Computational Intelligence - August 2012 - 56

Simulation

Optimization

Fuzzy Logic
MA Simulation

Fleet Evaluation

ADP
Monte Carlo Methods
Fleet Computation

Exact Optimization
Evolutionary Algorithms

Fleet Scheduling

Graph Optimization
Clustering

FIGURE 3 Components of the fleet mix problem and their corresponding solution methodology. Families of methods that include CI techniques are highlighted.

simulation using approximate dynamic
programming (ADP) [31] and simulation within optimization [13], [30], [36]-
[39]. Fleet mix computation is addressed
primarily using optimization although
simulation is increasingly being used
within an optimization framework to
improve model fidelity. Fleet evaluation
is addressed primarily using simulation
methodologies although optimization
has been used, with or without simulation, to improve fleet performance.
Finally, each of the three aspects of the
fleet mix problem, fleet evaluation, fleet
computation, and fleet scheduling, have
been used on its own or in conjunction
with one or both of the other aspects to
solve the given problem.
III. Fleet Mix Optimization

Fleet mix studies are typically concerned with the design of a fleet, or the
evaluation of the performance of a fleet
to fulfill specified requirements. Optimization has been used in fleet mix studies
to: (a) find a configuration c * that results
in the optimal fleet perfor mance
y = E (c *, r, f ) , (b) compute an optimal
fleet f * = G (c, r) with respect to one or
more objectives Fi ( f *) (in the case of
multiple objectives f * is a member of a
non-dominated set of fleets), and (c)
compute an optimal fleet schedule f *(t)
for the acquisition and retirement of
fleet platforms over a number of years
and even decades.
A. Configuration Optimization

Finding an optimal configuration c * to
use a fleet f to fulfill requirements r

56

entails routing and scheduling platforms
such that all tasks will be completed on
time. This problem was presented in the
context of military operations research
by Jaiswal [40].
Finding optimal assignments of platforms to a given task requires that the
task and platforms be appropriately
modeled. Transportation tasks (e.g.,
deployment, sustainment, prepositioning) are typically modeled as routing
[27] or bin packing [17] problems with
time windows to minimize the number
of platforms used to transport the
required cargo within a given time period. Other kinds of tasks (e.g., tactical,
patrol, search and rescue) are typically
modeled by the capabilities or "roles"
[28], [37] required to be supplied or
"filled" in order to accomplish the tasks
while minimizing the amount of excess
capability supplied by the assigned platforms. Often capability types are determined subjectively.
With many tasks to be accomplished
within their respective time windows,
configuration optimization is a complex
problem due to the possible overlapping
of the time windows in which tasks
must be accomplished [41]; therefore,
desired platform assignments may not be
available to accomplish a given task in a
timely manner. This is a scheduling
problem which has been the focus of
many industrial [42], [43], as well as military applications [8], [29].
Scenarios that require the transportation of cargo or passengers to many destinations from one or more sources can
be modeled as a type of vehicle routing

IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2012

problem (VRP) [44]. Many optimization
methods have been proposed to solve
the VRP and its many variants such as:
genetic programming [45], ant colony
optimization [46], tabu search [47], and
integer linear programming [48].
Military configuration optimization
has been carried out using mathematical programming scenario models [7],
[8], [29], [40], [49]-[52]. Prominent
among these models is a linear program
for military air mobility developed by
Baker et al. [8] that simultaneously
optimizes the loading, routing and
scheduling of a fleet of aircraft in order
to transport cargo to theatre while taking into consideration many detailed
operational constraints such as crew
availability, aircraft utilization, and airfield fuel capacity. This model computes y = E (c, r, f ) , where: y is a
weighted sum of total undelivered
cargo, total lateness of cargo delivery,
and other terms designed to discourage
undesirable behavior; c is the routing,
scheduling, and tasking of each aircraft
over a network of source, destination,
and en route airfields; r represents the
cargo that must be delivered from a
source to a destination; f is a fleet comprising military C-5s, C-141s, C-17s,
KC-10s, as well as aircraft of various
types from the civil reserve air fleet;
E (c, r, f ) is a linear program that computes y, and incorporates the many
operational constraints associated with
c, r, and f to determine their combined
feasibility. A bar r ier optimization
method is used to compute
y * = min c E (c, r, f ) .
An important aspect of military
applications that these models lack is any
accounting for uncertainty, hence the
development of stochastic programming
models for military transportation applications which incorporate uncertainty
in aircraft reliability [53], ground times
[54], and the possibility of attack on sea
ports [55].
It should be noted that the objective
function used in [8] is the sum of many
terms which can be considered as different characteristics of a solution such as
the lateness of cargo delivery and the
amount of cargo that is not delivered at



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