Customer Information Driven After Sales Service ... - RePub

5.3. Problem Formulation
and FSLs) is denoted by M (finite set of nonnegative integers) and indexed by i, where
i = 0 is fixed for the supplier location and i ∈ M \{0} for all FSL locations. We define
the set of candidate return stock locations L (finite set of nonnegative integers) with
each candidate stock location indexed as l. Similar to the source locations, l =0isfixed
for the supplier location and l ∈ L \{0} for all other candidate return locations. The
customers are dispersed geographically at various locations. Let j denote a customer in
the customer location set N (finite set of positive integers). The inventory at each FSL
is denoted by xi. The inventory at all FSL in the network can be denoted by a vector as
follows �x = {x1,x2, ..., xi}.
5.3.1 Demand Characteristics
We assume that demand requests from various customers follow an independent Poisson
process. The Poisson demand assumption is common in spare parts logistics. A demand
request originates due to a machine failure. It is plausible to assume that machine failures
at various customer locations occur independently and each machine failure requires a
single spare part unit. Note that the spare part may represent a carefully composed kit
of spare parts for satisfying (generic) demand requests. We denote the demand rate from
each customer j at each t as λj. We denote the probability that a demand request is
false by qj. Furthermore, there is a probability that no demand arrives during any time
period t which is denoted by λ0.
5.3.2 Cost Structures
A customer is charged a service price gj for the performed maintenance according to
the agreed contract. The transportation costs from each stock location i (either FSL or
supplier location) to each customer location j are known and denoted as f d ij (superscript
d identifies the forward transportation aspect). The transportation costs are linearly
increasing in distance with a fixed cost component. The return transportation costs
from customer j to return stock location l are denoted as f r lj (superscript r identifies the
return transportation aspect). Note that even if l = i, thenfr lj and f d ij may or may not
be equal. If the demand realization is served later than its service deadline, then penalty
costs are also incurred. The penalty cost function is zero for demand fulfillment within
the service deadline and exponentially increasing in the elapsed time after the service
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