Download Maximize the Expected Profit in Closed-loop Supply Chain System

Maximize the Expected Profit in Closed-loop Supply Chain System
DING Hua, ZHENG Biao
School of Economics and Trade, Henan University of Technology, P.R.China, 450052
[email protected]
Abstract: In this paper assuming price-sensitive stochastic demands and effort-sensitive stochastic
returns, a multi-echelon closed-loop supply chain with 3rd party reverse logistics is established. A target
rebate-punish contract between the manufacturer and the 3rd party reverse logistics is designed. The
second point is the closed-form analytic expressions for both united optimization strategies in
centralized closed-loop chain system and the one-leader-multiple-follower Stackelberg strategies in
decentralized system are deduced.
Keywords: Closed-loop, Supply chain, 3rd party reverse logistics, Stackelberg strategies
1 Introduction
Closed-loop supply chain, particularly those based on the third-party reverse logistics system for
material recycling, resource conservation and environmental protection is of momentous current
significance. "The eleventh Five-Year Plan" pointed out that we should vigorously accelerate the
development of recycling economy, focusing on research and development of new generation of
recycling manufacturing processes that includes manufacturing, energy conversion, and social waste
recycling. The 3rd party reverse logistics based on the material recovery mode helps to reduce
procurement costs and risks, improve efficiency of resource use in order to achieve the building of a
resource-saving and environment-friendly society. Research based on the third-party reverse logistics
optimization of closed-loop supply chain model has important theoretical and practical significance.
In recent years, closed-loop supply chain management in recycling and re-manufacture has become the
research hotspot at home and abroad. A number of scholars from different angles investigated some
aspects of the closed-loop supply chain, such as the structural design, inventory control, logistics, capital
flow and information flow management issues etc. On the closed-loop supply chain coordination, most
researches make use of microeconomics and game theory methods to build and inspect the
decision-making in reverse logistics, mainly in four aspects—the competition, information sharing, the
functions of integration and stimulation. Representative study relatively is described below. Savaskan
and Vanassenhove, Groenevelt and Majumder respectively investigated problems that the incentive and
coordination effect of competition and market structure re-create to the closed-loop supply chain. The
former model is built on the re-manufacturing retailers pricing competitive, while the latter concerned
about the first three parties re-manufacturer, and device manufacturers conflict. Through massive
literature summary of reverse supply chain, Prahinski proposed ten propositions obtained through the
empirical study method.
2 Model Descriptions
2.1 Model frame
This passage is considering the multi-echelon closed-loop supply chain which is composed by the
manufacturer and the vendor together with 3rd party reverse logistics, the framework of it suggest as
figure1.In the traditional forward supply chain, the manufacturer organizes the production according to
the order form of the vendor, and wholesales the production to the vendor, then sells to the consumers
by the vendor. In the reverse supply chain the manufacturer recycles the materiel from the third party
reverse logistics provider in order to reproduce. Otherwise, in order to encourage the third party
engaging recycling, the manufacturer firstly subscribes a contract. That is, if the quantity which the
third party recycled exceed lack the quantity which the contract prescribed, the manufacturer will
，
（ ）
691
（
）
reward or punish according the recycling quantity. In the closed-loop supply chain system, assume
that the manufacturer, vendor and third party are risk-neutral and absolutely rational, that is they make
decisions in accordance with the principle of profit maximization.
Wholesale
Manufacturer
Vendor
Customer base
Recycle
Rewards and punishment contract
τ 1 ( R − T ) + −τ 2 (T − R) +
Rate of recycle
3rd party logistics
Reverse supply chain
Forward supply chain
2.2 Variable definition and symbol description
In order to facilitate description, it is necessary to introduce a variety of variables and functions
description.
p—unit retail price of products, which is vendor’s decision variable, Q—quantity of orders, which is
also vendor’s decision variable. w—the unit wholesale price of products, which is manufacturer’s
decision variable. t—manufacturer’s recycling price for units of waste products which paid to third
party ,which is also manufacturer’s decision variable.
e—the unit effort of third party’s collection, which is third party’s decision variable, T—the quantity of
incentive contract, which is third part’s decision variable. cn—the unit cost of manufacturer’s production.
co—the unit cost of manufacturer’s recycling scrap manufacturing. V—the unit salvages value of unsold
products. r—the third party’s unit cost of recycling and treatment. τ 1 τ 2 —the unit awards of incentive
τ1 ＞ τ 2 . α
contract/ penalty factor , suppose
β —remanufacturing rate of recycled products,
／
—the rate of product recovery (0≤
α
＜ 1).
（0≤ β ＜1）.D (p)—the random demand which depend
on retail price, R (e)—the random quantity of recovery which depend on recovery efforts, π (*)—profit
function. The subscript r, m, t and s are respectively vendors, manufacturers, 3rd party and closed-loop
supply chain system variables and parameters related. Superscript S is measures solution of
decentralized systems of Stackelberg, I is centralized system of joint optimal solution. In order to ensure
the internal consistency of the supply chain, here suppose co ≤cn; β R (e)
D(p).
This paper applies the price of reliance on random demand which is similar to Emmons. The actual
demand Can be expressed as X (p, ξ ) = ξ D (p) (D (p) is a deterministic function, and negatively
≤
correlated with the retail price p).
ξ
is the random variable which is independent of p. Density
function and distribution function respectively are f( ξ ) and F( ξ ).Demand function has the form of
multiplication and addition, but it does not affect the conclusions . The density function of stochastic
 x 
1
f
 ( x ≥ 0) .Distribution function can be
D( p )  D( p ) 
 x 
∂F ( xp )
expressed as Fx ( xp ) = F 
> 0 . Similarly, suppose the expectation of the
 and
∂p
 D( p) 
demand can be expressed as f x ( xp) =
692
amount recovered is R (e), Then the actual recovery of the amount can be expressed as Y (e, ξ ) = ξ R (e)
and the mean is µY =
µξ R ( e ) . Which ξ
is independent of the random variables e. Density function
and distribution function respectively are g( ξ ) and G( ξ ).The density function of random quantity of
recovery can be expressed as gY ( ye) =
 y 
1
g
 ( y ≥ 0) . Distribution function can be
R(e)  R(e) 
 y 
∂GY ( ye)
<0.
 and
∂e
 R (e ) 
expressed as GY ( ye) = G 
In order to describe conveniently, it is necessary to have a brief description of the sub-function.
The vendor’s expected sales function: S ( Q , p ) , and then
S (Q, p ) = E [ min( x, Q )]
= ∫ Q0 xf ( xp )dx + ∫ Q∞ Qf ( xp )dx
= ∫ Q0 xf ( xp )dx + Q − ∫ Q0 Qf ( xp)dx 
Q
= ∫ Q0 xf ( xp )dx + Q −  ∫ Q0 F ( xp )dx + ∫ xdF ( xp ) 
0


Q
= Q − ∫ F ( xp )dx
0
(1)
The manufacturer's total production cost function: Cm (Q, y ) . Assume that the manufacturers make
the products remanufacturing combined with the original production process and try their best to use the
recycling product to remanufacture for minimizing production costs. Then
Cm (Q, y ) = cn (Q − β y )+ + co min(Q, β y )
(2)
Third party total recycling cost function: Cr ( y, e) , and then
Cr ( y, e) = eα 2 + ry (e, ε )
(3)
Rewards and punishment function: P ( y, Τ) in the target rebate-punish contract between the
manufacturer and the third party, and then,
P ( y, Τ) = τ 1 ( y − T )+ − τ 2 (T − y )+
(4)
The third party’s efforts of cost function: I (e) (when the third party’s collection effort is e), and then
I (e ) =
ae2
2
( a > 0) (5) ‘a’ is represents the third-party’s cost of collection effort coefficient
3 Closed-loop supply chain model
3.1 The united optimal strategies in centralized closed-loop chain system
In closed-loop supply chain system as the Figure 1 showing, we can get the expected profit function of
the manufacturer, seller and third parties. As formula (6) (8) shown.
～
Ε [π m (w, t; Q, T , e)] = E [ wQ − Cm (Q, y ) − ty (e, ε ) − P ( y, T )]
E [π r (Q, p; w)] = E  pS (Q, p ) − wQ + v (Q − x)+ 
693
(7)
(6)
E [π t (T , e; t )] = E [ty (e, ε ) − Cr ( y, e) + P ( y, T ) − I (e)]
(8)
Because the centralized closed-loop chain system is composed by the manufacturer and the vendors
together with the third party, and then the wholesale price (w) and recovery price (t) are internal
parameters that do not affect the overall decision of centralized system. Join formula 6 and 7 we can
get the expected profit of the centralized closed-loop chain system.
（ ） （ ），
E [π s (Q, p, e)] = pS (Q, p) + v (Q − x) + − Cm (Q, y ) − Cr ( y, e) − I (e)
= ( p − co )Q − ( p − v ) D ( p) ∫
Q
D( p)
0
(cn − co ) β R(e) ∫
Q
0
β R(e)
F ( x )dx −
G ( y )dy − eα 2 − r µY − I (e)
(9)
As the decision maker of the centralized system, in order to maximize the expected profit (Formula 9) in
closed-loop supply chain, the manufacturer chose the optimal ordering quantity, retail price and
collection effort.
，
In order to analyze conveniently Suppose
ξ
−
obey uniform distribution, interval [0 − ξ ] . Then we
−1
get f (ξ) = 1 ξ , F (ξ) = ξ ξ , F (ξ) = ξξ . The demand and the price functional relations are
D ( p) = k p ( p − η p ) ( k p is price elasticity of demand, k p < 0,η p > w ≥ v ). Similarly, suppose
ε
−
_
obey uniform distribution, interval [0 − ε ] .Then we get uε = ε 2 , g (ε ) = 1
_
_
ε , G (ε ) = ε ε ,
_
G −1 (ε ) = ε ε . The quantity of recovery and the collection effort functional relations are
R(e) = ke (e − ηe )(ke > 0,η e > 0)
3.2 One-leader-multiple-follower Stackelberg strategies in decentralized system
The rational decision-makers would like to pursue their own benefits in a multi-echelon closed-loop
supply chain system composed of manufacturers, distributors and the third party without considering the
benefits of the whole supply chain system. As shown in Figure 1, in the framework of closed-loop
supply chain system, the manufactures are the main party and the vendors and the third-party are the
side parties because manufacturers are resources owners, and at the same time they have respectively
signed whole sales contracts and objective incentive contract with the vendors and 3rd parties. The
vendor and the third party take the non-cooperative Nash measures. The vendor and the third party make
the optimal response strategy according to the manufacturer's optimal decision respectively. Concrete
model is as follows:
 ( L ) : max E [π m ( w, t ; Q , t , e ) ] = E [ wQ − C m ( Q , y ) − ty ( e , ε ) − P ( y , T ) ]
w ,t


s .t . Q = arg max E [π r ( Q , p ; w ) ]
Q,p


Τ = arg max E [π t (T , e ; t ) ]
Τ, e


e = arg max Ε [π t (T , e ; t ) ]

T ,e

( F1) : max Ε [π r ( Q , p ; w ) ] = Ε  pS ( Q , p ) − wQ + v ( Q − x ) + 

Q,p


( F 2) : max E [π t (T , e ; t ) ] = E [ty ( e , ε ) − C r ( y , e ) + P ( y , T ) − I ( e ) ]
T ,e

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(10)
(11)
(12)
(13)
(14)
(15)
4 Conclusion
This article proposed propositions in order to maximize the expected profit in closed-loop supply chain
system which is composed by the manufacturer and the vendor together with the third party. Participants
chose their optimal strategies to achieve the maximum profit. Because Closed-loop supply chain system
has a high degree of complexity and uncertainty, this paper only conducted a preliminary exploration
from the perspective of supply chain systems. How to analyze the uncertainty of supply, manufacture,
distribution, re-manufacturing and re-distribution in Closed-loop supply chain system is the study of
future.
References
[1]. Nunen J A E E V, Zuidwijk R O.E-enable closed-loop supply chains [J]. California Management
Review, 2004, 46(2)40-54
[2]. Prahinskia C, Kocabasoglub C.Empirical research opportunities in reverse supply chains[J]Omega,
2006, 34(6):519-532
[3]. Dobos I. Optimal production-inventory strategies for a HMMS-type reverse logistics systems [J].
International Journal of Production Economics, 2003, 81-82(1):351-360
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