Customer Information Driven After Sales Service ... - RePub

4.4. Structural Analysis and Heuristic Solution
values. Note that the MDP formulation could be viewed as an online version of the knapsack
problem. It is due to this reason that many approximations to efficiently compute
marginal values attempt to utilize the associated knapsack problem (see Section 4.2.2).
Similar to the RM literature, the primary idea is to utilize the expectation of demand
to devise the knapsack formulation. In comparison to the single choice knapsack formulations
utilized in RM literature, the horizontal nature of a spare parts logistics network
yields a multiple choice knapsack formulation. The devised formulation enables an approximation
of Vt(�x). We define a decision variable Uij that represents the stock units
reserved for customer j at stock location i. The reserved stock units for all customers at
a specific FSL i should not exceed the total capacity of the FSL, which is represented by
following supply constraint:
�
Uij ≤ xi ∀ i ∈ M (Supply Constraint)
j∈N
In addition, the reserved stock units for any customer j should not exceed the total
incoming demand from customer j. Here (T− t) denotes the length of the remaining
horizon.
�
Uij ≤ λj(T − t) ∀ j ∈ N (Demand Constraint)
i∈M
With these constraints, our objective is to maximize the total profits for the remaining
horizon, while considering service prices gj, transportation costs f d ij , and penalty costs
f p
j . We formulate our objective as follows:
Obj. fn. = � �
(gj − f d ij)Uij − �
(λj(T − t) − �
j∈N i∈M
j∈N
i∈M
Uij)f p
j
The first term represents the profits attained by serving the customer from FSL i and
the second term represents the penalty costs incurred for not serving the customer. By
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