Information Sharing in a Multi-Echelon Inventory System

ZHAO Xiaobo (赵晓波) et al:Information Sharing in a Multi-Echelon… 469
{ ( ′ ′ ) ( ) ( ) }
P x0, x1 | x0, x1 , a1 x0, x1
=
ϕ0
+
⎧ 1 { x′ 0 = ( x0 + a0 ( x0) − a1( x0, x1)) } ⋅ P{ D = x1 + m1 − x′ 1} ,
⎪
ϕ0
+
⎨1
{ x′ 0 = ( x0 + a0 ( x0) − a1( x0, x1)) } ⋅ P{ D≥x1 + m1} ,
⎪
⎪⎩
0, if 0 < x′ 1≤x1 + m1;
if x′
1 = 0;
if x′ 1 > x1 + m1
(14)
m = min a ( x , x ) , x
ϕ0
+ a ( x ) . The transi- retailer’s policy ϕ 1 = { a1( x0, x1) ∈ A1( x0, x1) , ( x0, x1)
∈
where 1 { 1 0 1 0 0 0 }
tion probability of the retailer only depends on the current
state and action.
From the definition of the DTMDP, with a given policy
ϕ 0 of the supplier, { ( x0, x1) , a1( x0, x1)
} is a DTMDP.
The one-step transition cost can be calculated as discussed
in Section 2. The DTMDP solution gives the
P ( x′ , x′ ) | ( x , x ) , a ( x , x ) =
{ }
0 1 0 1 0 0 1
{ ϕ1
+ } { }
{ ϕ1
+ } { ≥ }
1 } S . Since full information sharing is available, this
policy ϕ 1 is also known to the supplier. The one-step
transition probability of the supplier from state
( x0, x1)
to state ( x′ 0, x′
1)
with action a0( x0, x1)
is
given by
⎧ 1 x′ 0 = ( x0 + a0( x0) − a1 ( x0, x1)) ⋅ P D = x1 + m0 − x′ 1 , if 0 < x′ 1≤x1 + m0;
⎪
⎨1
x′ 0 = ( x0 + a0( x0) − a1 ( x0, x1)) ⋅ P D x1 + m0 , if x′
1 = 0;
⎪
⎪⎩
0, if x′ 1 > x1 + m0
With a given policy ϕ 1 of the retailer,
{ ( x x ) a ( x x ) }
0, 1 , 0 0, 1 is also a DTMDP, with the solution
giving a supplier’s policy ϕ′ 0 . If { ϕ0, ϕ1}
is the
equilibrium system policy, then
ϕ ϕ′ = (16)
0
An iterative algorithm can be developed to search
for the equilibrium policy { ϕ0, ϕ1}
. The algorithm is
as follows.
Algorithm 1
Step 1: Set an initial supplier’s policy ϕ 0 .
Step 2: Solve the DTMDP for the retailer given ϕ 0
to get the retailer’s policy ϕ 1 .
Step 3: Solve the DTMDP for the supplier given ϕ 1
to get the supplier’s policy ϕ′ 0 .
Step 4: If ϕ0 = ϕ′ 0 , stop; otherwise let ϕ′ 0 be the
new supplier’s policy and go to Step 2.
3.2 Non-information sharing
This model has no information sharing in the system
due to technical reasons or business confidentiality.
The supplier cannot know the states and demands of
the retailer, and the retailer cannot know the states and
orders of the supplier. Therefore, both the supplier and
0
(15)
the retailer can only observe their own inventory states.
The state spaces and the action spaces are
S = 012 , , , , U
(17)
{ }
0 0
{ }
A ( x ) = 01 , , 2,
, U −x
(18)
0 0 0 0
{ 012 }
S = ,, , , U
(19)
1 1
{ }
A1( x1) = 0,1,2, , U1 −x1
(20)
If the retailer orders a1( x 1)
from the supplier, he
may receive less than a1( x 1)
from the supplier. An
order-delivery rate matrix, M, is used to describe the
relationship between the order and the delivery to the
retailer,
⎡ m(0,
0)
⎤
⎢
m(1,0) m(1,1)
⎥
M = ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣mU ( 1, 0) mU ( 1,1) mU ( 1, U1)
⎦ (21)
where 0 ≤mab ( , ) ≤ 1 is the probability that the delivery
to the retailer is b if he orders a , and
a
( , ) 1
b 0
mab ∑ = , i.e., the summation of any row in the
=
matrix is equal to 1.
With matrix M, the one-step transition probability
for the retailer from state x 1 to state x′ 1 with action
a ( x ) ∈ A( x ) is given by
1 1 1 1