Information Sharing in a Multi-Echelon Inventory System

472
{ | , ( ) }
Tsinghua Science and Technology, August 2007, 12(4): 466-474
P x′ 0 x0 a0 x0
=
ϕ0 ϕ0
π0 (1) , , π0
( U0)}
for this policy are obtained by
⎧ o0( x0, x0 + a0( x0) − x′ 0), ⎪ U1
⎪
⎨ ∑ o0( x0, b), ⎪b=
x0+ a0( x0)
⎪
⎩ 0,
if x0 + a0( x0) ≥ x′
0 > 0;
if x′
0 = 0;
if x′ 0 > x0 + a0( x0)
solving the DTMDP.
The retailer’s state space and action space are
S1 = { ( x0, x1) | 0≤xi ≤ Ui, i = 01 , } (38)
A1( x0, x1) = { 0,1, , U1 − x1}
(39)
(37) For the RDIS scenario, the retailer knows the sup-
Therefore, with a given retailer’s order distribution
matrix O 1 , { x0, a0( x0)
} is a DTMDP. Analogously,
the ordering policy ϕ 0 = { a0(0) , a0(1) , , a0( U0)
} and
plier’s ordering policy ϕ 0 and stationary state distri-
ϕ0
bution Φ 0 . Given policy ϕ 0 , the retailer’s one-step
transition probability from state ( x0, x1)
to ( x′ 0, x′
1)
the state stationary distribution
ϕ0 Φ0 ϕ0
= { π0
(0) , with action a1( x0, x1)
is
P{ ( x′ 0, x′ 1) | ( x0, x1) , a1( x0, x1)
} =
ϕ0
+
⎧ 1 { x′ 0 = ( x0 + a0 ( x0) − a1( x0, x1)) } ⋅ P{ D = x1 + m− x′ 1} ,
⎪
ϕ0
+
⎨1
{ x′ 0 = ( x0 + a0 ( x0) − a1( x0, x1)) } ⋅ P{ D≥x1 + m} ,
⎪
⎪⎩
0,
if 0 < x′ 1≤x1 + m;
if x′
1 = 0;
if x′ 1 > x1 + m
(40)
ϕ0
where m min { a1( x0 x1) x0 a0 ( x0)
}
= , , + .
ϕ 0 1
For a given supplier’s ordering policy 0 , {( x , x ) ,
a1( x0, x 1)}
is a DTMDP. The DTMDP solution gives
the ordering policy ϕ 1 = { a1( x0, x1) ∈ A1( x0, x1) , ( x0, x1)
∈
S and the stationary state distribution ϕ1
Φ =
1 }
ϕ1
{ π1 ( x0, x1),( x0, x1) ∈ S1}
of the retailer. If { ϕ0, ϕ 1}
is
an equilibrium point policy, then the definition of the
ordering distribution matrix, O 1 , leads to
o (, i j)
=
1
U1
∑
b=
0
ϕ1 ϕ1
1 1
U1
b=
0
ϕ1
1
{ }
π ( ib , ) ⋅ 1 a ( ib , ) = j
∑
π ( ib , )
for all i∈S0, j∈ A1
(41)
Therefore, the iterative algorithm to search for the
ordering distribution matrix O 1 is as follows.
Algorithm 4
Step 1: Set initial values of O 1 andε .
Step 2: Solve the supplier’s DTMDP to obtain the
supplier’s ordering policy ϕ 0 .
Step 3: Solve the retailer’s DTMDP to obtain the
retailer’s ordering policy ϕ 1 and the stationary
ϕ1
distribution Φ 1 .
,
1
Step 4: Update the ordering distribution matrix 1 ′
O
using Eq. (41).
Step 5: If ∆( O1 − O ′ 1)
< ε , stop; otherwise let O ′ 1
be the new ordering distribution matrix and return to
Step 2, where ∆( O1 − O′ 1) =|| O ′ 1 − O 1||
.
4 Numerical Comparisons and
Analyses
The algorithms for the single supplier-single retailer
supply chains with various information sharing scenarios
were developed to determine their inventory policies.
The different information sharing scenarios will
result in different total expected long-run-average costs
of the supply chains. The algorithms were studied numerically
to show the valuation of the total cost with
respect to the information sharing scenarios.
The analyses calculated the equilibrium points using
randomly generated groups of system parameters. The
demand process at the retailer was assumed to follow a
Poisson process with parameter λ . The parameters
were generated within the ranges listed in Table 1.
Table 1 Parameters’ range
h0 h1 p0 p1 K0 K1 λ
[0.1, 3.0] [h 0, h 0+3.0] [h 0,15h 0] [h 1,15h 1] [0, 60h 0] [0, 20h 1] [0.5, 6.5]