Decision-making on Reverse Logistics for Manufacturers: an Evolutionary Game Theory Perspective Lirong Wei
Wei Gu
Donlinks school of economics and management University of Science and Technology Beijing Beijing, China
[email protected]
Donlinks school of economics and management University of Science and Technology Beijing Beijing, China
[email protected]
Abstract—Reverse logistics has become one of the most important competitive strategies for enterprises as environmental problems and resource issues have been serious. In this paper, an evolutionary game model is developed to analyze the reverse logistics decision for manufacturers. The decision-making process for implementin0067 the reverse logistics of the manufacturers under competitive market environment has been investigated in this paper. In addition, a dynamic evolutionary system is formulated to study the cooperation trends of manufacturers with recyclers. To verify the theoretical results, simulation experiments are conducted on MATLAB simulation platform. The results showed that manufacturer’s decisionmaking process is influenced by people’s attention on Enterprise Reverse Logistics in long-term perspective. In the implementation of the reverse logistics process, manufacturers can sign a contract with recyclers with incentives or penalty to ensure the supply of waste products. The results can guide manufacturers to make better decisions to promote the recycling of resources. Keywords—Reverse logistics; evolutionary game theory; replicator dynamics;decision making
I. INTRODUCTION Environmental issues and resource shortage are increasing awareness of importance by people. European Union, Japan and the US has published regulations or directives on waste management for a long time. Recently, business is showing great interest in reverse logistics or taking it as a strategic activity. The reverse logistics is also a profitability activity from direct and indirect ways. According to Wassenhove Thierry, there are five reasons for reverse logistics: sales return, maintenance, re-use, remanufacture and recycle [1]. Alumur focused on the remanufacturing reverse logistics network including computers, washing machine and take consideration of EPR in decision making [2]. Zarei et al. addressed the problem of network design for scrapped cars and he integrated distribution of new cars with collection network [3]. Distributors of new cars are also responsible for collection of scrapped cars. Li et al. conducted a location-inventory-vehicle routing model for sales return problem in e-commerce [4]。
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Lee et al. developed a Stackelberg model that manufacturers as leader and retailer as follower. His research indicated that the decision of retailers are making based on the frequency of the return of products. Products with short lifecycle will get higher profits[5]. In this paper, we adopt evolutionary game theory (EGT) for decision making in reverse logistics for manufacturers. The EGT is specialized in dealing with competition and cooperation of two participators. Dong and Song developed the evolutionary game model based on ESS and replicator dynamic for reverse supply chain [6]. However, in their research, they consider every supplier’s decision making process separately. The objective of this paper is to study the decision making of reverse logistics for manufacturers and the EGT has been adopted to describe the competition between manufacturers and the cooperation between manufacturers and collectors. The results of this paper provide suggestions for the manufacturers who are willing to carry out reverse logistics. II. COMPETITION MODEL A. Model description Some manufacturers (ma) will lead to carrying out reverse logistics as they discover the profit from recycling waste products, as well as take into account of the responsibility for the environment. Assuming that the manufacturers who collect waste products save costs by material recycling or components remanufacturing. What’s more, the implementation of reverse logistics helps the manufacturers promote corporate image and win market for their products. However, the waste products need to be preprocessed before recycling or remanufacturing by professional equipment. The manufacturers need a sum of fixed input to establish reverse logistics network and face uncertainty of reverse logistics. Therefore, some conservative manufacturers (mb) are not willing to develop reverse logistics. In the evolutionary game theory, manufacturers can change their strategy through learning and imitation. The proportion of manufacturers who develop reverse logistics changes according to replicator dynamic process. In order to describe the competition or cooperation between
manufacturers and collectors. We construct evolutionary game model. The notations are as bellow. i
index for manufacturers, i=1, 2, 3, …,l
k
index for products, k=1, 2, 3, …, n
dynamics system. The replicator dynamics can be described by a differential equation as below. = F（ p） = p(f s1 - f s ) p(t)
(9)
Where,
p
price of product k of manufacture i
f s1 = pΠ a1 + (1- p)Π a2
(10)
cik
unit production cost of product k of manufacture i
f s2 = pΠ a3 + (1 - p)Π a4
(11)
f s = pf s1 + (1- p) f s2
(12)
k i
( ) volume of sales of product k of manufacture i
D pik
Qik additional profit when manufacture i develop reverse logistics of product k
ξik
unit recycling price of product k of manufacture i
According to replicator dynamic system in the above, two theorems can be established. 1) Theorem 1 (i) For the replicator dynamic system (9) the points p* = 0 , p* = 1 are its equilibrium points.
( ) recycling volume of product k of manufacture i
D ξ ik
ζ ik preprocessing cost of product k of manufacture i ωik unit value of product k for manufacture i U i fixed input of manufacture i of reverse logistics implementation In this section, manufacturers have the same strategies {S1 ，S2}. Assuming that there are two manufacturers are selected randomly to play the game under bounded rationality. The pure strategies are: (1) S1: the manufacturer choose to develop reverse logistics; (2) S2: the manufacturer are not willing to recycle. Table 1 depicts the payoff matrix of the manufacturers which is symmetrical: Π a1 = Π b1 , Π a3 = Π b2 , Π b3 = Π a2 , Π a4 = Π b4 . TABLE I.
PAYOFF MATRIX mb
ma
S2
S1
Π a1 , Π b1
Π a2 , Π b2
S2
Π a3 , Π b3
Π a4 , Π b4
Πa2
k i
a3
k i
k i
k i
k i
k i
k i
k i
k i
k i
k i
k i
k i
( )( p
Π a4 = D pik
k i
k i
k i
- cik
k i
k i
)
k i
k i
i
k i
then
p* =
( )( p
2D pik
k i
)
(
)
( )(ω )
- cik + Qik pik - cik + D ξik
(p
k i
k i
2Q
k i
-c
k i
)
(13)
− ξik − ζ ik − U i
is an equilibrium point. Proof. F（ p） = p(f s1 - f s ) can be sorted out as below: ⎡2D
F ( p ) = p ( 1 - p ) ⎢⎢
( p )( p k i
-U i − 2 ⎣⎢
k i p Q ik
)
( )(ω
- c ik + D ξ ik
(
p ik
-
)
c ik
k i
)
(14)
− ξ ik − ζ ik ⎤⎥ ⎥ ⎦⎥
The stability condition of differential equations is to satisfy F ( p ) = 0 , x ∈ [0,1] , it is clear that p* = 0 , p* = 1 are two equilibrium points.
( )( p
( )( p - c ) + D (ξ )(ω − ξ − ζ ) − U = D( p )( p -c ) + Q ( p -c ) + D(ξ )(ω −ξ −ζ ) −U Π = D ( p )( p - c ) - Q ( p - c ) k i
and 2D ( pik )( pik - cik ) + D (ξik )( ωik − ξik − ζ ik ) -U i < Qik ( pik - cik )
A = 2 D pik
i
k i
)
(
)
( )(ω
- cik + Qik pik - cik + D ξik
(
B = 2Qik pik - cik
Where k i
(ii)if ⎡ 2D ( pik ) + Qik ⎤ ( pik - cik ) + D (ξik )(ωik − ξik − ζ ik ) > U i ⎣ ⎦
Set
S1
Π a1 = D pik
f s1 and f s2 represent expected fitness payoff of the manufacturers who recycling or not respectively. The average fitness payoff is f s .
(1) (2) (3) (4)
B. Replicator dynamic system and ESS The manufacturers play the game repeatedly in free market. Among the manufacture group, some accept the strategy S1 and the proportion is p. Others accept the strategy S2 and the proportions 1-p, p ∈ [0,1] . The proportion changes as a player will adjust her strategy according to replicator
k i
)
− ξik − ζ ik − U i (15)
)
(16)
and when 0 < A < B means that
( )
(
)
)
( )(
( )(ω
⎡ 2 D pik + Qik ⎤ pik - cik + D ξik ⎣ ⎦ (17) and
( )( p
2D pik
(18)
k i
k i
)
− ξik − ζ ik > U i
)
(
- cik + D ξik ωik − ξ ik − ζ ik -U i < Qik pik - cik
)
A is another equilibrium point of replicator B dynamic system (9). 2) Theorem 2 then p* =
(i) If ⎡ 2D ( pik ) + Qik ⎤ ( pik - cik ) + D (ξ ik )(ωik − ξ ik − ζ ik ) < U i , ⎣ ⎦
ξ1k ξ0k
p* = 0 is ESS.
unit recycling cost of product k of informal processors
(ii) If 2D ( pik )( pik - cik ) + D (ξik )(ωik − ξik − ζ ik ) -Qik ( pik - cik ) > U i ,
ξijk
p* = 1 is ESS.
( )
(
)
( )(ω
(iii)If ⎡ 2D pik + Qik ⎤ pik - cik + D ξ ik ⎣ ⎦
k i
−ξ −ζ k i
k i
) >U
i
,
and 2D ( pik )( pik - cik ) + D (ξik )(ωik − ξik − ζ ik ) -Qik ( pik - cik ) < U i , *
p =
( )( p
2D pik
k i
wholesale unit price of collected waste product k
)
(
)
( )( )
)
- cik + Qik pik - cik + D ξik ωik − ξ ik − ζ ik − U i
(
2Qik pik - cik
is ESS.
Proof. The ESS point require dF(p) < 0 ,
unit recycling price of product k from recycler j for
manufacture i ( ξ1k < ξijk < ξ0k )
B penalty cost for break the contract The pure strategies of recyclers(m) are: (1) S1: implement reverse logistics; (2) S2: not willing to recycle materials or remanufacture of product. The pure strategies of recyclers(r) are: (1) C: willing to cooperate with manufacturers and provide waste products; (2) U: not to supply waste products to the manufacture. The payoff matrix is showed in table 2.
dp
TABLE II.
dF(p) = 3Bp 2 - 2(A+ B)p + A . Theorem 1 can be checked by dp the requirement. When A < 0 . p* = 0 is an ESS. If B < A ,
p* = 1 is an ESS. When A > 0 and B > A , p* =
A is an ESS B
and then the two strategies are simultaneous. III. COOPERATAION MODEL
A. Model description In order to study how to promote the cooperation between manufacturers and collectors, we construct an evolutionary game model. As successful cooperation will provide insurance of resource supply and avoid waste collection market disruption. In China, most collectors are individual operators. Without market surveillance and order, amount of waste product flow to small unlicensed processing plants who may pollute the environment and raid the market. In the cooperation model, assuming that recyclers and manufacturers establish a relationship by way of recycling contracts. In the case of cooperation, manufacturers pay fixed costs to encourage recyclers, and recyclers transfer the collected waste products to manufacturers with the transfer price. In order to obtain maximum profits and prepare for the response of market uncertainty, they will develop a contingency plan to minimize the risk of reverse logistics of implementation. Whichever player betrayed the contract will cause a loss of the player, and then there must be a penalty for the player who breach the contract. According to the hypothesizes, the new parameters are described as below.
λ
C
U
S1
Π m1 , Π r1
Π m2 , Π r2
S2
Π m3 , Π r3
Π m4 , Π r4
Where
( )( p
Πm1 = D pik
k i
)
( )(
)
- cik + D ξ kj ωik − ξijk − ζ ik − Ui - Gij
( )( p
- cik -U i + B
( )( p
- cik - Gij - B
Π m2 = D pik
k i
Π m3 = D pik
k i
( )( p
Π m4 = D pik
k i
(15)
)
(16)
)
(17)
- cik
)
(18)
The fitness payoffs of recyclers are as bellows.
( )(ξ − λ − ξ ) + G Π = D (ξ )(ξ − λ − ξ ) - B = D (ξ )(ξ − λ − ξ ) +G + B Π = D ( ξ )(ξ − λ − ξ )
Π r1 = D ξ jk
k ij
k j
r2
k j
Π r3
r4
k j
k 0
k j
k j
ij
k j
k 1
k j
k j
k j
k 0
k j
ij
k j
(19) (20) (21) (22)
unit recycling price of product k of recycler j from unit recycling cost of product k of recycler j
The replicator dynamic equations of S1 took by manufacturers and C took by recyclers are as follows.
market k j
r
M
B. Replicator dynamic system and ESS The proportion of manufacturers take the strategy S1 is x , and 1- x is the proportion who take the strategy S2. Similarly y represents proportion of recyclers who are willing to cooperate, and the other part of recyclers who do not cooperate is 1 - y , thus (x, y ) ∈ [0,1] × [0,1] .
j index for collectors , j=1, 2, 3, …, m
ξ jk
PAYOFF MATRIX
( ) volume of recycled product k of recycler j
D ξ kj
Gij fixed incentives for manufacturers from collector j
= x(f m1 - f m ) x(t)
(23)
= y(f r1 - f r ) y(t)
(24)
Where: f m1 is expected fitness payoff of the manufacturers who implement reverse logistics. f m2 is expected fitness payoff of the manufacturers who are not willing to recycle or remanufacture. f m is average fitness payoff of manufacturers. f r1 is expected fitness payoff to manufacturers who select cooperation strategy. f r2 is expected fitness payoff who are not willing to cooperate. f r is average fitness payoff of recyclers.
, y * ) is a saddle point.
F ( x ) = 0 ， F ( y ) = 0 , (x, y ) ∈ [0,1] × [0,1]
f r1 = xΠ r1 + (1 − x)Π r3
(28)
f r2 = xΠ r2 + (1 − x)Π r4
(29)
f r = yf r1 + (1 − y ) f r2
(30)
k j
k i
−ξ −ζ k ij
) y -U + B ⎤⎦
(31)
)
(32)
k i
i
)
and ( )(ω − ξ − ζ ) * * + B < D (ξ )(ξ − ξ ) , ( x , y ) is an
0 < U i - B < D ξ jk
)
D ξ jk ξ0k − ξ ijk < Gij
k j
k i
k ij
k 0
k i
k 1
equilibrium point where:
( )(ξ − ξ ) − G D (ξ )(ξ − ξ ) k 0
k 1
k j
k ij
ij
-B
( )(ω
k i
− ξ ijk − ζ ik
and
)
( )(ξ
Gij + B < D ξ
k j
k 0
−ξ
k 1
)
D ξ jk ωik − ξijk − ζ ik -U i < D ξ jk ξ1k − ξ0k +Gij ,
the
equilibrium point (0, 0)is the ESS.
)
equilibrium point (0,1) is the ESS. and
( )(
)
The determinant (detJ) and the trace (trJ) are as follows.
( )( ( )(
( )(
( )(
)
)
D ξ jk ξ0k − ξ1k < Gij + B ,
equilibrium point (1,0) is the ESS.
)
det J = (1-2x)(1− 2y) ⎡D ξ kj ωik −ξijk −ζik yV ⎤ ⎣ ⎦ ⎡D ξjk ξijk −ξ1k x+D ξ jk ξ1k −ξ0k + Gij +B⎤ ⎣ ⎦ 2 2 -x2 (1-x) ⎡D ξ jk ⎤ ωik −ξijk −ζik ξijk −ξ1k ⎣ ⎦ k k ⎡ tr J = (1-2x ) D (ξ j )(ωi − ξijk − ζ ik ) y -Ui +B⎤ ⎣ ⎦ When + (1− 2 y ) ⎡ D (ξ jk )(ξijk − ξ1k ) x+D (ξ jk )(ξ1k − ξ0k ) +Gij +B⎤ ⎣ ⎦
( )(
) ( ) (
)(
)
)
Ui - B > 0 , Gij + B < D (ξ kj )(ξ0k − ξ1k ) , x = 0 and y = 0 , det J>0 and tr J D ξ jk ξ0k - ξ ijk - B , the equilibrium point (1,1) is the
(25)
x (t ) = F ( x ) = x (1 − x ) ⎡ D ξ ⎣
*
( )(
)
f m1 = yΠ m1 + (1 − y )Π m2
The replicator dynamic equation can be arranged as follows:
( )(
( )(
B > U i - D ξ jk ωik − ξijk − ζ ik
If
*
The above ESS is obtained under different parameters theoretically. In order to better understand the dynamic development of the system, the simulation experiments were conducted on the platform of MATLAB2013a in the next section. IV. SIMULATION AND DISCUSSION
the
The Fig. 1 shows manufacturers’ strategy’s convergence paths in different initial conditions. (The initial p=0.1, p=0.5 and p=0.8 respectively) Finally, the ESS p=1 is obtained. Under the initial parameters, the manufacturers compete with
each other, and they learn from the experience of a long run. Finally all the manufacturers will choose to implement reverse logistics. Though the fixed input for remanufacturing or recycling is high, the loss in the market in the long run will cause failure to the business in the long run. So, the manufacturers who are not willing to implement reverse logistics at first will convert their strategy finally. Fig. 1. The evolutionary process under different initial conditions
Fig. 2 and Fig. 3 show cooperation convergence paths of manufacturers and recyclers in reverse logistics. (The initial (0.2,0.8), (0.5,0.5), (0.8,0.2) respectively) As shown in Fig. 2 under the condition, manufacturers and recyclers are inclined to cooperate finally. The cooperation contract weakened the impact of fixed investment, what means that the mechanism of incentives and penalty can control the cooperation effectively. The proper contract will promote the cooperation between manufacturers and recyclers which is good for the development of circular economy. Fig. 2. The evolutionary process under different initial condition
V. CONCLUSIONS This paper addresses the problem of decision making on reverse logistics for manufacturers from an evolutionary game theory perspective. The results indicate that: (1) In the decision making process when the manufacturers’ objective is profit maximization, the fixed investment influence the decision most. The manufacturers who take responsibility and implement reverse logistics first will win the market. In a long run the loss of the market for those who are not willing to recycle or remanufacture will loss which will urge them to change the decision on reverse logistics. It is means that to implement the reverse logistics is a better choice in a long run in the market competition. (2) In the cooperation of manufacturers and recyclers the incentives and penalty in the contract are active constraint which will promote the success of cooperation persistently. From the conclusions, we can also confirm that consumers’ attention and preference are very helpful for the reverse logistics and resource recycling. Furthermore, the government can encourage to import or research advanced recycling technology. Policy support and consumer attention will promote the development of environment friendly enterprises. ACKNOWLEDGMENT This research is supported by the National Natural Science Foundation of China (71172168), the fundamental Research Founds for the Central Universities (FRF-SD-13-003B). REFERENCE
Fig. 3. The evolutionary process of the replicater dynamic system (initial x=0.8 y=0.2)
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