Environmental Engineering Reference
In-Depth Information
matrix in according to qualitative evaluation of
its appearance probability, effect severity and
recovery time.
Required variables
x
i,j
can be calculated in the
following way:
(
)
=
f x
, ...,
x
, ...,
x
,
x
, ...,
x
,
x
, ...,
x
1 1
,
1
,
n
2
,
n
i
,
1
i n
,
m
,
1
m n
,
m
(
)
∑
1
f x
i
i
,
j
i
=
(6)
1
,
if j th countermeasure is applied
to decrease criticality o
−
To solve this problem it is appropriate to use
a method of dynamical programming.
Second problem requires formulating of object
function. It is possible using mean arithmetical
value of noncriticality, which, in turn, can ad-
ditionally be weighted depending on the impor-
tance of the attack type. Initial generalized level
of noncriticality of I&C system can be defined
using Equation (7) and (7.8) to take into account
weights of the attack:
f i th attack type
if j th countermeasure is not applied
to decre
−
;
x
=
i j
,
0
−
ase criticality of i
−
th attack type
.
Such a problem is one of the combinatorial
problems class, when optimization function is
defined at finite set with elements represented by
samples of
m
x
n
elements (
n
countermeasures,
which can be applied for each of
m
possible attack
types). However, in some cases, values of unknown
variables
x
i,j
can be determined within the set of
nonnegative integer numbers
x
m
∑
1
i j
,
∈
+
. It is ap-
propriate, for example, when redundancy of ele-
ments is used to decrease the criticality.
In this case value of
x
i,j
defines redundancy
rate; appropriately, costs of such countermeasures
application
c
i,j
increases with multiply number. In
some cases, the redundancy rate can be limited
in an explicit way.
In such definition, the optimization problem
for choice of security countermeasures falls into
wide class of integer problems of linear program-
ming. Nevertheless, it can be reduced to a subclass
of combinatorial problems. In this case, different
redundancy rate should be represented as separate
method containing estimation of costs and effec-
tiveness that depends on the rate.
According to (7.4) and (7.5), global optimiza-
tion of objective function
f(x)
, can be reduced to
phased optimization. Hence, optimal minimization
of costs, associated with security countermeasures
application, is additive object function for which
the effect of a such decision for a single attack
type is corresponding (6).
Z
d
i
NCR
=
i
=
,
(7)
m
m
m
∑
∑
NCR
=
a d
,
a
=
1
.
(8)
i
i
i
i
=
1
i
=
1
Then, a problem of criticality decreasing with
specified limitations of costs can be formulated
in the following way:
m
n
m
( )
=
∑
∑
∑
f x
a
e x
d
max,
a
1
x D
+ →
=
∈
i
i j
,
i j
,
i
i
i
=
1
j
=
1
i
=
1
(9)
m
n
∈
{ }
∑
∑
D
=
x R
∈
mn
|
c x
≤
C
;
x
0 1
,
i j
,
i j
,
max
i j
,
i
=
1
j
=
1
(10)
where
d
i
is an initial level of noncriticality for
i
-th
attack type;
C
max
is maximal acceptable costs for
all the countermeasures applied to decrease the
criticality of attacks;
e
i,j
is an effectiveness of
j
-th
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