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( )
i
making for solving global optimization problems
(Plevris et al., 2010).
N
n
IG
SB
∑
1
[
]
min
d SB C D k
(
,
)
⋅
( )
k
i
i
=
1
k
=
( )
i
n
1
∑
SB
Problem Formulation
x
=
x
C
k
( )
i
(4)
n
i
k
=
1
SB
( )
A general formulation of a nonlinear optimization
problem can be stated as follows
i
n
1
SB
∑
y
=
y
C
( )
i
k
n
i
k
=
1
SB
D k
( )
=
A k BP k
( )
⋅
( )
T
min ( )
f
x x
=
[
x
,
,
x
]
1
n
n
x R
∈
Subject to
( )
is the number of structural blocks al-
located to the
i
th
inspection crews,
d
(
SB
k
,C
i
) is the
distance between the
SB
k
building block from the
centre of the
i
th
group of structural blocks (with
coordinates
x
C
and
y
C
), while
D
(
k
) is the demand
for the k
th
building block defined as the product
of the building block total area times the built-up
percentage (i.e. percentage of the area with a
structure). This is defined as a discrete optimiza-
tion problem since the design variables
x
are in-
teger numbers denoting the inspection crews to
which each built-up block has been assigned and
thus the total number of the design variables is
equal to the number of structural blocks and the
range of the design variables is [1,
N
IG
].
where
n
S
(3)
g
x
x x x
( )
≤
0
k
=
1
,
,
m
k
L
U
≤ ≤
where
x
is the design variables vector of length
n
,
f
(
x
): R
n
→R is the objective function to be
minimized, the vector of
m
inequality constraint
functions
g
(
x
): R
n
→R
m
and
x
L
,
x
U
are two vectors
of length
n
defining the lower and upper bounds
of the design variables, respectively.
The main objective of this work is to formulate
the problem of inspecting the structural systems
of a city/area as an optimization problem. This
objective is achieved in two steps: in the first step,
the structural blocks to be inspected are optimally
assigned into a number of inspection crews (as-
signment problem), while in the second step the
problem of hierarchy is solved for each group of
blocks (inspection prioritization problem). In the
formulation of the optimization problems consid-
ered in this work, the city/area under investigation
is decomposed into
N
SB
structural blocks while
N
IG
inspection crews are considered for inspect-
ing the structural condition of all structural and
infrastructure systems of the city/area.
STEP 2: INSPECTION
PRIORITIZATION PROBLEM
The definition of this problem is a typical
Travel-
ling Salesman Problem
(TSP) (Colorni et al., 1992)
which is a problem in combinatorial optimization
studied in operations research and theoretical
computer science. In TSP a salesman spends his
time visiting
N
cities (or nodes) cyclically. Given
a list of cities and their - pair-wise - distances,
the task is to find a Hamiltonian tour of minimal
length, i.e. to find a closed tour of minimal length
that visits each city once and only once. For an
N
city asymmetric TSP if all links are present then
there are (
N
-1)! different tours. TSP problems are
also defined as integer optimization problems,
STEP 1: OPTIMUM
ASSIGNMENT PROBLEM
The assignment problem is defined as a nonlinear
programming optimization problem as follows
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