Information Technology Reference
In-Depth Information
where
f
is the
objective function
to be minimised by varying a vector of
decision
, or
design
,
variables
T
n
.
In some cases, all the decision variables are
continuous
, being described by lower
and upper bounds,
x
=
(
x
1
…
,
,
x
)
.
Single variable
optimisation corresponds to
=
1
n
L
i
U
i
. In other cases the variables are
discrete
,
able to take on only particular distinct values,
x
∈
ℜ
:
x
≤
x
≤
x
i
x
∈
X
=
{
X
,
,
X
}
, which in-
…
i
i
i
,
i
,
k
cludes the special case of binary variables:
x
.
Mixed variable
problems have
both discrete and continuous variables in
x
. The objective function
f
itself may be
continuous or discontinuous.
Many practical problems involve
constrained optimisation
where
x
needs to sat-
isfy additional equality and/or inequality relationships:
p
∈
{
h
i
(
x
)
=
0
i
=
1
,
;
…
(2)
x
.
(3)
Multi-objective
optimisation aims to minimise several objective functions simul-
taneously:
g
i
(
)
≥
0
i
=
1
,
q
…
min
f
i
(
x
);
i
=
1
,
r
.
…
(4)
x
HS has been used to solve all of these different types of optimisation problem.
1.2 The Motivation for Undertaking Optimisation
Table 1 shows some common reasons for conducting optimisation studies. This serves
to clarify some of the applications discussed later in this chapter. More detailed
examples may be found in the following chapters of this topic.
Table 1.
Why is Optimisation Performed?
Role of Optimisation
Description
Benchmarking
Some optimisation problems have become de facto standards that
are used to compare existing optimisation methods and test new
ones
Design
This is the process of selecting materials, configurations, sizes and
conditions for a man-made system that best satisfy some design re-
quirements
Calibration or parameter
estimation
Mathematical models of physical systems often contain parameters
that need to be adjusted to optimise the fit between the model pre-
dictions and real-world data
Scheduling
These problems involve finding the best sequence of events or tasks
to be allocated to resources or equipment, typically to minimise total
production time or cost
Route finding
In any network, such as a road system or the Internet, there is a need
to find the best way to transport items between different locations
Analysis
Systems tend to seek a state of lowest energy, which is a form of
natural optimisation—to help understand these systems we can set
up and solve optimisation problems
Search WWH ::
Custom Search