Journal of Computers - Academy Publisher

JOURNAL OF COMPUTERS, VOL. 6, NO. 9, SEPTEMBER 2011 1887
activities are managed by the retailer. This paper considers
an asymmetric information about the cost of returned
goods.
The paper proceeds as follows. The next section
presents the assumptions and notations. In Section 3, the
integrated model is discussed firstly. In Section 4, the
return policy under symmetric information situation is
investigated. Section 5 focuses on the return policy for an
asymmetric information relationship. Section 6 gives the
numerical analysis. Section 7 summarizes the findings.
II. ASSUMPTIONS AND NOTATIONS
The demand D is a random within [0, b ] . We denote
by f, F , µ the density function, distribution function
y
of D , respectively. Let E( y) = ∫ xf( x) dx . The retail
−∞
price p and the supplier cost c are exogenous variable
and the wholesale price of the supplier w is endogenous
variable. The salvage values of the supplier and the
retailer are different and denoted by m v and v r ,
respectively. In this paper, we assume the retailer takes
back work and pays the logistic cost, denoted by l .
If vr ≥vm − l , from the supply chain point of view,
returning goods is unreasonable. This paper
assumes vr < vm −l and considers an asymmetric
information about the cost of reverse logistics. We
assume the real value of l is the retailer’s private
knowledge and we call this retailer l -type retailer for
convenience in presentation. The supplier does not make
sure the type of the retailer, but he deems the value of l is
either l with probability of ρ or l with probability
of1− ρ . The buyback contract is a practical method for
the supplier to share risks and losses of the retailer. We
denote r as the buyback price, which is the decision
variable of the supplier as well as w . In asymmetric
information situation, the supplier should offer retailers a
menu of returns policies trading off l -type retailer
with l -type retailer. The one goal of the supplier’s
contract is to coordinate the supply chain and the other is
to maximize the supplier profit.
Let l -type retailer’s ordering size is Ql () , the
expected surplus and sale are Ol () and Sl () . Simply
calculating gives
Ql ()
Ol () = ∫ F( xdx ) , Sl () = Ql () −Ol
() (1)
0
The total expected profit of the channel is
∏ m+ r() l = ( p−c) Q() l −( p− vm+ l) O() l (2)
The profit of l -type retailer is
∏ r () l = ( p−w) Q() l −( p− r+ l) O() l
(3)
The profit of the supplier is
∏ () l = ( w−c) Q() l −( r− v ) O() l
(4)
m m
III. THE INTEGRATED MODEL
The goal of this paper is to develop a return policy to
coordinate the supply chain. The coordination of supply
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chain means the decision in decentralized enable the
channel to obtain the same profits as a vertically
integrated firm’s. In order to give a benchmark for follows
discussion, in this section, we first focus on an integrated
structure in which both the supplier and the retailer agree
to take decisions to maximize the total channel profits
(joint profit maximization).
We denote the optimal order size and the maximum
expected profit of the channel by * *
Q () l , m r() l ∏ + . Using
Leibniz’s rule to obtain the first and second derivatives
shows that m r() l ∏ + is concave. The sufficient optimality
condition is the well-known formula:
*
F( Q ( l)) = ( p− c)/( p+ l− vm).
(5)
Using the relationship
Q
∞
xf( x) dx = µ − xf( x) dx
∫ ∫
0
and substituting from (5) into (2) and simplifying gives
the optimal expected profit:
* *
∏ m+ r() l = ( p+ l− vm) E( Q ()). l
(6)
* *
Proposition 1. Q () l and m r() l ∏ + decrease as l increases
*
Proof. From (5), we have ∂Q ()/ l ∂ l < 0.
Taking the first-
*
order derivative of m r() l ∏ + , one has
* * * *
∂ ∏m+ r()/ l ∂ l = E( Q ()) l − Q () l F( Q ()) l < 0.
The higher l means the higher the cost, thus this
conclusion is intuitional.
For the convenience in presentation is follows
subsections, let
*
Q () l
*
O () l = ∫ F( x) dx.
(7)
0
IV. THE RETURN POLICY UNDER SYMMETRIC
INFORMATION SITUATION
For the sake of comparing, before investigate the
asymmetric information situation, now we discuss the
problem of channel coordination by return policy with
common knowledge about l . When the supplier know
the retailer’s cost l , the supplier first declares the
wholesale price w and buyback price r . The retailer, as
s
the follower sets the decision of ordering size Q () l . It is
straightforward to find that only if r > vm+ l , then the
retailer sends back the excess goods.
Using the same method gives
s
F( Q ( l)) = ( p− w)/( p+ l− r).
(8)
s
l -type retailer’s expected profit, denoted by r () l ∏ , is
s s
∏ r ( l) = ( p+ l− r) E( Q ( l))
(9)
From (5) and (8), we get the observation as in Proposition
2.
Proposition 2. If wrsatisfy ,
w= βc+ (1 − β) p, r = (1 − β)( p+ l) + βvm
, (10)
where l /( l + c −vm) ≤β≤1 the combined contract of wholesale price and buyback
policy can coordinate the supply chain and has the
follows properties:
Q