Customer Information Driven After Sales Service ... - RePub

5.4. Markov Decision Process Formulation
and return decisions. Although the sequential approach is computationally less intensive;
one may argue that the one-directional consideration of the sourcing and return decisions
may lead to suboptimal decisions at times. In other words,
Proposition 5.1
V in
t (�x) ≥ V sq
t (�x)
Proof We show that the inequality holds by induction. Assume that the inequality
V in
t+1(�x) ≥ V sq
t+1 (�x) ∀ �x ∈ Z+
M holds. Now we show that it holds for t. Weknow,
V in
t (�x) = � �
λj max (1 − qj){gj − f
i∈M,l∈L
j∈N
d ij − f p in
ij + Vt+1(�x − �ei)}
� in
+ qj Vt+1(�x − �ei + �el) − f d ij − f r �
lj
�
≥ � �
λj max (1 − qj){gj − f
i∈M,l∈L
j∈N
d ij − f p sq
ij + Vt+1 (�x − �ei)}
� sq
+ qj Vt+1 (�x − �ei + �el) − f d ij − f r �
lj
�
≥ � �
�
λj (1 − qj)max gj − f
i∈M
j∈N
d ij − f p sq �
ij + Vt+1 (�x − �ei)
+ qj max
l∈L|i∗ � sq
Vt+1 (�x − �ei + �el) − f d ij − f r �
lj
�
=V sq
t (�x). �
Proposition 5.1 implies that bi-directional integration would lead to a higher expected
value than the sequential formulation. In other words, we may infer that the decisions
made while accounting for the bi-directional interaction between sourcing and returns
should lead to a higher value. To analyze the extent of losses incurred by sequentially accommodating
the interaction between sourcing and return decisions, we perform detailed
numerical experiments.
5.4.1 Integrated vs.
ments
Sequential Decisions - Numerical Experi-
We first describe the simulation test bed used to numerically analyze the integrated and
sequential formulations. We consider a logistics network consisting of three customers,
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