Information Sharing in a Multi-Echelon Inventory System

468
ordering costs are charged at both the supplier and the
retailer, whereas fixed ordering costs only occur at the
highest echelon in Clark and Scarf’s model. The assumption
of fixed ordering costs is realistic, but results
in analysis difficulties. The assumption causes the
model to no longer possess structural properties for optimal
inventory policies.
The system was modeled as a discrete time Markov
decision process (DTMDP) to obtain the optimal inventory
policies. The state space of the centralized system
is
S = { ( x0, x1) | 0≤xi ≤ Ui, i = 01 , } (1)
The action space of the system at state ( x0, x1)
is
{ }
Ax ( 0, x1) = ( a0, a1)|0 ≤ai ≤ Ui − xi, i=
0,1 (2)
The one-step transition probability from state ( x0, x1)
to state ( x′ 0, x′
1)
with action ( a0, a1)
is
P{ ( x′ 0, x′ 1) | ( x0, x1) ; ( a0, a1)
} =
P{ ( x′ 0, x′ 1) | ( x0, x1)
} =
P{ x′ 0 | x0} ⋅ P{ x′ 1| x
1}
(3)
where x i ( i = 01) , is the state of stage i after the
shipment from the supplier to the retailer and before
the customer receives his order. Since x1≤x1≤ x1 + a1,
then
⎧1,
Px { ′ 0 | x
0}
=⎨
⎩0,
For the retailer,
if x′ 0 = x
0;
else
(4)
⎧P{
D = x1 − x′ 1} ,
⎪
P{ x′ 1| x1} = ⎨P{
D≥ x1} ,
⎪
⎩0,
if 0 < x′ 1≤x1; if x′
1 = 0;
if x′ 1 > x1
(5)
The transition probability only depends on the current
state for a given policy.
Denote function 1{} ⋅ as an indicator function. The
one-step cost is
c(( x′ 0, x′ 1) | ( x0, x1) ; ( a0, a1))
=
c0(( x′ 0, x′ 1) | ( x0, x1) ; ( a0, a1))
+
c1(( x′ 0, x′ 1) | ( x0, x1) ; ( a0, a1))
(6)
where
c0(( x′ 0, x′ 1) | ( x0, x1) ; ( a0, a1))
=
hx′ 0 0 + p0( a1 − x1 + x1)
+
qa + K ⋅ 1 a > 0
(7)
0 0 0 0
{ }
Tsinghua Science and Technology, August 2007, 12(4): 466-474
c1(( x′ 0, x′ 1) | ( x0, x1) ; ( a0, a1))
=
hx′ 1 1+ q1( x1− x1) + K1⋅1 x1− x1>
0 +
p max 0, d x 1 x 0
{ }
{ − } ⋅ { ′ = }
1 1 1
(8)
The long-run average cost of the system, given policy
ϕ = { a0( x0, x1) , a1( x0, x1) ∈ A( x0, x1) , ( x0, x1) ∈ S}
, is
then given by
C( ϕ) = ∑ π(
x0, x1) ⋅ EC(( x0, x1) ; ( a0, a1))
(9)
where ( x0 ( x0, x1) ∈S
x1)
( x0, x1)
for policy ϕ and x0 x1 a0 a1
π , is the stationary distribution of state
EC(( , ) ; ( , )) is the
expected one-step cost at state ( x0, x1)
.
EC(( x0, x1) ; ( a0, a1))
=
P ( x′ , x′ ) | ( x , x ) ; ( a , a ) i
∑
( x0′ , x1′ ) ∈S
{ }
0 1 0 1 0 1
c(( x′ , x′ ) | ( x , x ) ; ( a , a ))
0 1 0 1 0 1
(10)
Thus the optimal policy of the centralized system is the
policy that minimizes the cost in Eq. (10). According to
Putterman [11] , the policy can be obtained using a standard
value iteration algorithm or a policy iteration algorithm.
3 Decentralized System
In a decentralized system, the supplier and the retailer
make their decisions independently. The decision objective
of both stages is to minimize their own expected
long-run-average costs.
3.1 Full information sharing
If full information sharing is available in the system,
the supplier’s ordering policies and state are known to
the retailer, and vice versa. Therefore, the state spaces
of the supplier and the retailer are the same.
S = S = ( x , x ) | 0≤x ≤ U , i = 01 , (11)
{ i i }
{ ≤ ≤ }
0 1 0 1
A ( x , x ) = a ( x , x ) | 0 a ( x , x ) U −x
0 0 1 0 0 1 0 0 1 0 0
{ ≤ ≤ }
A ( x , x ) = a ( x , x ) | 0 a ( x , x ) U −x
1 0 1 1 0 1 1 0 1 1 1
(12)
(13)
Assume that the supplier adopts a stationary policy
ϕ 0 = { a0( x0, x1) ∈ A0( x0, x1) , ( x0, x1) ∈ S0}
, which is
known to the retailer. At state ( x0, x1)
, the retailer orders
a1( x0, x1)
and the retailer knows that the supplier
ϕ0
will order a0 ( x0, x1)
. Therefore, the one-step transition
probability of the retailer from state ( x0, x1)
to
( x′ x′
)
a ( x , x ) is determined by
state 0, 1 with action 1 0 1