Information Sharing in a Multi-Echelon Inventory System

470
{ ( ) }
P x′ 1| x1, a1 x1
=
a1
⎧
⎪∑
P{ D= x1 + b− x′ 1} m( a1, b), ⎪b
= 0
⎨ a1
⎪
P{ D x1 + b} m( a1, b) ,
⎪∑
≥
⎩b
= 0
if x′
1 > 0;
if x′
1 = 0
(22)
With a given order-delivery rate matrix M, { x , a ( x ) }
1 1 1
is a DTMDP.
The retailer’s policy ϕ 1 = { a1(0) , a1(1) , , a1( U1)
} is
obtained by solving the DTMDP to minimize the ex-
ϕ1
ϕ1
pected long-run average cost. Denote Φ 1 = { π 1 (0) ,
ϕ1 ϕ1
π1 (1) , , π1
( U1)}
as the stationary distribution for policy
ϕ 1 . Furthermore, denote the order distribution of the
retailer for policy ϕ 1 by O1 = { o1(0) , o1(1) , , o1( U1)
} .
The order distribution can be calculated from
1( ) =
U1
k = 0
ϕ1 π1
( ) ⋅ 1 { ϕ1
1 ( ) = } , = 01 , , , 1
o j ∑ k a k j j U (23)
In the system, the retailer’s order distribution is
known to the supplier. With the retailer’s order distribution,
the one-step transition probability for the supplier
from state 0 x to state x′ 0 with action
a ( x ) ∈ A ( x ) is given by
0 0 0 0
U
1
+
{ ′ } ∑ { ′
}
P x | x , a ( x ) = 1 x = ( x + a ( x ) − z) ⋅o
( z)
0 0 0 0 0 0 0 0 1
z=
0
With a given retailer’s order distribution
(24)
1 , O
{ x0 a0( x0)
}
icy ϕ = { a (0) , a (1) , , a ( U ) }
, is a DTMDP. The supplier’s ordering pol-
and the stationary
0 0 0 0 0
ϕ0 Φ0 =
ϕ0 π0 ϕ0 , π0 ϕ0
, , π0
U0
state distribution { (0) (1) ( ) }
for this policy are obtained by solving the DTMDP.
The order-delivery rate matrix is then updated as
⎡ m′
(0, 0)
⎤
⎢
m′ (1, 0) m′
(1, 1)
⎥
M ′ = ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣m′ ( U1, 0) m′ ( U1, 1) m′ ( U1, U1)
⎦ (25)
where
m′ ( a , b)
=
1
U0
ϕ0 ϕ1
1{ b min a1 x0 a0 ( x0) } π 0( x0) o1 ( a1)
x0
0
∗
∑ = ⎡
⎣ , + ⎤
⎦ ⋅ ⋅ (26)
=
that satisfies
a1
∑
0 ≤m′ ( a , b) ≤ 1 , m′ ( a , b)
= 1.
1 1
b=
0
Tsinghua Science and Technology, August 2007, 12(4): 466-474
If the system reaches equilibrium, then for
all a1 = 01 ,, , U1,
b= 01 ,, , a1,
m′ ( a1, b) = m( a1, b)
,
i.e., M = M ′ . Therefore, an iterative algorithm can be
developed to search for the equilibrium order-delivery
rate matrix which will then give the equilibrium policy
ϕ , ϕ .
{ }
0 1
Algorithm 2
Step 1: Set initial values of M and ε .
Step 2: Solve the retailer’s DTMDP to obtain the op-
ϕ1
timal policy ϕ 1 , the stationary distribution Φ and
the order distribution O1 of the retailer.
Step 3: Solve the supplier’s DTMDP to obtain the
optimal policy ϕ 0 and the stationary distribution
ϕ0
Φ of the supplier.
Step 4: Update the order-delivery rate matrix M ′
according to Eq. (26).
Step 5: If ∆( M − M ′ ) < ε , stop; otherwise let M ′
be the new order-delivery rate matrix and return to
Step 2.
In Step 5, ∆( M − M′ ) =|| M − M ′ || , where || X ||
max x − min x .
is the norm with { ij} { ij}
i, j
i, j
3.3 Supplier-dominated information sharing
In this case, the supplier knows the retailer’s state at
each decision epoch, but the retailer only knows its
own state. Assume that the supplier also knows the
consumer demands to the retailer. The retailer’s state
space and action space are
S = 012 ,, , , U
(27)
{ }
1 1
{ }
A ( x ) = 01 , , 2,
, U −x
(28)
1 1 1 1
As mentioned in Section 3.2, the retailer policy can
be obtained from the order-delivery rate matrix if the
retailer only knows its own state. Hence, the solution is
similar to that in Section 3.2. However, since the supplier
knows the retailer’s state, the probability that the
retailer orders a1( x 1)
at state x 1 and receives b in
the next period depends on the retailer’s current state
x 1 . Therefore, the relationship between the order and
the delivery to the retailer can be described with a vec-
tor of matrices M = ⎡
⎣
M ⎤ x , x
1 ⎦ 1∈S1, where