771
1 INTRODUCTION
The general semi‐Markov model of critical
infrastructure accident consequences including the
superposition of three models, the process of the
initiatingeventsgeneratedbyacriticalinfrastructure
accident,theprocessof the environment threats and
theprocess of environment degradation, isdesigned
and then adopted to the maritime
transport critical
infrastructure. The proposed model, methods and
toolsareappliedtothiscriticalinfrastructureaccident
with chemical release consequences modeling and
identification, on the basis of the statistical data
coming from reports of chemical accidents at the
BalticSeaandworldseawaters,andprediction.The
model also includes
the cost analysis of losses
associated with those consequences of chemical
releases.Further, under the assumption ofthe stress
ofweatherinfluenceontheshipoperationcondition
intheformofmaritimestormand/orotherhardsea
conditions existence, critical infrastructure accident
consequences are examined and the results are
comparedwith
thepreviousones.Finally,thecritical
infrastructure accident losses optimization is
performedandpractical suggestions and procedures
oftheselossesmitigationaregiven.
2 GENERALMODELOFCRITICAL
INFRASTRUCTUREACCIDENT
CONSEQUENCES
Thegeneralmodelofacriticalinfrastructureaccident
consequences including the process of initiating
events, the process of environment
threats and the
processofenvironment degradationisdesignedand
describedindetailin(Bogalecka&Kołowrocki2016,
2017d,2018a).
2.1 Processofinitiatingevents
Weassume,asin(Bogalecka&Kołowrocki2017a,d)
thattheprocessofinitiatingeventsistaking
,
N,
different initiating events states e
1
,e
2
,…,e
. Next, we
mark by E(t), t<0,+), the process of initiating
events, that is a function of a continuous variable t,
taking discrete values in the set {e
1
,e
2
,…,e
} of the
Consequences of Maritime Critical Infrastructure
Accidents with Chemical Releases
M.Bogalecka
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT: The probabilistic general model of critical infrastructure accident consequences including three
modelsoftheprocessoftheinitiatingeventsgeneratedbyacriticalinfrastructureaccident,theprocessofthe
environment threats and the process of environment degradation is created and adopted to the maritime
transport
criticalinfrastructureunderstoodasashipnetworkoperatingattheseawatersandthenappliedto
accidentconsequencesmodeling,identificationandtotheseconsequencesoptimizationandmitigation.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 13
Number 4
December 2019
DOI:10.12716/1001.13.04.09
772
initiating events states. We assume a semi‐Markov
model(Grabski2015,Kołowrocki2014,Kołowrocki&
Soszyńska‐Budny 2011, Limnios & Oprisan 2005,
Macci2008,Mercier2008)oftheprocessofinitiating
events E(t), and we mark by
lj
its random
conditional sojourn times at the initiating events
states e
l
, when its next initiating events state is e
j
,
l,j1,2,…,
,lj.
Under these assumption, the process of initiating
events may be described by the vector [p
l
(0)]1x
of
probabilitiesoftheprocessofinitiatingeventsstaying
at the particular initiating events states at the initial
moment t=0, the matrix [p
lj
(t)]
x
of probabilities of
transitionsbetweentheinitiatingeventsstatesandthe
matrix [H
lj
(t)]
x
of the distribution functions of the
conditionalsojourntimes
lj
oftheprocessE(t)atthe
initiatingevents states or equivalently by the matrix
[h
lj
(t)]
x
of the density functions of the conditional
sojourn times
lj
, l,j=1,2,…,
, lj of the process of
initiatingeventsattheinitiatingeventsstates.
The approximate limit values of transient
probabilities p
l
, l=1,2,…,
at the particular states of
the process of initiating events given by (3) in
(Bogalecka & Kołowrocki 2017d) can be either
calculatedanalyticallyusingtheaboveparametersof
the process of initiating events or evaluated
approximately by experts (Bogalecka & Kołowrocki
2017a,2018a).
2.2 Processofenvironmentthreats
Weassume,asin(Bogalecka&Kołowrocki2017b,d)
that the process of environment threats of the sub‐
region D
k, k=1,2,…,n3, is taking
k,
kN, different
states of environment threats
.,...,,
)(
2
)(
1
)(
k
k
kk
sss
Next,
wemarkbyS
(k/l)(t),t<0,+),k1,2,…,n3,l= 1,2,…,
,
theprocessofenvironmentthreatsofthesub‐region
D
k,k=1,2,…,n3whilethe process of initiatingevents
E(t)isatthestatee
l
,l=1,2,…,
.TheprocessS(k/l)(t)isa
function defined on the time interval t<0,+)
depending on the states of the process of initiating
events E(t) and taking discrete values in the set
},...,,{
)/(
2
)/(
1
)/(
k
lk
lklk
sss
of the environment threats
states. We assume a semi‐Markov model (Grabski
2015, Kołowrocki 2014, Kołowrocki & Soszyńska‐
Budny 2011, Limnios & Oprisan 2005, Macci 2008,
Mercier 2008) of the process of environment threats
S
(k/l)(t)andwemarkby
ij
lk )/(
itsrandomconditional
sojourntimesatthestates
,
)/(
i
lk
s
whenitsnextstate
is
,
)/(
j
lk
s
i,j=1,2,…,
k,i
j,k=1,2,…,n3,l=1,2,…,
.
Under these assumption, the process of
environment threats S
(k/l)(t), for each sub‐region Dk,
k=1,2,…,n
3, may be described by the vector
k
i
lk
p
x1)/(
)]0([
ofinitialprobabilitiesoftheprocessof
environment threats staying at particular
environmentthreatsstatesattheinitialmomentt=0,
thematrix
kk
ij
lk
p
x
)/(
][
ofprobabilities of transitions
between the environment threats states
i
lk
s
)/(
and
,
)/(
j
lk
s and the matrix
kk
tH
ij
lk
x
)/(
)]([ of the
distributionfunctionsoftheconditionalsojourntimes
ij
lk )/(
oftheprocessS(k/l)(t)attheenvironmentthreats
statesorequivalentlybythematrix
kk
th
ij
lk
x
)/(
)]([
of
thedensityfunctionsoftheconditionalsojourntimes
,
)/(
ij
lk
i,j=1,2,…,
k, i
j, k=1,2,…,n3, l=1,2,…,
of
theprocessofenvironmentthreatsattheenvironment
threatsstates.
The following characteristics of the process of
environment threats S
(k/l)(t) can be either calculated
analytically using the above parameters of the
conditional sub‐process of environment threats or
evaluated approximately by experts (Bogalecka &
Kołowrocki2017b,2018a):
approximatelimitvaluesoftransientprobabilities
,
)/(
i
lk
p i=1,2,…,
k, k=1,2,…,n 3, l=1,2,…,
at the
particular states of the process of environment
threats given by (8) in (Bogalecka & Kołowrocki
2017d),
limitformsoftotalprobabilities
,
)(
i
k
p
i=1,2,…,
k,
k=1,2,…,n
3 of the joined process of environment
threatsandprocessofinitiatingevents(Bogalecka
&Kołowrocki2017d)
i
k
p
)(
,
1
)/(
l
i
lk
l
pp
i=1,2,…,
k,k=1,2,…,n3. (1)
2.3 Processofenvironmentdegradation
Weassume,asin(Bogalecka&Kołowrocki2017c,d)
that the process of environment degradation of the
sub‐region D
k, k=1,2,…,n3 is taking
k
,
k
N
different environment degradation states
.,...,,
)(
2
)(
1
)(
k
k
kk
rrr
Next, we mark by R(k/
)(t), t<0,+),
k=1,2,…,n
3,
=1,2,…,
k,the process ofthe
environment degradation of the sub‐region D
k,
k=1,2,…,n
3whiletheprocessofenvironmentthreats
S
(k)(t) of the sub‐region Dk is in the state
,
)(
k
s
=1,2,…,
k.TheprocessR(k/
)(t)isafunctiondefined
on the time interval t<0,+), depending on the
statesoftheprocessofenvironmentthreatsS
(k)(t)and
takingdiscretevaluesintheset
},...,,{
)/(
2
)/(
1
)/(
k
k
kk
rrr
ofthe environmentdegradationstates.Weassume a
semi‐Markovmodel(Grabski2015,Kołowrocki2014,
Kołowrocki & Soszyńska‐Budny 2011, Limnios &
Oprisan 2005, Macci 2008, Mercier 2008) of the
process of environment degradation R
(k/
)(t) and we
markby
ij
k )/(
itsrandomconditionalsojourntimes
at the states
i
k
r
)/(
when its next state is
,
)/(
j
k
r
i,j=1,2,…,
,
k
ij,k=1,2,…,n3,
=1,2,…,
k.
Under these assumption, the process of
environment degradation R
(k/
)(t) for each sub‐region
D
k, k=1,2,…,n3 may be described by the vector
k
i
k
q
x1
)/(
)]0([
ofinitialprobabilitiesoftheprocessof
environment degradation staying at particular
environmentdegradationstatesattheinitialmoment
t=0, the matrix
kk
ij
k
q
x
)/(
][
of probabilities of
transitions between the environment degradation
states
i
k
r
)/(
and
,
)/(
j
k
r
and the matrix
kk
tG
ij
k
x
)/(
)]([
of the distribution functions of the
conditionalsojourntimes
ij
k )/(
oftheprocessR(k/
)(t)
attheenvironmentdegradationstatesorequivalently
bythematrix
kk
tg
ij
k
x
)/(
)]([
ofthedensityfunctions
of the conditional sojourn times
,
)/(
ij
k
i,j=1,2,…,
,
k
ij, k=1,2,…,n3,
=1,2,…,
k of the
process of environment degradation at the
environmentdegradationstates.
The following characteristics of the process of
environment degradation R
(k/
)(t) can be either
calculatedanalyticallyusingtheaboveparametersof
theprocessofenvironmentdegradationorevaluated
approximately by experts (Bogalecka & Kołowrocki
2017c,2018a):
773
approximatelimitvaluesoftransientprobabilities
,
)/(
i
k
q
i=1,2,…,
,
k
k=1,2,…,n3,
=1,2,…,
k at
theparticularstatesoftheprocessofenvironment
degradation given by (16) in (Bogalecka &
Kołowrocki2017d),
limit forms of total probabilities
,
)(
i
k
q
i=1,2,…,
,
k
k=1,2,…,n3 of the joined process of
environment degradation, the process of
environment threats and the process of initiating
events(Bogalecka&Kołowrocki2017d)
() () (/) (/) (/)
111
[]
kk
ii li
kkk klk
l
qpq ppq






(2)
fori=1,2,…,
,
k
k=1,2,…,n3.
3 CRITICALINFRASTRUCTUREACCIDENT
LOSSES
Wedenoteby(Bogalecka&Kołowrocki2017d,2018c)
),(
)(
tL
i
k
i=1,2,…,
,
k
k=1,2,…,n3, (3)
the losses associated with the process of the
environment degradation R
(k)(t), t<0,+),
k=1,2,…,n
3, in the sub‐region Dk, k=1,2,…,n3 at the
environment degradation state
,
)(
i
k
r
i=1,2,…,
,
k
k=1,2,…,n
3 in the time interval <0,t>. Thus, the
approximateexpectedvalueofthelossesinthetime
interval <0,t>, associated with the process of the
environment degradation R
(k)(t) of the sub‐region Dk
canbedefinedby
k
i
i
k
i
k
k
tLqtL
1
)()(
)(
)()(
fork=1,2,…,n3, (4)
where
i
k
q
)(
mean the limit transient probabilities of
the unconditional process of the environment
degradation at its particular states and are given by
(2),and
),(
)(
tL
i
k
t<0,+)aredefinedby(3).
Thelossesassociatedwithparticularenvironment
degradation states are involved with negative
consequences in the accident area. The types of
consequences are various for different kinds of
accident and accident area. For instance, in the
shipping, the closure of port, closure of
fishery area
and people death can be considered as the negative
consequences.Thelossescanbeexpressedbythecost
ofthe negativeconsequencesincaseliketheclosure
ofport,closureoffisheryarea(Etkin1999,Goldstein
&Ritterling2001,Kontovasetal.2011,Psaraftis2008).
In the case of
negative consequences like people
death, the losses can be expressed as the number of
lossoflife.Inthepaperweonlyconsidertheaccident
consequencesthatcanbeexpressedbycost.
Under these assumption, if we fix the number of
kinds of accident consequences by
and the cost
functionofthisconsequencelastingt
,)]([
)(
)(
ji
k
tK
j=1,2,…,
,i=1,2,…,
,
k
k=1,2,…,n3 (5)
thanthelossforthesub‐regionD
kisexpressedbythe
total cost of all consequences lasting t in the sub‐
regionD
k,andisgivenby
,)]([)(
1
)(
)()(
j
ji
k
i
k
tKtL
i=1,2,…,
,
k
k=1,2,…,n3.(6)
Hence,accordingto(4),lossesassociatedwiththe
processof the environment degradationR
(k)(t) of the
sub‐regionD
karegivenby
,)]([)(
11
)(
)()(
)(


k
ij
ji
k
i
k
k
tKqtL
k=1,2,…,n3. (7)
Furthermore,thetotalexpectedvalueofthelosses
forthefixedtime
,
0,associatedwiththeprocess
oftheenvironmentdegradationR(t)inallsub‐regions
of the considered critical infrastructure operating
environmentregionD,canbeevaluatedby
,)()(
3
1
)(
n
k
k
LL
(8)
whereL
(k)(
)aregivenby(7)fort=
.
4 CRITICALINFRASTRUCTUREACCIDENT
LOSSESWITHCONSIDERINGCLIMATE‐
WEATHERCHANGEPROCESSIMPACT
4.1 Criticalinfrastructureaccidentareaclimate‐weather
changeprocess
The critical infrastructure accident area climate‐
weather change process parameters are (Kołowrocki
et al. 2017): the number of climate‐weather states w,
the vector [q
b(0)]1xw of the initial probabilities of the
climate‐weather change process C(t) staying at
particular climate‐weather states c
b at the moment
t=0, the matrix [q
bl]wxw of the probabilities of
transitions q
bl, b,l=1,2,…,w, bl of the climate‐
weatherchangeprocessC(t)fromtheclimate‐weather
statec
btocl;andthematrix[Nbl]wxwofthemeanvalues
N
bl=E[Cbl], b,l=1,2,…,w, bl of the climate‐weather
change process C(t) conditional sojourn times C
bl at
the climate‐weather states c
b when its next climate‐
weatherstateisc
l.
The critical infrastructure operating area climate‐
weatherchange process characteristicis(Kołowrocki
etal.2017)thevector
[q
b]1xw=[q1,q2,…,qw] (9)
ofthelimitvaluesoftransientprobabilities
q
b(t)=P(C(t)=cb),t <0,+),b=1,2,…,w,
of the climate‐weather change process C(t) at the
particularoperationstatesc
b.
We consider that the climate‐weather change
processaffectsthelossesassociatedwith the process
of the environment degradation (Bogalecka &
Kołowrocki 2017e). We suppose that there are w=6
774
climate‐weather states c
b, b=1,2,…,w dependent on
the wave height and the wind speed, distinguished
fortheshipoperatingareaattheBalticSeaopenand
restrictedwatersandalsow =6climate‐weatherstates
c
b,b=1,2,…,wdependentonthewindspeedandthe
wind direction, distinguished for the ship operating
area at the Baltic Sea port waters. These climate‐
weather states c
b, b=1,2,…,w are detailed defined in
(Kuligowska2017).
4.2 Criticalinfrastructureaccidentlossesrelatedto
climate‐weatherimpact
We denote the losses associated with the process of
the environment degradation R
(k)(t), t<0,+),
k=1,2,…,n
3, in the sub‐region Dk, k=1,2,…,n3, at the
environment degradation state
,
)(
i
k
r
i=1,2,…,
,
k
k=1,2,…,n
3, in the time interval <0,t> while the
climate‐weather change process C(t) at the critical
infrastructureaccidentareaisatthe climate‐weather
state c
b, b=1,2,…,w, by (Bogalecka & Kołowrocki
2017e)
()
()
[()],
ib
k
Lt  (10)
t
<0,+),i=1,2,…,
,
k
k=1,2,…,n3,b=1,2,…,w.
Thelosses
)(
)(
)]([
bi
k
tL
aretheconditionallosseswhile
the climate‐weather change process C(t) is at the
climate‐weatherstatec
b,b=1,2,…,w,definedby
),(][)]([
)(
)(
)(
)(
)(
tLtL
i
k
bi
k
bi
k
 (11)
t
<0,+),i=1,2,…,
,
k
k=1,2,…,n3,b=1,2,…,w,
where
,][
)(
)(
bi
k
i=1,2,…,
,
k
k=1,2,…,n3,b=1,2,…,w, (12)
are the coefficients of the climate‐weather change
process impact on the losses associated with the
process of the environment degradation in the sub‐
region D
k, k=1,2,…,n3, at the environment
degradation state
,
)(
i
k
r
i=1,2,…,
,
k
k=1,2,…,n3, in
the time interval <0,t> while the climate‐weather
change process C(t) at the critical infrastructure
accident area is at the climate‐weather state c
b,
b=1,2,…,w. Thus, by (7) and (11) the conditional
approximateexpectedvalueofthelossesinthetime
interval <0,t>, associated with the process of the
environmentdegradationR
(k)(t),of the sub‐region Dk
while the climate‐weather change process C(t) is at
the climate‐weather state c
b, b=1,2,…,w, can be
definedby
k
i
bi
k
i
k
b
k
tLqtL
1
)(
)()(
)(
)(
)]([)]([
 (13)
fork=1,2,…,n
3,b=1,2,…,w,where
i
k
q
)(
aregivenby
(2) and
,)]([
)(
)(
bi
k
tL
t<0,+) are defined by (11)‐
(12).
Further,applyingtheformulafortotalprobability,
theunconditionalapproximateexpectedvalueofthe
losses, impacted by the climate‐weather change
processC(t),inthetimeinterval<0,t>,associatedwith
theprocess of the environment degradation R
(k)(t) of
thesub‐regionD
k,canbeexpressedby
,)]([)(
1
)(
)()(
w
b
b
k
b
k
tLqtL
k=1,2,…,n3, (14)
where q
b are given by (9) and
,)]([
)(
)(
b
k
tL
t<0,+)
aredeterminedby(13).
Hence,accordingto(13),wehave
,)]([)(
11
)(
)()(
)(


w
bi
bi
k
i
kb
k
k
tLqqtL
k=1,2,…,n3. (15)
Finally, the total expected value of losses,
impactedbytheclimate‐weatherchangeprocessC(t ),
in the fixed time interval <0,
>, associated with the
process of the environment degradation R(t), in all
sub‐regions of the considered critical infrastructure
operatingenvironmentregionD,canbeevaluatedby
,)()(
3
1
)(
n
k
k
LL
(16)
where
L
(k)(
)aregivenby(14)fort=
.
Thus, considering (11), the coefficient of the
climate‐weatherchange processimpactonthelosses
associated with the process of the environment
degradationinthesub‐region D
k,k=1,2,…,n3, in the
timeinterval<0,
>,maybedefinedas
(k)=
L
(k)(
)/L(k)(
),
<0,+),k=1,2,…,n3, (17)
where
L (k)(
) are the losses related to the climate‐
weatherimpact,determinedby(14)andL
(k)(
)arethe
losses without considering climate‐weather impact,
determinedby(4).
Similarly, the coefficient of the climate‐weather
change process impact on the total losses associated
with the process of the environment degradation in
the entire considered region D, in the time interval
<0,
>,maybedefinedas
=
L
(
)/L(
),
<0,+), (18)
where
L (
)arethetotallossesrelatedtotheclimate‐
weather impact determined by (16) andL(
) are the
total losses without considering climate‐weather
impactdeterminedby(8).
Other practically interesting characteristics of the
environment degradation caused by critical
infrastructure accident consequences related to the
climate‐weatheraretheindicatorsoftheenvironment
of the sub‐regions D
k, k=1,2,…,n3 resilience to the
losses associated with the critical infrastructure
accident related to the climate‐weather change that
areproposedtobedefinedby
RI
(k)(
)=1/
(k),
<0,+),k=1,2,…,n3, (19)
where
(k)aredeterminedby(17)andtheindicatorof
the environment of the entire region D resilience to
775
the total losses associated with the critical
infrastructure accident consequences related to the
climate‐weather change that are proposed to be
definedby
RI(
)=1/
,
<0,+), (20)
where
isdeterminedby(18).
5 APPLICATIONTOTHEDYNAMICSHIP
CRITICALINFRASTRUCTURENETWORK
OPERATINGATTHEBALTICSEAWATERS
On the basis of the statistical data, using the
procedures given in (Bogalecka & Kołowrocki 2016,
2017a, b, c, d, 2018a) we identify and predict the
processofenvironmentdegradation
fortheBalticSea
waters. Namely, we calculate unconditional
approximate transient probabilities
,
)(
i
k
q
k1,2,…,5,
i=1,2,…,
,
k
1
30,
2
28,
3
28,
4
31,
5
23 at the particular states of the process of
environmentdegradationgivenby(2),forparticular
sub‐regions D
k, k1,2,…,5 that are as follows (the
probabilitiesoftransitionsthatarenotequalto0are
presentedonly):
1
)1(
q
0.999872179003445,
2
)1(
q
0.000000005069726,
6
)1(
q
0.000054820128704,
11
)1(
q
0.000072995798125;
1
)2(
q
0.999871085266778,
6
)2(
q
0.000016170471066,
12
)2(
q
0.000032213681563,
16
)2(
q
0.000042280457051,
21
)2(
q
0.000032213681563,
25
)2(
q
0.000003353578877,
27
)2(
q
0.000002682863102;
1
)3(
q
0.999871085266778,
6
)3(
q
0.000016170471066,
12
)3(
q
0.000032213681563,
16
)3(
q
0.000042280457051,
21
)3(
q
0.000032213681563,
25
)3(
q
0.000003353578877,
27
)3(
q
0.000002682863102;
1
)4(
q
0.999871139828532,
12
)4(
q
0.000036818375059,
16
)4(
q
0.000048324117265,
21
)4(
q
0.000036818375059,
28
)4(
q
0.000003832946714,
30
)4(
q
0.000003066357371;
1
)5(
q
1. (21)
The general model of critical inf rastructure
accident consequences is applied to cost analysis of
lossesassociatedwithconsequences generatedbythe
critical infrastructure defined as a ship operating at
the Baltic Sea (Bogalecka & Kołowrocki 2018c).
Considering (21), according to (6)‐(7) and the
information coming from experts,
the losses
associated with the process of the environment
degradation R
(k)(t) of the particular sub‐region Dk,
k=1,2,…,5, during the time t=1 hour, amount (in
PLN):
attheopenandrestrictedwaters
L
(1)(1)0.785,L(2)(1)2.467,L(3)(1)3.091,
L
(4)(1)3.072,L(5)(1)=0; (22)
atGdyniaandKarlskronaports
L
(1)(1)1.457,L(2)(1)3.145,L(3)(1)3.769,
L
(4)(1)3.750,L(5)(1)=0. (23)
Considering the above results, after applying (8),
thetotalexpectedvalueoflossesassociatedwiththe
process of the environment degradation R(t) in all
sub‐regions of the considered critical infrastructure
operatingenvironmentregionD,duringthetimet=1
hour,amounts(inPLN)
attheopenandrestrictedwaters:
L(1)=9.415; (24)
atGdyniaandKarlskronaports:
L(1)=12.121. (25)
Moreover, the losses of critical infr astructure
accident consequences impacted by the climate‐
weatherchangeprocessarecalculated.
The approximate limit values of transient
probabilities q
b, b=1,2,…,6 of the climate‐weather
change process, at the climate‐weather states for the
operating area (GMU Safety Interactive Platform
2018)amount
attheopenwaters:
q
1=0.834,q2=0.149,q3=0,
q
4=0,q5=0.015,q6=0.002; (26)
attherestrictedwaters:
q
1=0.827,q2=0.155,q3=0.004,
q
4=0,q5=0.007,q6=0.007; (27)
attheGdyniaPort:
q
1=0.394,q2=0.010,q3=0.473,
q
4=0.006,q5=0.017,q6=0; (28)
attheKarlskronaPort:
q
1=0.364,q2=0.005,q3=0.417,
q
4=0.016,q5=0.197,q6=0.001. (29)
Accordingtotheinformationcomingfromexperts,
the coefficients
,][
)(
)(
bi
k
b=1,2,…,6, k=1,2,…,5,
i=1,2,…,
,
k
1
30,
2
28,
3
28,
4
31,
5
23oftheclimate‐weatherimpactonlossesatthe
climate‐weather change process states c
b, b=1,2,…,6
are
attheopenandrestrictedseawatersarea:
)(
)1(
][
bi
=1.0,b=1,2,3,i=1,2,…30,
)(
)1(
][
bi
=2.0,b=4,5,6,i=1,2,…30,
)(
)2(
][
bi
=1.0,b=1,i=1,2,…28,
)(
)2(
][
bi
=2.0,b=2,i=1,2,…28,
)(
)2(
][
bi
=2.5,b=3,5,i=1,2,…28,
)(
)2(
][
bi
=1.8,b=4,i=1,2,…28,
776
)(
)2(
][
bi
=3.0,b=6,i=1,2,…28,
)(
)3(
][
bi
=1.0,b=1,4,i=1,2,…28,
)(
)3(
][
bi
=2.0,b=2,5,i=1,2,…28,
)(
)3(
][
bi
=3.0,b=3,6,i=1,2,…28,
)(
)4(
][
bi
=1.0,b=1,2,…,6,i=1,2,…31,
)(
)5(
][
bi
=1.0,b=1,2,…,6,i=1,2,…23; (30)
attheGdyniaandKarlskronaports:
)(
)1(
][
bi
=1.0,b=1,3,5,i=1,2,…30,
)(
)1(
][
bi
=2.0,b=2,4,6,i=1,2,…30,
)(
)2(
][
bi
=1.0,b=1,3,5i=1,2,…28,
)(
)2(
][
bi
=2.0,b=2,4,6,i=1,2,…28,
)(
)3(
][
bi
=1.0,b=1,3,5,i=1,2,…28,
)(
)3(
][
bi
=2.0,b=2,4,6,i=1,2,…28,
)(
)4(
][
bi
=1.0,b=1,2,…,6,i=1,2,…31,
)(
)5(
][
bi
=1.0,b=1,2,…,6,i=1,2,…23. (31)
Hence, according to (11) and (13)‐(15), the
unconditional approximate expected value of the
environmental losses
),(
)(
tL
k
during the time t=1
hour,associatedwiththeprocessoftheenvironment
degradation R
(k)(t) of the sub‐region D k, k=1,2,…,5
while the climate‐weather change process C(t) is at
theclimate‐weatherstatec
b,b=1,2,…,6,areasfollows
(inPLN)
attheopenseawaters:
(1)
8(1) 0.79 ,L
(2)
0(1) 2.90 ,L
(3)
1(1) 3.61 ,L
(4)
2(1) 3.07 ,L
(5)
;(1) 0L
(32)
attherestrictedseawaters:
(1)
6(1) 0.79 ,L
(2)
5(1) 2.92 ,L
(3)
0(1) 3.66 ,L
(4)
2(1) 3.07 ,L
(5)
;(1) 0L
(33)
attheGdyniaPort:
(1)
1(1) 1.48 ,L
(2)
5(1) 3.19 ,L
(3)
0(1) 3.8 0 ,L
(4)
0(1) 3.75 ,L
(5)
;(1) 0L
(34)
attheKarlskronaPort:
(1)
9(1) 1.48 ,L
(2)
4(1) 3.21 ,L
(3)
1(1) 3.8 1 ,L
(4)
0(1) 3.75 ,L
(5)
.(1) 0L
(35)
Considering (32)‐(35) respectively and applying
(16), the total expected value of the losses
),(tL
impactedbytheclimate‐weatherchangeprocessC(t ),
duringthetimet= 1hour,associatedwiththeprocess
oftheenvironmentdegradationR(t)inallsub‐regions
of the considered critical infrastructure operating
environmentregionD,amounts(inPLN)
attheopenseawaters:
)1(L
10.381; (36)
attherestrictedseawaters:
)1(L
10.453; (37)
attheGdyniaPort:
)1(L
12.226; (38)
attheKarlskronaPort:
)1(L
12.264. (39)
Thus, considering (22)‐(23) and (32)‐(35)
respectively, and according to (17) and (19), the
indicators RI
(k)(t) of the environment of the sub‐
regions D
k, k=1,2,…,5, resilience to the losses
associated with the critical infrastructure accident
relatedtotheclimate‐weatherchangeare
attheopenseawaters:
RI
(1)(1)=98.4%,RI(2)(1)=85.1%,
RI
(3)(1)=85.6%,RI(4)(1)=100%,
RI
(5)(1)–n/aasL(5)(1)=0and L (5)(1)=0; (40)
attherestrictedseawaters:
RI
(1)(1)=98.6%,RI(2)(1)=84.3%,
RI
(3)(1)=84.5%,RI(4)(1)=100%,
RI
(5)(1)–n/aasL(5)(1)=0and L (5)(1)=0; (41)
attheGdyniaPort:
RI
(1)(1)=98.4%,RI(2)(1)=98.4%,
RI
(3)(1)=99.2%,RI(4)(1)=100%,
RI
(5)(1)–n/aasL(5)(1)=0and L (5)(1)=0; (42)
attheKarlskronaPort:
RI
(1)(1)=97.8%,RI(2)(1)=97.8%,
RI
(3)(1)=98.9%,RI(4)(1)=100%,
RI
(5)(1)–n/aasL(5)(1)=0and L (5)(1)=0. (43)
Next, considering (24)‐(25) and (36)‐(39)
respectively, and according to (18) and (20), the
indicatorRI(t )oftheenvironmentoftheentireregion
D resilience to the losses associated with the critical
infrastructureaccidentrelatedtotheclimate‐weather
changeis
attheopenseawaters:
RI(1)=90.7%; (44)
attherestrictedseawaters:
RI(1)=90.1%; (45)
777
attheGdyniaPort:
RI(1)=99.1%; (46)
attheKarlskronaPort:
RI(1)=98.8%. (47)
The above results point the more significant
impactofthe climate‐weatherchangeprocesswithin
the open and restricted waters than Gdynia and
Karlskronaports.Thereasonforthiscanbeexplained
that the wave height and the wind speed are
parameters
considered in the state of the climate‐
weatherchangeprocessattheopenandrestrictedsea
waters, whereas the wind speed and the wind
directionareparametersconsideredinthestateofthe
climate‐weather change process at Gdynia and
Karlskrona ports. It confirms that a wind direction
that is
consider in the states of the climate‐weather
changeprocessonlyforGdyniaandKarlskronaports
has a little significant impact on a value of losses
associated with the process of the environment
degradation.
Finally, these results are applied to the accident
consequencescostoptimizationthroughtheaccident
losses minimizing. From
the linear equation (4), we
can see that the mean value of expected critical
infrastructure accident losses L
(k)(t), t<0,+),
associated with the process of the environment
degradationR
(k)(t)ofthesub‐regionDk,k1,2,…,5is
determined by the limit value of transient
probabilities
,
)(
i
k
q
i=1,2,…,
,
k
k1,2,…,5 of the
process of the environment degradation at the state
,
)(
i
k
r
i=1,2,…,
,
k
k1,2,…,5andthe meanvalueof
the critical i nfrastructure accident losses
)(
)(
tL
i
k
associated with the process of the environment
degradationR
(k)(t)ofthesub‐regionDk,k1,2,…,5,at
the state
,
)(
i
k
r
i=1,2,…,
,
k
k1,2,…,5. Similarly,
from the linear equation (15), we can see that the
meanvalueofexpectedcriticalinfrastructureaccident
losses
),(
)(
tL
k
t<0,+),associatedwiththeprocess
of the environment degradation R
(k)(t) of the sub‐
region D
k, k1,2,…,5, impacted by the climate‐
weather change process C(t) is determined by the
limitvalueoftransientprobabilitiesq
b,b=1,2,…,6of
the climate‐weather change process C(t) at the
particular climate‐weather state c
b, b=1,2,…,6, the
limit value of transient probabilities
,
)(
i
k
q
i=1,2,…,
,
k
k1,2,…,5 of the process of the
environment degradation at the state
,
)(
i
k
r
i=1,2,…,
,
k
k1,2,…,5andbythemeanvalueofthe
critical infrastructure accident losses
bi
k
tL )]([
)(
associated with the process of the environment
degradationR
(k)(t)ofthesub‐regionDk,k1,2,…,5at
the state
,
)(
i
k
r
i=1,2,…,
,
k
k1,2,…,5 impacted by
theclimate‐weatherchangeprocessC(t).
Therefore, the optimization based on the linear
programming(Kołowrocki&Soszyńska‐Budny2011,
Klabjan&Adelman2006,Vercellis2009)ofthecritical
infrastructure accident losses associated with the
processof the environment degradationR
(k)(t) of the
sub‐region D
k, k1,2,…,5 without and with
considering the climate‐weather change process C(t)
can be proposed. Namely, we may look for the
corresponding optimal values
,
)(
i
k
q
i=1,2,…,
,
k
k
1,2,…,5 of the limit transient probabilities
,
)(
i
k
q
i=1,2,…,
,
k
k1,2,…,5 of the process of the
environment degradation at the state
,
)(
i
k
r
i=1,2,…,
,
k
k1,2,…,5tominimizethemeanvalue
of critical infra structure accident losses L
(k)(t) in the
sub‐region D
k, k1,2,…,5 (Bogalecka & Kołowrocki
2018b) or optimal values
,
)(
i
kb
qq
i=1,2,…,
,
k
b=1,2,…,6, k
1,2,…,5 of the limit transient
probabilities
,
)(
i
kb
qq
i=1,2,…,
,
k
b=1,2,…,6,
k
1,2,…,5 of the process of the environment
degradation atthe state
,
)(
i
k
r
i=1,2,…,
,
k
k
1,2,…,5 to minimize the mean value of critical
infrastructureaccidentlosses
)(
)(
tL
k
impactedbythe
climate‐weatherchangeprocessC(t)inthesub‐region
D
k,k1,2,…,5(Bogalecka&Kołowrocki2018b).Now,
we can obtain the optimal solution, using the
proceduregivenin(Bogalecka &Kołowrocki 2018b).
Namely, we can find the optimal values
,
)(
i
k
q
i=1,2,…,
,
k
k1,2,…,5 of the limit transient
probabilities
,
)(
i
k
q
i=1,2,…,
,
k
k1,2,…,5, or
,
)(
i
kb
qq
i=1,2,…,
,
k
b=1,2,…,6, k1,2,…,5 of the
transient probabilities
,
)(
i
kb
qq
i=1,2,…,
,
k
b=1,2,…,6, k
1,2,…,5 that minimize the objective
functionsgivenby(4)and(15)respectively.
The inventory of losses associated with the
shipping critical infrastructure accident without and
with considering the climate‐weather change impact
andresilienceindicatorsfortheselossesimpactedby
theclimate‐weatherchange,basedondatacollectedat
the
Baltic Sea waters, before and after optimization
arepresentedinTables1‐4.
The performed comparison of values of losses
associated with the shipping critical infrastructure
accident without and with considering the climate‐
weather change impact and resilience indicators for
theselossesimpactedbytheclimate‐weather change
confirms and justifies
the reasonableness of the
critical i nfrastructure accident losses optimization. It
maybethebasisofsomesuggestionsonnewstrategy
assuring lower environment losses concerned with
chemical releases generated by an accident of ships
operating within the shipping critical infrastructure
network.
Table1. Shipping critical infrastructure accident losses (in
PLN) and resilience indicators for open sea waters before
andafteroptimization
_______________________________________________
Beforeoptimization
_______________________________________________

(1)
5(1) 0.78L
(1)
8(1) 0.79L
(1)
0.984RI

(2)
7(1) 2.46L
(2)
0(1) 2.90L
(2)
0.851RI

(3)
1(1) 3.09L
(3)
1(1) 3.61L
(3)
0.856RI

(4)
2(1) 3.07L
(4)
2(1) 3.07L
(4)
1.000RI

(5)
0(1)L
 
(5)
0(1)L
n/a
_________________________________________
total
)1(L
9.415
)1(L
10.381
R
I
0.907
_______________________________________________
Afteroptimization
_______________________________________________

)1(
)1(
L
0.570
)1(
)1(
L
0.575
)1(
IR
0.991

)1(
)2(
L
1.437
)1(
)2(
L
1.570
)2(
IR
0.915

)1(
)3(
L
1.980
)1(
)3(
L
2.141
)3(
IR
0.925

)1(
)4(
L
1.930
)1(
)4(
L
1.930
)4(
IR
1.000

)1(
)5(
L
0
)1(
)5(
L
0n/a
_________________________________________
total
)1(L
5.917
)1(L
6.216
I
R
0.952
_______________________________________________
778
Table2. Shipping critical infrastructure accident losses (in
PLN) and resilience indicators for restricted sea waters
beforeandafteroptimization
_______________________________________________
Beforeoptimization
_______________________________________________

(1)
5(1) 0.78L
(1)
6(1) 0.79L
(1)
0.986RI

(2)
7(1) 2.46L
(2)
5(1) 2.92L
(2)
0.844RI

(3)
1(1) 3.09L
(3)
0(1) 3.66L
(3)
0.845RI

(4)
2(1) 3.07L
(4)
2(1) 3.07L
(4)
1.000RI

(5)
0(1)L  
(5)
0(1)L
n/a
_________________________________________
total
)1(L
9.415
)1(L
10.453 R
I
0.901
_______________________________________________
Afteroptimization
_______________________________________________

(1)
0(1) 0.57L
(1)
0(1) 0.57L
(1)
1.000RI

(2)
7(1) 1.43L
(2)
7(1) 1.57L
(2)
0.911RI

(3)
0(1) 1.98L
(3)
7(1) 2.15L
(3)
0.918RI

(4)
0(1) 1.93L
(4)
0(1) 1.93L
(4)
1.000RI

(5)
0(1)L

(5)
0(1)L
n/a
_______________________________________
total
)1(L
5.917
)1(L
6.234
I
R
0.949
_______________________________________________
Table3. Shipping critical infrastructure accident losses (in
PLN)and resilience indicators for Gdynia Port before and
afteroptimization
_______________________________________________
Beforeoptimization
_______________________________________________

(1)
7(1) 1.45L
(1)
1(1) 1.48L

(1)
0.984RI

(2)
5(1) 3.14L
(2)
5(1) 3.19L
(2)
0.884RI

(3)
9(1) 3.76L
(3)
0(1) 3.80L
(3)
0.992RI

(4)
0(1) 3.75L
(4)
0(1) 3.75L
(4)
1.000RI

(5)
0(1)L  
(5)
0(1)L
n/a
_________________________________________
total
)1(L
12.122
)1(L
12.225 R
I
0.992
_______________________________________________
Afteroptimization
_______________________________________________

(1)
9(1) 1.14L
(1)
5(1) 1.15L
(1)
0.994RI

(2)
6(1) 2.01L
(2)
1(1) 2.03L
(2)
0.992RI

(3)
9(1) 2.55L
(3)
8(1) 2.56L
(3)
0.996RI

(4)
9(1) 2.50L
(4)
9(1) 2.50L
(4)
1.000RI

(5)
0(1)L

(5)
0(1)L
n/a
_________________________________________
total
)1(L
8.231
)1(L
8.262
I
R
0.952
_______________________________________________
Table4. Shipping critical infrastructure accident losses (in
PLN) and resilience indicators for Karlskrona Port before
andafteroptimization
_______________________________________________
Beforeoptimization
_______________________________________________

(1)
7(1) 1.45L
(1)
9(1) 1.48L
(1)
0.978RI

(2)
5(1) 3.14L
(2)
4(1) 3.21L
(2)
0.978RI

(3)
9(1) 3.76L
(3)
1(1) 3.81L
(3)
0.989RI

(4)
0(1) 3.75L
(4)
0(1) 3.75L
(4)
1.000RI

(5)
0(1)L
 
(5)
0(1)L n/a
_________________________________________
total
)1(L
12.122
)1(L
12.225 R
I
0.988
_______________________________________________
Afteroptimization
_______________________________________________

(1)
9(1) 1.14L
(1)
7(1) 1.15L
(1)
0.992RI

(2)
6(1) 2.01L
(2)
7(1) 2.03L
(2)
0.989RI

(3)
9(1) 2.55L
(3)
2(1) 2.57L
(3)
0.995RI

(4)
9(1) 2.50L
(4)
9(1) 2.50L
(4)
1.000RI

(5)
0(1)L

(5)
0(1)L
n/a
_________________________________________
total
)1(L
8.231
)1(L
8.275
I
R
0.995
_______________________________________________
Fromthe performed analysis of the results of the
chemical spills at sea consequences optimization it
can be suggested to modify the process of accident
initiating events and the process of environment
threats, and the process of environment degradation
inthewaythatcausesthereplacing(approximately)
the conditional
mean sojourn times of the
environment degradation process at its particular
statesbeforetheoptimizationbytheiroptimalvalues
aftertheoptimization.
Instead of this practically difficult modification it
seems to be easier to change the process of accident
initiating events and the process of environment
threats characteristics that results in
replacing
(approximately) the unconditional mean sojourn
times of the environment degradation process at its
particular states before the optimization by their
optimalva luesaftertheoptimization.Theeasiestway
ofthesetwoprocessesmodificationisthatleadingto
thereplacing(approximately)thetotalsojourn times
of the process of accident
initiating events and the
process of environment threats at their particular
states during the fixed time before the optimization
by their optimal va lues after the optimization.
Comingdirectlyfromthepracticesuggestionsonthe
way of minimizing the environment losses are the
basis for creating the general procedures and new
strategies
assuringthecriticalinfrastructuresaccident
consequences decreasing the environment losses. In
practice it includes the following proactive and
reactive strategies (HELCOM 2002, IMO 2002,
Kristiansen2005,Mamacaetal.2009):
prevention measures to elimination or reduction
accidentsatsea(establishandrevisionofnational
laws and regulations, IMO conventions and
resolutions, inspection, certification and auditing,
maintenanceoftheshipandequipment,reduction
trafficcongestion),
investigation of accidents and learn from
experience(identifycausesandpotentialmeasures
thatwillreducethethreatsanddegradationeffects
ofsimilaraccidentinthefuture),
identification of hazard and possible events that
may cause threats and result in severity of
degradationeffects,
emergency preparedness (preparation and
revision of emergency action plan, high quality
equipment for combating released substances,
rescuerstraining),
reductionthetimeofemergencyresponseprocess,
quicklyundertakingaproperdecisionandaction,
selection the best response method
(recommendation for decision‐ma king, decision
support, cooperation with external parties and
exchangeofinformation,toolsforforecastingand
equipment for monitoring the spread or drift of
releasedsubstances).
6 CONCLUSION
Presented
in the paper model, methods, procedures
andtoolsaresupposedtobeveryusefulinthecritical
infrastructureaccidentconsequencesmodeling,
identification,prediction,optimizationandmitigation
the losses associated with these consequences. The
constructedmodel is appliedtothemaritimecritical
infrastructure accident consequences caused by the
ship operating at the
sea waters and chemical
releases. The papers contains results obtained when
the model was applied to the critical infrastructure
accidentconsequencescausedbytheshipoperatingat
theBalticSea.However,theproposedgeneralmodel
of critical infrastructure accident consequences is a
779
universal tool that can have wide applications in
various industrial sectors. In spite of the model has
beendesignedforthemaritimecriticalinfrastructure,
it can be applied to identification, prediction,
optimization and mitigation of the losses associated
withchemicalreleasesgeneratedbyanyothercritical
infrastructures, industrial installations and
systems.
Next, based on the results, a new strategy assuring
low consequences of any critical infrastructure
accidentcanbecreatedthroughtheinitiatingevents,
environment threats and environment degradation
processes modification related to minimizing critical
infrastructureaccidentlosses.
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