1 þcm Þ
ð1þrm Þðbm þcr Þ
: r ¼ 1 2
1; ð17Þ
bðcm cr A2 Þk : 2bbm
In reality, if the retailer is no profit, no sales and collecting. Therefore, if formula (17) is meaningful, it must satisfy the following conditions.
rm 6
a þ bðA1 þ cm Þ 1; 2b bm þ cr
bm P
bðcm cr A2 Þ k : 2b
The basis for decision making is the same as above. The manufacturer makes decision based on limited rational expectations. She adjusts the game process on the basis of marginal gains. If the marginal profits of t period are positive, in the t + 1 period
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Fig. 6. (a) a = 120, a1 at [0, 0.002], a2 = 0.001, rm variation diagram. (b) a = 120, a1 at [0, 0.002], a2 = 0.001, bm variation diagram. (c) a = 120, a1 is at [0, 0.002], a2 = 0.001, r1 variation diagram. (d) a = 120, a1 is at [0, 0.002], a2 = 0.001, r2 variation diagram.
this strategy action will be used. And if the t period marginal profits are negative, in the t + 1 period the collection price will be lowered. The repeated game model of dynamic adjustment is as follows.
8 > r ðt þ 1Þ ¼ r m ðtÞ þ a1 r m ðtÞððbm ðtÞ þ cr Þð1 þ r 1 ðtÞÞða þ bðA1 þ cm ÞÞ 2bð1 þ r 1 ðtÞÞ2 ð1 þ r m ðtÞÞðbm ðtÞ þ cr Þ2 Þ > > m > > 2 2 > > < bm ðt þ 1Þ ¼ bm ðtÞ þ a2 bm ðtÞðð1 þ r m ðtÞÞð1 þ r 1 ðtÞÞða þ bðA1 þ cm ÞÞ 2bðbm ðtÞ þ cr Þð1 þ r 1 ðtÞÞ ð1 þ rm ðtÞÞ 2 þð1 r 2 ðtÞÞðbðcm cr A2 Þ kÞ 2bð1 r2 ðtÞÞ bm ðtÞÞ; > > > > r 1 ðt þ 1Þ ¼ r1 ðtÞ þ a3 r 1 ðtÞðð1 þ r m ðtÞÞðbm ðtÞ þ cr Þða þ bðA1 þ cm ÞÞ 2bð1 þ r 1 ðtÞÞð1 þ r m ðtÞÞ2 ðbm ðtÞ þ cr Þ2 Þ; > > > : r 2 ðt þ 1Þ ¼ r2 ðtÞ þ a4 r 2 ðtÞðbm ðtÞðbðcm cr A2 Þ kÞ þ 2bð1 r 2 ðtÞÞbm ðtÞ2 Þ:
ð18Þ
In the formula (18), a1 is adjustment coefficient of rm a2 is adjustment coefficient of bm a3 is adjustment coefficient of r1(t). a4 is adjustment coefficient of r2(t). 4.2. Numerical simulation The initial set up and various parameter values with the decentralized decision-making control model, a = 100, cm = 60, cr = 30, b = 1.1, b = 0.6, k = 1, A1 = 0.5, A2 = 0.5, rm(1) = 0.4, bm(1) = 20, r1(1) = 0.3, r2(1) = 0.3. According to the numerical simulation, not only system variables graph in the phase space along with time can be drew, but also power spectrum graph of system variables can be estimated by period chart method. When a1 ¼ 0:001; a2 ¼ 0:002; a3 ¼ 0:001; a4 ¼ 0:001; rm ; bm e and en+1 = xn+1 dn+1 are shown as Fig. 9(a)–(d). According to the numerical results, no matter how large the collection price adjustment parameters, the collection price traverse the whole value area over time, but r m; ; bm e and en+1 = xn+1 dn+1pm in the system (18) always limit in a certain range, which also verifies boundedness and ergodicity of chaos.
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Fig. 7. (a) a = 120, a1 is at [0, 0.002], a2 = 0.001, Pm variation diagram. (b) a = 120, a1 is at [0, 0.002], a2 = 0.001, Pr variation diagram. (c) a = 120, a1 is at [0, 0.002], a2 = 0.001, P variation.
Fig. 8. a is at [100, 120], a1 is at [0, 0.002], a2 = 0.001, Pm variation diagram.
The most profits of the supply chain system shows as Fig. 10. The most profits of the manufacturer, the retailer and the supply chain are respectively 221.3, 162.6 and 384.2. Corresponding to the optimal decision variables are respectively 0.338, 20.01, 0.0996 and 0.3474. The collecting price is 13.0585, and product sale price is 73.5780.
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Fig. 9. (a) a1 = 0.001, a2 = 0.002, a3 = 0.001, a4 = 0.001, power spectrum of rm. (b) a1 = 0.001, a2 = 0.002, a3 = 0.001, a4 = 0.001, power spectrum of bm. (c) a1 = 0.001, a2 = 0.002, a3 = 0.001, a4 = 0.001, power spectrum of r1. (d) a1 = 0.001, a2 = 0.002, a3 = 0.001, a4 = 0.001, power spectrum of r2.
5. Comparative analysis and conclusions The results in Stackelberg decision-making and in centralized decision-making are shown as in the Table 1. Three conclusions from Table 1 are as follows. Firstly, the collecting price in decentralized control (non-cooperation) is lower than in the centralized control (cooperation). Secondly, the profits of the manufacturer and the retailer in the decentralized control are lower than in the centralized control. Lastly, in the centralized control, collecting price, collecting amounts, sale price, sale amounts and profits of the manufacturers, retailers and supply chain systems are increase. It is means that the manufacturers and retail return profits to consumers by raising the collecting price of used-products. The manufacturer and the retailer to achieve win–win situation, also consumers have also benefited.
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Fig. 10. a1 = 0.001, a2 is at [0, 0.002], a3 = 0.001, a4 = 0.001, P variation diagram.
Table 1 Comparative results.
rm bm r1 r2 br pr
Pm Pr P
Stackelberg decision-making
Comparing
Centralized decision-making
0.6085 14.67 0.0884 0.5517 6.5766 78.2034 194.4 96.6 291
> < < > < < < < <
0.338 20.01 0.0996 0.3474 13.0585 73.5780 221.3 162.9 384.2
Through comparison centralized decision-making with decentralized decision-making, we have concluded that the collecting price and system profits of the latter are lower. This conclusion for the supply chain guide enterprises sure each phase of the best provide decision basis for the collecting price, in order to achieve maximize returns. This work was supported by the Foundation: the Doctoral Scientific Fund Project of the Ministry of Education of China under contract 20090032110031. It also supported by National Natural Science Foundation of China 61273231. References [1] M. Fleischmann, J.M. Bloemhof-Ruwaard, R. Dekker, E. van derLaan, J.A.E.E. van Nunen, L.N. Van Wassenhove, Quantitative models for reverse logistics: a review, Eur. J. Oper. Res. 103 (1997) 1–17. [2] D. Guide, V. Jayraman, R. Srivastava, W.C. Benton, Supply chain management for recoverable manufacturing systems, Interfaces 30 (3) (2000) 125–142. [3] T. McGuire, R. Staelin, An industry equilibrium analysis of downstream vertical integration, J. Market. Sci. 2 (1983) 161–192. [4] V.D.R. Guide, V. Wassenhove, Managing product returns for remanufacturing, Wording paper, 2000/75/TM, INSEAD, Fontainblau, France. [5] R.C. Savaskan, S. Bhattacharya, L.N. Van Wassenhove, Channel choice and coordination in a remanufacturing environment, R. INSEAD.2000, Working Paper. [6] Qiaolun Gu, Tie-Gang Gao, Lian-shuan Shi, Price decision analysis for reverse supply chain based on game theory, J. Syst. Eng. Theory Pract. 3 (2005) 20–25. [7] Yu-yan Wang, Bang-yi Li, Fei-fei Yue, The research on two price decision models of closed-loop supply chain, J. Forecast. 25 (6) (2006) 70–73. [8] Jing-yan Ge, Pei-qing Huang, Juan Li, Social environmental consciousness and price decision analysis for closed-loop supply chains—based on vertical differentiation model, J. Indust. Eng. Manag. 4 (2007) 6–10. [9] Xiao-min Zhao, Zhi-jun Feng, Pei-qing Huang, Closed-loop supply chains management-managerial innovation on meeting weee eu directive in our electronic industries, J. China Indus. Econom. (8) (2004) 48–55. [10] Weixin Yao, To Study atomic models for closed-loo supply chain in e-commerce, Manag. Sci. China 16 (1) (2003) 65–68. [11] V.D.R. Guide Jr., V. Jayarman, J.D. linton, Building contingence planning for closed-loop supply chains with product recovery, J. Oper. Manag. (2003). [12] SjurDidrikFlhm, Yuri M. Kaniovski, Price expectations and cobwebs under uncertainty, J. Ann. Oper. Res. 114 (2002) 167–181.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Research on game model and complexity of retailer collecting and selling in closed-loop supply chain Yuehong Guo a,b, Junhai Ma a,⇑ a b
College of Management Economic, Tianjin University, Tianjin 300072, China School of Economics and Management, Tianjin University of Technology and Education, Tianjin 300222, China
a r t i c l e
i n f o
Article history: Received 2 March 2012 Received in revised form 7 August 2012 Accepted 11 September 2012 Available online 21 September 2012 Keywords: Repeated games Closed-loop supply chain Complexity Chaos
a b s t r a c t A closed-loop supply chain is a complex system in which node enterprises play important roles and exert great influence. Firstly, this paper established a collecting price game model for a close-loop supply chain system with a manufacturer and a retailer who have different rationalities. It assumed that the node enterprises took the marginal utility maximization as the basis of decision-making. Secondly, through numerical simulation, we analyzed complex dynamic phenomena such as the bifurcation, chaos and continuous power spectrum and so on. Thirdly, we analyzed the influences of the system parameters; this further explained the complex nonlinear dynamics behavior from the perspective of economics. The results have significant theoretical and practical application value. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction The importance of the environmental performance of products and processes for sustainable manufacturing and service operations increasingly is being recognized in practice. While legislation introduced in Europe, North America, and Japan encourages this awareness, many corporations have recognized reducing production costs and improving profits from used products. China has required that electrical products manufacturer must be responsible for collecting waste products since 2003. Remanufactured products are typically upgraded to quality standards of new products, so that they can be sold in new product markets. (see Fig. 1). This paper draws on and contributes to several streams of literature, each of which we review below. A growing literature in operations management addresses reverse logistics management issues for remanufactured products. We refer the reader to Fleischmann et al. [1] and Guide et al. [2] for complete literature reviews. They defined as the supply chain to reverse supply chain which formed a complete closed-loop system (closed-loop supply chain, hereinafter referred to as CLSC). McGuire and Staelin [3] studied the pricing methods under three collecting channels which manufacturer, retailer and third-party collector collected products respectively. Guide and Wassenhove [4] to Xerox Corporation and Kodak Corporation studied that the manufacturers are responsible for the management of the collecting. Savaskan and Bhattacharya [5] studied three kinds of collecting channels of manufacturers collecting, retailer collecting and the third-party collecting in closed-loop supply chain. They found that the collecting distance was closer from consumers, the collecting efforts was more effective. In the literature describing reverse logistics management mode of the third-party. In this collecting mode, manufacturers entrust a third-party to perform extended producer responsibility and manage used-products.
⇑ Corresponding author. E-mail address: [email protected] (J. Ma). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.09.034
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Manufacturer
Retailer
Customer
Fig. 1. A manufacturer subcontract the sale and collection activity to an retailer.
Gu et al. [6] and Wang et al. [7] studied pricing analysis of reverse supply chain using the game theory method under retailers collecting mode and concluded the optimal strategy of single stage. Ge [8] established the demand model of new products and remanufacturing products from social environmental protection consciousness, consumers’ utility function to new products and remanufacturing products, and drew the conclusions. The first is the optimal pricing strategy in different collecting channels in decentralized decision-making. The second is the influence of social environmental protection consciousness. Xiao-min Zhao et al. [9] studied closed-loop supply chains management-managerial innovation on meeting weee eu directive in Chinese electronic industries. Weixin Yao [10] established atomic models for closed-loop supply chain in e-commerce. V.D.R. Guide Jr et al. [11] had a research on contingence planning for closed-loop supply chains with product recovery, and got meaningful conclusion. Sjur. Didrik. Flhm et al. [12] analyzed price expectations and cobwebs under uncertainty. 2. Model assumptions and notation 2.1. Assumption This model is based on these following assumptions. (1) This paper considers remanufactured products and new products don’t form a competitive market. When Remanufactured products do not meet the market demand, it supplemented by new products. New products and remanufactured products are no difference. The manufacturer manufactures new products and remanufacturing products. (2) The closed-loop supply chain consists of a manufacturer and a retailer, the retailer collects waste products from consumers and returns the manufacturer. The manufacturer transfers payments to the retailer, the retailer sales products to consumers. The manufacturers and the retailer are independent decision makers, and their strategic space is to choose the best collecting price. Their goal is to maximize returns in discrete time period as t = 0, 1, 2 . . . . (3) The number of the collecting is increasing function of collecting price. The collecting capability and manufacturing capability are unlimited. In order to simplify the problem and emphasize main parameters influence of the system, all the collecting products can be manufactured. 2.2. Notation description bm(t)—It is the transfer collecting price which the manufacturer paying to the retailer, the manufacturer’s decision variable. br(t), r2(t) — br(t) is the unit collecting price of the retailer paying to consumer markets. r2(t) is the profitability of the retailer, and which is the retailer’s decision variable. The formula is as follows.
br ðtÞ ¼ bm ðtÞð1 r2 ðtÞÞ;
ð1Þ
cm — It is the marginal cost of manufacturing a new product with raw materials, to be a constant. cr — It is the marginal cost of remanufacturing a used-product into a new one, to be a constant. pm(t), rm(t) — pm(t) is the unit wholesale price in t period which is the manufacturer’s decision variable. rm(t) is the profitability of the manufacturer, and which is the manufacturer’s decision variable. The formula is as follows.
pm ðtÞ ¼ ðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ;
ð2Þ
pr(t), r1(t) — pr(t) is the unit selling price in t period which is the retailer’s decision variable. r1(t) is the selling profitability of the retailer which is the retailer’s decision variable. The formula is as follow:
pr ðtÞ ¼ pm ðtÞð1 þ r1 ðtÞÞ:
ð3Þ
Ai — It is the collecting cost of retailer which including logistics cost etc., a constant, where i = 1, 2, A1 are selling costs, A2 are collecting costs. D–D is the demand for products in the market as a function of product price, i.e., D = a bpr(t) where a is market scale, b is price sensitivity of consumers. The value of b depends on many factors, including economic factors, social factors and psychological factors. It reflects in three aspects: firstly, it is the quantity and similar degree of products, in general, the higher the replacement of goods, the higher the value of b, on the other hand, the lower the replacement of goods, the lower the
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value of b. Secondly, it is the importance of goods, the value of b is lower of basic necessities of life, while the value of b is higher of luxury goods that is non-essential. Thirdly, it is adjustment time of consumers to demand, in general, the shorter the adjustment time, the lower of b; the longer the adjustment time, the higher of b. G — It is the market supply for the used-product in the market as a function of collection price. We assume that supply associated with collection price only and it is linear increasing function of the collection cost, i.e., G = k + bbr(t)k, b > 0. k is the quantity of volunteer to return used-products from consumers, and on another level as a measure of social awareness; b is the price sensitivity of consumers for collection price. Furthermore, D is greater than G. Pm — It is the profit function of the manufacturer. The function is as follows.
Pm ¼ ½pm ðtÞ bm ðtÞ cr ½k þ bbr ðtÞ þ ½pm ðtÞ cm ½a bpr ðtÞ k bbr ðtÞ;
ð4Þ
Pr —It is the profit function of the retailer. The function is as follows.
Pr ¼ ½pr ðtÞ pm ðtÞ A1 ½a bpr ðtÞ þ ½bm ðtÞ br ðtÞ A2 ½k þ bbr ðtÞ;
ð5Þ
P— It is the profit function of the closed-loop supply chain. The function is as follows.
P ¼ Pm þ Pr :
ð6Þ
Will be substituted into Eqs. (4)–(6), Pm(t), Pr(t) and P(t) are as follows.
8 Pm ðtÞ ¼ ½cm cr bm ðtÞ½k þ bbm ðtÞð1 r 2 ðtÞÞ þ ½ð1 þ r m ðtÞÞðcr þ bm ðtÞÞ cm > > > > > > ½a bð1 þ r m ðtÞÞðcr þ bm ðtÞÞð1 þ r 1 ðtÞÞ; > > > > > > > Pr ðtÞ ¼ ½r 1 ðtÞðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ A1 ½a bðbm ðtÞ þ cr Þð1 þ r m ðtÞÞð1 þ r1 ðtÞÞ > < þ½bm ðtÞr 2 ðtÞ A2 ½k þ bbm ðtÞð1 r 2 ðtÞÞ; > > > > PðtÞ ¼ Pm ðtÞ þ Pr ðtÞ > > > > > > ¼ ð1 þ rm ðtÞÞðcr þ bm ðtÞÞð1 þ r1 ðtÞÞða þ bðcm þ A1 ÞÞ bð1 þ r m ðtÞÞ2 ðcr þ bm ðtÞÞ2 ð1 þ r 1 ðtÞÞ2 > > > > : aðcm þ A1 Þ þ bm ðtÞð1 r 2 ðtÞÞðbðcm cr A2 Þ kÞ bbm ðtÞ2 ð1 r 2 ðtÞÞ2 þ kðcm cr A2 Þ:
ð7Þ
3. Stackelberg model 3.1. Model and analysis Suppose that the manufacturer and the retailer are principal and subordinate relationship, and the manufacturer is Stacklelberg leader, the retailer is follower, then they process sequential dynamic game, the game equilibrium is Stacklelberg equilibrium. In this game, the manufacturer make the decision of sales price and collection price according to the market information, then the retailer make decision according to the decision-making of the manufacture. Using backward induction, first find the second stage of the game model response function, to get Stackelberg balanced. First of all requirements is as follows.
max Pr ¼ ða bðbm ðtÞ þ cr Þð1 þ r m ðtÞÞð1 þ r1 ðtÞÞÞðr1 ðtÞðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ A1 Þ þ ðk þ bbm ðtÞð1 r 2 ðtÞÞÞðbm ðtÞr 2 ðtÞ A2 Þ ð8Þ The optimal marginal profit of the retailer can be concluded by the first-order conditions of formula (8). The result is as follows.
( @ Pr @r 1
¼ ðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ½a bð2r1 ðtÞ þ 1Þð1 þ r m ðtÞÞðbm ðtÞ þ cr Þ þ bA1 ;
@ Pr @r 2
¼ bm ðtÞ½k þ bbm ðtÞ 2br2 ðtÞbm ðtÞ þ bA2 :
ð9Þ
The retailer’s reaction functions are as follows by solving formula (9).
8 < r 1 ¼ 2b
aþbA1
ðbm þcr Þð1þrm Þ
: r ¼ bbm þbA 2 þk : 2 2bb
12 :
ð10Þ
m
Formula (10) is the optimal decision-making of the retailer on the premise of the manufacturer’s selling price and collecting price are certain, the retailer obtains the decision after she observed the manufacturer behavior. It is the retailer’s reaction function. Substitution formula (10) into formula (7), by @ Pm/@rm = 0, @ Pm/@bm = 0, the optimal result of the manufacturer can be obtained. Assume that the parameters values for a = 100, cm = 60, cr = 30, b = 1.2, b = 1, k = 0.1, A1 = 0.5, A2 = 0.5. Therefore, the Stac kelberg equilibrium is r m ; bm ; r 1 ; r 2 ¼ ð0:36; 12:25; 0:11; 0:59Þ. When making price decision, restricted by the objective conditions such as the decision-making ability and so on, each decision maker cannot get the whole market information and his price decision is also not completely rational. Also, because
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of the differences such as in the input on information collections and in the experiences of decision maker and so on, the information grasped by each node enterprise is not necessarily same exactly in the condition of incomplete market information, and they cannot have the exact same rational levels, so the final price decision and its setting rule of each maker are not exactly the same. Therefore, the manufacturer makes decision based on limited rational expectations. She adjusts the game process on the basis of marginal profits. If the marginal profits in period t are positive, then they will continue their output adjustment strategy in period t + 1. The model can be constructed as follows.
8 < r m ðt þ 1Þ ¼ r m ðtÞ þ a1 rm ðtÞ @@rPm ; m
: b ðt þ 1Þ ¼ b ðtÞ þ a b ðtÞ @ Pm ; m m 2 m @bm
ð11Þ
where a1 is the adjustment coefficient of rm. a2 is adjustment coefficient of bm. Then, the dynamic adjustment of the closed-loop supply chain system can be written as follows.
8 > r m ðt þ 1Þ ¼ r m ðtÞ þ a1 rm ðtÞ ðbm ðtÞ þ cr Þ abA12þbcm bðbm ðtÞ þ cr Þð1 þ r m ðtÞÞ ; > > > > > > > > b ðt þ 1Þ ¼ b ðtÞ þ a b ðtÞ ð1þrm ðtÞÞðaþbcm bA1 Þ bðb ðtÞ þ c Þð1 þ r ðtÞÞ2 > m m 2 m m r m > 2 > > < r ÞþbA2 ; bbm ðtÞ kbðcm c 2 > > > > > > aþbA1 > 12 ; r 1 ðtÞ ¼ 2bðbm ðtÞþc > > r Þð1þr m ðtÞÞ > > > > : r ðtÞ ¼ bbm ðtÞþbA2 þk 2 2bbm ðtÞ
ð12Þ
From system (12) can be concluded that the manufacture firstly makes decision, and the decision variables directly relate to a1 and a2. But the retailer’s decision variables directly relate to rm and bm. The following simulates complex dynamics characteristics of system (12) through numerical simulation. 3.2. Numerical simulation Using MATLAB, influence of the parameters on the system (12) can be analyzed through numerical simulation. (1) The influence of a to the supply chain Assume that a1 is at [0, 0.002] and a2 is fixed, the initial value rm and bm are 0.4 and 20, rm, bm, r1 and r2 are as shown in Fig. 2 (a)–(d). The Figure shows that when 0 < a1 < 0.0014, the system is in stable state, after multi game, rm, bm, r1 and r2 is stable at the point of (0.6085, 14.67, 0.0884, 0.5517). When a1 = 0.0014, the system (12) occurs the first bifurcation, then after cycle 2, 4 cycle, etc., the system is gradually into the chaotic state. The influence of a to profits of the supply chain shows as Fig. 3(a)–(c). In stable state, the most profits of the manufacturer, the retailer and the supply chain are respectively 194.4, 96.6 and 291(Due to the numerical simulation and values, there is a normal error.). Any participant in the closed-loop supply chain, not to break this pattern, the stability of the market can be maintained. But the ultimate goal of enterprises is to win the best interests. Fig. 3(b) shows that there are two situations after the first bifurcation. The first situation is that the retailer profits increase significantly, the second situation is less profitable. If the retailer choose to adjust along the direction of increased profits from the supply chain point of view, to improve the system’s total profits, but at the manufacturer’s point of view allows manufacturers a substantial decline in profits, apparently the manufacturer does not allow this happens, then the best choice for manufacturers to maintain the same steady state, while the manufacturer is a leader in the supply chain, the ability to maintain the steady state of the system. But with the changes of products life cycle, the manufacturer as the leader becomes the follower, resulting in loss of balance, confusion, and ultimately into the chaotic state. Therefore, in the supply chain system, the game parties must not only consider their own profits, taking into account also the game the opponent’s response to achieve the optimal state of the system. (2) System variables power spectrum According to the numerical simulation, not only system variables graph in the phase space along with time can be drew, but also power spectrum graph of system variables can be estimated by period chart method. When a1 ¼ 0:002; a2 ¼ 0:001; rm ; bm e and en+1 = xn+1 dn+1 are shown as Fig. 4 (a)–(d). According to the numerical results, no matter how large the collection price adjustment parameters, the collection price traverse the whole value area over time, but r m ; bm e and en+1 = xn+1 dn+1 pm in the system (12) always limit in a certain range, which also verifies boundedness and ergodicity of chaos. (3) The influence of initial setup The butterfly effect is an important symbol of chaotic motions, which is sensitive to initial state, and small changes of the initial conditions may cause the adjacent orbital evolution to index form separate after multiple episodes. For the above conditions, when a1 = 0.002, a2 = 0.001, sensitive dependence on initial value of rm ; bm e and en+1 = xn+1 dn+1 are as shown in Fig. 5(a)–(d). rm values are respectively taken 0.4 and 0.4001, after 59 cycles iteration, rm difference
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Fig. 2. (a) a1 is at [0, 0.002], a2 = 0.001, rm variation diagram. (b) a1 is at [0, 0.002], a2 = 0.001, bm variation diagram. (c) a1 is at [0, 0.002], a2 = 0.001, r1 variation diagram. (d) a1 is at [0, 0.002], a2 = 0.001, r2 variation diagram.
values is 0.48 which is 4800 times than the initial difference value. bm values are respectively taken 20 and 20. 001, after 92 cycles iteration, bm difference values is 0.7904 which is 790.4 times than the initial difference value. Derived from Fig. 5, we can conclude that system variables value are approximate in the first period, with increase of iterations the difference increase obviously. The system variables at the beginning of the period of time relatively close, with the increase in the number of iterations, the difference became clear intuitively that the system sensitive dependence on initial; manufacturer is the leader, retailers to follow, manufacturers decision-making, rational decision-making behavior retailers in the observed behavior of the manufacturer, the retailer’s strategy is superior to the manufacturer’s strategy, leading manufacturer of the initial impact is far much larger than the retailer. (4) The influence of a to the system Other parameters constant, when a = 120, a1 is at [0, 0.002], a2 is 0.001,the initial value of rm and bm are respectively 0.4 and 20, rm, bm, e and en+1 = xn+1 dn+1 are as shown in Fig. 6(a)–(d). The Figure shows that when 0 < a1 < 0.0014, the system is in stable state. after multi game, rm, bm, r1 and r2 is stable at the point of (0.7978, 14.69, 0.1301, 0.5516). When a1 = 0.0014, the system (12) occurs the first bifurcation, then after cycle 2, 4 cycle, etc., the system is gradually into the chaotic state. The influence of a to profits of the supply chain shows as Fig. 7(a)–(c). In stable state, the most profits of the manufacturer, the retailer and the supply chain are respectively 350.2, 179.1 and 530. The comparison analysis of Figs. 6 and 2 shows that with the increase of market demand, rm and e significantly increase, but bm and en+1 = xn+1 dn+1 have no significant change. The comparison analysis of Figs. 8 and 3 shows that profits of the manufacturer and the retailer increase significantly with the increase of a. The total system profit is shown in Fig. 8 when a is gradually from 100 to 120. The conclusion is drawn that to changes in market demand did not have a direct impact in the collecting market, and this reflects the current level of collecting is relatively low in China, and a good collecting system have not be formed; In addition, it reflects the enterprises meet market demand through new materials.
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Fig. 3. (a) a1 is at [0, 0.002], a2 = 0.001 Pm variation diagram. (b) a1 is at [0, 0.002], a2 = 0.001 Pr variation diagram. (c) a1 is at [0, 0.002], a2 = 0.001 P variation diagram.
4. Centralized control decision-making model 4.1. Model and analysis Centralized control is that the manufacturers and the retailer codetermine to realize profits maximization of the supply chain system.
max P ¼ Pm þ Pr
ð13Þ
We get formula (4) and (5) into the formula (13), and attain the profits function of supply chain which as follows
2 2 2 max P ¼ bm þ cr 1 þ r m 1 þ r 1 ða þ bðcm þ A1 ÞÞ b 1 þ r 1 1 þ r m bm þ cr aðcm þ A1 Þ 2 2 þ bm 1 r 2 ðbðcm cr A2 Þ kÞ bbm 1 r2 þ kðcm cr A2 Þ;
ð14Þ
PðtÞ ¼ Pm ðtÞ þ Pr ðtÞ ¼ ð1 þ rm ðtÞÞðcr þ bm ðtÞÞð1 þ r1 ðtÞÞða þ bðcm þ A1 ÞÞ bð1 þ r m ðtÞÞ2 ðcr þ bm ðtÞÞ2 ð1 þ r 1 ðtÞÞ2 aðcm þ A1 Þ þ bm ðtÞ ð1 r 2 ðtÞÞðbðcm cr A2 Þ kÞ bbm ðtÞ2 ð1 r 2 ðtÞÞ2 þ kðcm cr A2 Þ: The second-order derivatives about rm ; bm e and en+1 = xn+1 dn+1 are as follows in formula (15).
8 2 @ PðtÞ=@r m ðtÞ2 ¼ 2bð1 þ r 1 ðtÞÞ2 ðbm ðtÞ þ cr Þ2 < 0; > > > > < @ 2 PðtÞ=@b ðtÞ2 ¼ 2bð1 þ r ðtÞÞ2 ð1 þ r ðtÞÞ2 2bð1 r ðtÞÞ2 < 0; m 1 m 2 > @ 2 PðtÞ=@r 1 ðtÞ2 ¼ 2bð1 þ r m ðtÞÞ2 ðbm ðtÞ þ cr Þ2 < 0; > > > : 2 @ PðtÞ=@r 2 ðtÞ2 ¼ 2bbm ðtÞ2 < 0:
ð15Þ
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Fig. 4. (a) a1 = 0.002, a2 = 0.001, power spectrum of rm. (b) a1 = 0.002, a2 = 0.001, power spectrum of bm. (c) a1 = 0.002, a2 = 0.001, power spectrum of r1. (d) a1 = 0.002, a2 = 0.001, power spectrum of r2.
P is strictly concave function about rm ; bm e and en+1 = xn+1 dn+1, therefore, the optimal solutions can be obtained by @ P=@r m ðtÞ ¼ 0; @ P=@bm ðtÞ ¼ 0; @ P=@r 1 ¼ 0; @ P=@r 2 ¼ 0. The process is as follows. 8 ða þ bðA1 þ cm ÞÞ 2bð1 þ r 1 ðtÞÞð1 þ r m ðtÞÞðbm ðtÞ þ cr Þ ¼ 0; > > > > < ð1 þ r ðtÞÞð1 þ r ðtÞÞða þ bðA þ c ÞÞ 2bð1 þ r ðtÞÞ2 ð1 þ r ðtÞÞ2 ðb ðtÞ þ c ÞÞ; m 1 1 m 1 m m r 2 > > þð1 r ðtÞÞðbðc c A Þ kÞ 2bb ðtÞð1 r ðtÞÞ ¼ 0; 2 m r 2 m 2 > > : bðcm cr A2 Þ þ k þ 2bbm ðtÞð1 r 2 ðtÞÞ ¼ 0:
ð16Þ
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Fig. 5. (a) a1 = 0.002, a2 = 0.001, the initial value sensitivity of rm. (b) a1 = 0.002, a2 = 0.001, the initial value sensitivity of bm. (c) a1 = 0.002, a2 = 0.001, the initial value sensitivity of r1. (d) a1 = 0.002, a2 = 0.001, the initial value sensitivity of r2.
8 < r 1 ¼ 2b
aþbðA1 þcm Þ
ð1þrm Þðbm þcr Þ
: r ¼ 1 2
1; ð17Þ
bðcm cr A2 Þk : 2bbm
In reality, if the retailer is no profit, no sales and collecting. Therefore, if formula (17) is meaningful, it must satisfy the following conditions.
rm 6
a þ bðA1 þ cm Þ 1; 2b bm þ cr
bm P
bðcm cr A2 Þ k : 2b
The basis for decision making is the same as above. The manufacturer makes decision based on limited rational expectations. She adjusts the game process on the basis of marginal gains. If the marginal profits of t period are positive, in the t + 1 period
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Fig. 6. (a) a = 120, a1 at [0, 0.002], a2 = 0.001, rm variation diagram. (b) a = 120, a1 at [0, 0.002], a2 = 0.001, bm variation diagram. (c) a = 120, a1 is at [0, 0.002], a2 = 0.001, r1 variation diagram. (d) a = 120, a1 is at [0, 0.002], a2 = 0.001, r2 variation diagram.
this strategy action will be used. And if the t period marginal profits are negative, in the t + 1 period the collection price will be lowered. The repeated game model of dynamic adjustment is as follows.
8 > r ðt þ 1Þ ¼ r m ðtÞ þ a1 r m ðtÞððbm ðtÞ þ cr Þð1 þ r 1 ðtÞÞða þ bðA1 þ cm ÞÞ 2bð1 þ r 1 ðtÞÞ2 ð1 þ r m ðtÞÞðbm ðtÞ þ cr Þ2 Þ > > m > > 2 2 > > < bm ðt þ 1Þ ¼ bm ðtÞ þ a2 bm ðtÞðð1 þ r m ðtÞÞð1 þ r 1 ðtÞÞða þ bðA1 þ cm ÞÞ 2bðbm ðtÞ þ cr Þð1 þ r 1 ðtÞÞ ð1 þ rm ðtÞÞ 2 þð1 r 2 ðtÞÞðbðcm cr A2 Þ kÞ 2bð1 r2 ðtÞÞ bm ðtÞÞ; > > > > r 1 ðt þ 1Þ ¼ r1 ðtÞ þ a3 r 1 ðtÞðð1 þ r m ðtÞÞðbm ðtÞ þ cr Þða þ bðA1 þ cm ÞÞ 2bð1 þ r 1 ðtÞÞð1 þ r m ðtÞÞ2 ðbm ðtÞ þ cr Þ2 Þ; > > > : r 2 ðt þ 1Þ ¼ r2 ðtÞ þ a4 r 2 ðtÞðbm ðtÞðbðcm cr A2 Þ kÞ þ 2bð1 r 2 ðtÞÞbm ðtÞ2 Þ:
ð18Þ
In the formula (18), a1 is adjustment coefficient of rm a2 is adjustment coefficient of bm a3 is adjustment coefficient of r1(t). a4 is adjustment coefficient of r2(t). 4.2. Numerical simulation The initial set up and various parameter values with the decentralized decision-making control model, a = 100, cm = 60, cr = 30, b = 1.1, b = 0.6, k = 1, A1 = 0.5, A2 = 0.5, rm(1) = 0.4, bm(1) = 20, r1(1) = 0.3, r2(1) = 0.3. According to the numerical simulation, not only system variables graph in the phase space along with time can be drew, but also power spectrum graph of system variables can be estimated by period chart method. When a1 ¼ 0:001; a2 ¼ 0:002; a3 ¼ 0:001; a4 ¼ 0:001; rm ; bm e and en+1 = xn+1 dn+1 are shown as Fig. 9(a)–(d). According to the numerical results, no matter how large the collection price adjustment parameters, the collection price traverse the whole value area over time, but r m; ; bm e and en+1 = xn+1 dn+1pm in the system (18) always limit in a certain range, which also verifies boundedness and ergodicity of chaos.
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Fig. 7. (a) a = 120, a1 is at [0, 0.002], a2 = 0.001, Pm variation diagram. (b) a = 120, a1 is at [0, 0.002], a2 = 0.001, Pr variation diagram. (c) a = 120, a1 is at [0, 0.002], a2 = 0.001, P variation.
Fig. 8. a is at [100, 120], a1 is at [0, 0.002], a2 = 0.001, Pm variation diagram.
The most profits of the supply chain system shows as Fig. 10. The most profits of the manufacturer, the retailer and the supply chain are respectively 221.3, 162.6 and 384.2. Corresponding to the optimal decision variables are respectively 0.338, 20.01, 0.0996 and 0.3474. The collecting price is 13.0585, and product sale price is 73.5780.
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Fig. 9. (a) a1 = 0.001, a2 = 0.002, a3 = 0.001, a4 = 0.001, power spectrum of rm. (b) a1 = 0.001, a2 = 0.002, a3 = 0.001, a4 = 0.001, power spectrum of bm. (c) a1 = 0.001, a2 = 0.002, a3 = 0.001, a4 = 0.001, power spectrum of r1. (d) a1 = 0.001, a2 = 0.002, a3 = 0.001, a4 = 0.001, power spectrum of r2.
5. Comparative analysis and conclusions The results in Stackelberg decision-making and in centralized decision-making are shown as in the Table 1. Three conclusions from Table 1 are as follows. Firstly, the collecting price in decentralized control (non-cooperation) is lower than in the centralized control (cooperation). Secondly, the profits of the manufacturer and the retailer in the decentralized control are lower than in the centralized control. Lastly, in the centralized control, collecting price, collecting amounts, sale price, sale amounts and profits of the manufacturers, retailers and supply chain systems are increase. It is means that the manufacturers and retail return profits to consumers by raising the collecting price of used-products. The manufacturer and the retailer to achieve win–win situation, also consumers have also benefited.
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Fig. 10. a1 = 0.001, a2 is at [0, 0.002], a3 = 0.001, a4 = 0.001, P variation diagram.
Table 1 Comparative results.
rm bm r1 r2 br pr
Pm Pr P
Stackelberg decision-making
Comparing
Centralized decision-making
0.6085 14.67 0.0884 0.5517 6.5766 78.2034 194.4 96.6 291
> < < > < < < < <
0.338 20.01 0.0996 0.3474 13.0585 73.5780 221.3 162.9 384.2
Through comparison centralized decision-making with decentralized decision-making, we have concluded that the collecting price and system profits of the latter are lower. This conclusion for the supply chain guide enterprises sure each phase of the best provide decision basis for the collecting price, in order to achieve maximize returns. This work was supported by the Foundation: the Doctoral Scientific Fund Project of the Ministry of Education of China under contract 20090032110031. It also supported by National Natural Science Foundation of China 61273231. References [1] M. Fleischmann, J.M. Bloemhof-Ruwaard, R. Dekker, E. van derLaan, J.A.E.E. van Nunen, L.N. Van Wassenhove, Quantitative models for reverse logistics: a review, Eur. J. Oper. Res. 103 (1997) 1–17. [2] D. Guide, V. Jayraman, R. Srivastava, W.C. Benton, Supply chain management for recoverable manufacturing systems, Interfaces 30 (3) (2000) 125–142. [3] T. McGuire, R. Staelin, An industry equilibrium analysis of downstream vertical integration, J. Market. Sci. 2 (1983) 161–192. [4] V.D.R. Guide, V. Wassenhove, Managing product returns for remanufacturing, Wording paper, 2000/75/TM, INSEAD, Fontainblau, France. [5] R.C. Savaskan, S. Bhattacharya, L.N. Van Wassenhove, Channel choice and coordination in a remanufacturing environment, R. INSEAD.2000, Working Paper. [6] Qiaolun Gu, Tie-Gang Gao, Lian-shuan Shi, Price decision analysis for reverse supply chain based on game theory, J. Syst. Eng. Theory Pract. 3 (2005) 20–25. [7] Yu-yan Wang, Bang-yi Li, Fei-fei Yue, The research on two price decision models of closed-loop supply chain, J. Forecast. 25 (6) (2006) 70–73. [8] Jing-yan Ge, Pei-qing Huang, Juan Li, Social environmental consciousness and price decision analysis for closed-loop supply chains—based on vertical differentiation model, J. Indust. Eng. Manag. 4 (2007) 6–10. [9] Xiao-min Zhao, Zhi-jun Feng, Pei-qing Huang, Closed-loop supply chains management-managerial innovation on meeting weee eu directive in our electronic industries, J. China Indus. Econom. (8) (2004) 48–55. [10] Weixin Yao, To Study atomic models for closed-loo supply chain in e-commerce, Manag. Sci. China 16 (1) (2003) 65–68. [11] V.D.R. Guide Jr., V. Jayarman, J.D. linton, Building contingence planning for closed-loop supply chains with product recovery, J. Oper. Manag. (2003). [12] SjurDidrikFlhm, Yuri M. Kaniovski, Price expectations and cobwebs under uncertainty, J. Ann. Oper. Res. 114 (2002) 167–181.