Computational Intelligence - August 2012 - 57

all. Another approach to modeling this
problem would be to use multi-objective optimization. In this way, the model
would generate many non-dominated
solutions that could yield information
on the trade-off among the objectives. A
drawback to multi-objective optimization is that finding the pareto front is
much more demanding computationally
than finding a single optimal solution;
therefore, meta-heuristic methods such
as evolutionary algorithms have been
developed to find pareto-optimal solutions [56].
Multi-objective optimization has
been applied by Zhao et al. [17] to
solve the basic problem of arranging
the sorties of a set of identical aircraft
such that the number of aircraft used
and the risk of failing to accomplish all
sorties by a given deadline is minimized. The risk in this model is due to
the uncertainty in the duration of each
sortie. They use the non-dominated
sorting genetic algorithm II (NSGA II)
[57] to search for pareto-optimal schedules. More complex fleet mix studies
were undertaken by Kaluzny et al. [20]
and Wesolkowski et al. [21], which also
use evolutionary approaches to solve a
multi-objective optimization model for
the loading of multiple types of aircraft
for a given task. The objectives of this
problem are to minimize the number of
each type of aircraft used to transport
all of the cargo.
B. Optimal Fleet Computation

The goal of fleet computation is to
determine the optimal number of each
platform type that should be acquired in
order to fulfill a given set of task
requirements r [58]. Fleet computation is
typically data driven, and, therefore, carried out by first gathering specifications
for considered platforms, required tasks,
and the conditions under which the
fleet will operate. This data is then input
into a model that outputs one or more
fleets f * . Each fleet can then be analyzed in order to assess the validity of
the model and the data before the results
are sent to the decision maker. We will
focus on the modeling aspect of fleet
mix computation.

Fleet mix scheduling is especially useful when a force
undergoes a transition from an outdated fleet to a more
modern fleet. ... it is important that the fleet, at any
given point in time, has sufficient capability to fulfill
all ... of its specified roles.
A Mixed Integer Linear Program
(MILP) was developed by Walmsley and
Hearn [59] to compute the number of
ar mored combat support vehicles
required to fulfill a number of roles
while minimizing cost. This model
computes f = G (c, r) , where f is the
number of each type of required vehicles; r is the set of roles that the vehicles
are required to fulfill; c is the set of vehicle to role assignments; G (c, r) is a
MILP with the only constraint being
the fulfillment of all roles r. MILP techn i q u e s we re u s e d t o c o m p u t e
min c Fcos t (G (c, r)) , where Fcost ( f ) is the
cost of the fleet.
An early optimization model for
computing a fleet for commercial cargo
transport was developed by Etezadi and
Beasley [60] that approximates a VRP
with a MILP that computes the number of each platform type that would
be needed to meet all of the demands
for cargo transportation while minimizing cost.
Taking into consideration that there
is usually a trade-off between cost and
performance, a multi-objective fleet
computation model was developed by
Ghanmi et al. for the tactical vehicle fleet
mix problem [28]. In a similar fashion as
in Walmsley and Hearn [59], the fleet is
computed to fulfill a number of required
roles. The model has two optimization
sub-models, each corresponding to a different objective: minimize cost, maximize performance. Stuive et al. [61] used
NSGA II to solve this model as a multiobjective optimization problem.
C. Fleet Mix Scheduling

An important research area is that of
determining the fleet mix schedule f (t) .
A fleet mix schedule1 is a multi-year
1

Also referred to as a fleet plan.

fleet mix optimization problem which
includes retirement and acquisition of
assets over a large number of years (usually 10 years or more) [9]. Fleet mix
scheduling is especially useful when a
force undergoes a transition from an
outdated fleet to a more modern fleet.
During this transition, it is important
that the fleet, at any given point in time,
has sufficient capability to fulfill all or
most of its specified roles. The U.S. Army
helicopter fleet has undergone this kind
of transition, necessitating the development of fleet mix scheduling tools [34].
Brown et al. [34] formulate the problem of scheduling the update of an
army's helicopter fleet over a given time
as a MILP. The model includes constraints on the age of helicopters in the
fleet, budget constraints, the missions
that the fleet is expected to perform, the
level of technology of the fleet, and various aspects of helicopter production
such as cost and yearly production rates.
On the other hand, given that a fleet
mix may be augmented over time within budgetary constraints, Wesolkowski
et al. [32] studied the adaptability of a
given fleet to future scenarios by generating a tree of possible fleets that can be
composed from the initial fleet by adding platforms under budgetary constraints. Barlow et al. [33] also construct
a graph of possible fleet augmentations
over a planning horizon based on budget, temporal, and resource constraints.
They use graph optimization to choose the
best path through the graph. Figure 4(a)
shows a simple example of a fleet
growth tree with an initial fleet, the two
fleet options for the first acquisition
phase, the four fleet options for the second phase, and so on. Note that only
one path from root to leaf node can be
taken given that acquisition decisions are
usually irreversible.

AUGUST 2012 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE

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