Customer Information Driven After Sales Service ... - RePub

5.4. Markov Decision Process Formulation
the probability of a false demand signal, the spare part logistics need to decide the
appropriate stock location for spare part return. Note that the spare part return decision
depends on the earlier spare part sourcing decision. The costs of a false demand signal
are f d ij + f r lj ,whereirepresents the source stock location and l represents the candidate
return stock location. Suppose that we decided to source the spare part from stock
location i and a false demand occurred, then we need to maximize the following function
to identify the candidate stock location for spare part return. Similar to the previous
section, �el is unit vector notation with inconsideration l equals one and others as zero.
�
max Vt+1(�x − �ei + �el) − f
l∈L
d ij − f r �
lj
5.4 Markov Decision Process Formulation
Figure 5.1 highlights the interaction between sourcing and return decisions. In this
section, we present two MDP formulations to accommodate this interaction. In the first
formulation, we account for the probability of false demand and consequent spare part
return while deciding for the forward sourcing decision. Similarly, for the return decision,
we account for the preceding sourcing decision from the current time instance t. Inthe
subsequent text, we term this as a bi-directional interaction of the forward sourcing and
return decision. To formalize this interaction, we present the following two stage MDP
formulation. The value of stock �x at time t equals the expected revenue given the optimal
use of the available resources.
V in
t (�x) = � �
λj max (1 − qj){gj − f
i∈M,l∈L
j∈N
d ij − f p
ij
+ V in
t+1(�x − �ei)} + qj{V in
t+1(�x − �ei + �el) − f d ij − f r �
lj}
Observe that we may write,
V in
t (�x) = � �
λj max (1 − qj){gj − f
i∈M
j∈N
d ij − f p in
ij + Vt+1(�x − �ei)}
� in
Vt+1(�x − �ei + �el) − f d ij − f r �
lj
�
+ qj max
l∈L
137
(5.1)