REVIEW OF MATRIX METHODS FOR RISK
ASSESSMENT IN WATER SUPPLY SYSTEM
Rak J., Tchórzewska-Cieślak B.
Zakład Zaopatrzenia w Wodę i Odprowadzania Ścieków, Politechnika Rzeszowska
Al. Powstańców Warszawy 6, 35-950 Rzeszów, Poland
Abstract: In this work the matrix methods for risk assessment in water supply system SZW have
been presented. The three risk levels were assumed: tolerable, controlled and unacceptable. The
two parametric matrix for risk assessment combines the point scale of the probability that the
threat appears (P) with the point scale of the consequences ( C ). The novelty in the attitude to
risk assessment is taking into account that the system protection (O) is inversely proportional to
the parameters P, C, E and N.
1. Introduction
The objective realities in SZW operating are the losses caused by the breaks in water
supply or the low quality of supplied water. The related risk can rise protests of drinking
water consumers. Nowadays the water-pipe companies try to get quality management
certificates according to the international standard ISO9001:2000, that requires the
procedures to estimate widely understood risk. Etymology of the word risk has
multiaspects meaning. In Arabic risq means fate, act of God. In Spanish ar-risko means
courage, danger. In English, however, the synonym of risk is the word hazard that is
understood as danger or a potential source of danger. In Greek riza means sharp cliff, reef.
In Latin riscare means to dodge something. P.L. Bernstein in his work titled ”Against
Gods –The Unusual History of Risk ” (1997) says that risk comes from an old Italian
word risicare which means to have courage to do something. The problem of risk in civil
engineering was introduced by prof. E. Kempa in his work titled “ Risk Analysis in Water
Treatment Systems” that was published in 1993. [2] Later this subject was developed,
among others, by the authors of works [1,3,4,5,6]. Risk is also a subject of the monograph
titled “Problem of Risk in Water Supply System Operating” published in 2004 [5].
2. General characteristic of matrix methods for risk assessment in
SZW
Procedures for risk analysis cover the whole activity aiming to identify threats, to estimate
risk and its size [9,10]. The appearance of the extraordinary event produces the state of
68
Rak J., Tchórzewska-Cieślak B.
________________________________________________________________________
emergency to which some potential of danger is assigned. The release of this potential
leads to failure and failure related losses (financial) and even to the loss of health and
human death.[4]. Determination of the acceptable risk level relies on an introduction of
the criteria values [7] according to the rules given in fig.1
R
E
S
P
O
N
S
I
B
I
L
I
T
Y
S
C
A
L
E
CRIMINAL
RESPONSIBILITY
UNACCEPTABLE
RISK
CIVIL
RESPONSIBILITY
CONTROLLED
RISK
COMPANY
RESPONSBILITY
R
I
S
K
L
E
V
E
L
TOLERABLE
RISK
LACK OF
RESPONSIBILITY
Fig.1 The illustration of the possibilities that the given risk level occurs
As an example we can suggest to introduce the following categories of probability –
frequency of the undesirable events occurrence and the categories of their consequences
[7], presented in table 1.
Table 1. The list of the categories of probabilities and consequences
A
B
C
D
E
Category of probability- frequency
Often
Probable
Occasional Little
probability
Improbable
F
G
H
I
J
Category of consequences
Catastrophic
Serious
Significant
Marginal
Negligible
Each time risk ( r ) is determined according to the formula:
r=X.Y
where :
X – frequency of the undesirable events occurrence,
Y – consequences of the undesirable events,
(1)
Using the formula (1) we can obtain the following possibilities of the undesirable events
combinations shown as the risk matrix in fig. 2
Review of matrix methods for risk assessment...
69
________________________________________________________________________
AxF ;
AxG ;

AxH ;

AxI ;
AxJ ;

BxF ; CxF ; DxF ; ExF 
BxG ; CxG ; DxG ; ExG 
BxH ; CxH ; DxH ; ExH 

BxI ; CxI ; DxI ; ExI 
BxJ ; CxJ ; DxJ ; ExJ 
Fig 2. Risk matrix
The unacceptable risk: [A x F], [A x G] , [ A x H], [ B x F], [B x G] , [C x F],
The controlled risk: [A x I], [A x J], [B x H], [B xI], [C x G], [C x H], [D x F],
[D x G], [E x F]
The tolerable risk: [B x J], [C x I], [C x J], [D x H], [D x I], [D x J], [E x G],
[E x H], [E x I], [E x J].
The procedure presented above gives a general characteristic of the essence of matrix
methods for risk assessment [6].The risk matrix presented in fig. 2 has a character of
matrix to which the undesirable events are referred.
3. The two parametric risk matrix
The presented matrix is one of the simplest. From the mathematical point of view risk (r)
is defined as following :
r=P.C
(2)
where:
P – a measure of the system operating unreliability corresponding with category of
probability - frequency,
C – a measure of the consequences corresponding with category of consequences –
damages, expressed in financial units.
In tab.2 the two parametric risk matrix is presented, assuming the following risk scales
and corresponding point weights:
 probability (P): little – 1, medium – 2, large – 3.
 consequences (C): little – 1, medium – 2, large – 3.
Table 2. The two parametric risk matrix
C
1
2
P
3
r
1
2
3
1
2
3
2
4
6
3
5
9
70
Rak J., Tchórzewska-Cieślak B.
________________________________________________________________________
According to the basic matrix for risk assessment given above we can analyse different
undesirable events assuming the following scale of risk:
 the tolerable risk – a number of points from 1 to 2,
 the controlled risk – a number of points from 3 to 4,
 the unacceptable risk – a number of points from 6 to 9.
4. The three parametric risk matrix
Taking into account that SZW is a complex technical system built from subsystems and
elements that are firmly interconnected it makes sense to expand the SZW operating risk
matrix by next parameters influencing risk size. The three parametric matrix for risk
assessment is proposed. The parameters are following : the frequency of the threat
occurrence (P), threat consequences (C) and the exposure to threat (E). The exposure to
threat should be related to the period of time when the public water pipe has been used as
a source of drinking water. The numerical risk assessment is a product of the above
mentioned parameters [7]:
r=PCE
(3)
The following scales and weights of the particular parameters are assumed:
 scale of threat frequency (P):
- almost impossible incidents ( 1 in 100 years ); with weight 0.1
- occasionally possible incidents ( 1 in 20 years ); with weigh 1.0
- little probable incidents ( 1 in 10 years ), with weigh 2.0
- quite probable incidents ( once a year ), with weigh 5.0
- very probable incidents ( 10 times a year ), with weigh 10.0
 scale of threat consequences size (C):
- little loss up to 5103 EUR ; with weight 1.0
- medium loss from 5103 to 5104 EUR, with weight 3.0
- large loss 5104 EUR – 105 EUR; with weight 7.0
- very large loss 105 – 106 EUR, with weight 15.0
- serious disaster , losses over 106 EUR; with weight 50.0
 scale of exposure to threat (E):
- slight, once a year or less often , with weight 0.5.
- minimal, a few times a year; with weight 1.0
- occasionally, several times a month, with weight 2.0
- often, several times a week, with weight 5.0
- constant, with weight 10.0
The numerical risk assessment determined in this way takes the values within the range
0.05 to 5103. The levels of risk in the five stage scale are shown in table 3 .
Review of matrix methods for risk assessment...
71
________________________________________________________________________
Table 3. The levels of risk
Class
1
2
3
4
5
Description
very little
little
medium
large
very large
Numerical values
0,05 < r  5 ....
5 < r  50
50 < r  200
200 < r  400
400 < r  5000
Risk level
tolerable
controlled
unacceptable
5. The four parametric matrix for risk assessment
Every modern SZW should be provided with different protection and monitoring systems
which increases its operating and safety reliability. That is why the fourth parameter
characterising the size of this protection has been introduced to the risk matrix connected
with SZW operating [8]. The four parametric matrix for risk assessment has been
proposed, according to the formula:
PC N
(4)
r
O
where :
P - point weight connected with the probability that the representative undesirable event
appears,
C - point weight connected with the size of losses,
N - point weight connected with a number of the endangered inhabitants,
O - point weight connected with SZW protection against extraordinary threat ( protective
barriers, clean water reservoirs, monitoring, etc )
Parameter (O) is inversely proportional to the size of risk. Analogically as in the two and
three parametric methods every time the size of parameters P,C,N and O are described
according to the following point scale: low – L= 1, medium – M = 2, high – H = 3. In this
way the point scale to measure risk in the numerical form within the range [0,33  27] has
been obtained. In table 4 the four parametric risk matrix is shown; the particular
numerical values were obtained using the formula (4).
The description of the risk components:
 category for a number of the endangered inhabitants – N,
- low – a number of the endangered inhabitants less than 5 000 – N=1,
- medium - a number of the endangered inhabitants from 5 001 to 50 000 – N=2,
- high - a number of the endangered inhabitants higher than 50 001 – N=3,
72
Rak J., Tchórzewska-Cieślak B.
________________________________________________________________________
Table 4. The four parametric risk matrix
P
L=1
C
L=1
N
M=2
H=3
O
H=3
M=2
L=1
H=3
M1 = 2
L=1
H=3
M=2
L=1
L=1
LLLH
0,33
LLLM
0,5
LLLL
1
LMLH
0,66
LMLM
1
LMLL
2
LHLH
1
LHLM
1,5
LMLL
3
M=2
LLMH
0,66
LLMM
1
LLML
2
LMMH
1,33
LMMM
2
LMML
4
LHMH
2
LHMM
3
LMLM
6
H=3
LLHH
1,5
LLHM
1,5
LLHM
3
LMHH
2
LMHM
3
LMHL
6
LHHH
3
LHHM
4,5
LMLH
9
P
M=2
C
L=1
N
M=2
H=3
O
H=3
M=2
L=1
H=3
M=2
L=1
H=3
M=2
L=1
L=1
MLLH
0,66
MLLM
1
MLLL
2
MMLH
1,33
MMLM
2
MMLL
4
MHLH
2
MHLM
3
MHLL
6
M=2
MLMH
1,33
MLMM
2
MLML
4
MMMH
2,66
MMMM
4
MMML
8
MHMH
4
MHMM
6
MHML
12
H=3
MLHH
2
MLHM
3
MLHL
6
MMHH
4
MMHM
6
MMHL
12
MHHH
6
MHHM
9
MHHL
18
P
H=3
C
M=2
O
L=1
N
H=3
H=3
M=2
L=1
H=3
M=2
L=1
H=3
M=2
L=1
L=1
HLLH
1
HLLM
1,5
HLLL
3
HMLH
2
HMLM
3
HMLL
6
HHLH
3
HHLM
4,5
HHLL
9
M=2
HLMH
2
HLMM
3
HLML
6
HMMH
4
HMMM
6
HMML
12
HHMH
6
HHMM
9
HHML
18
H=3
HLHH
3
HLHM
4,5
HLHL
9
HMHH
6
HMHM
9
HMHL
18
HHHH
9
HHHM
13,5
HHHL
27
Review of matrix methods for risk assessment...
73
________________________________________________________________________
category for the probability that failure occurs – P,
- low – unlikely – once in 10  50 years - P=1,
- medium – quite likely – once in 1  10 years - P=2,
- high – likely - 1  10 times a year or more - P=3.
 category for consequences – C
 little – perceptible organoleptic changes in water, isolated consumer complaints ,
financial losses up to 5 . 103 EUR - C=1,
 medium – considerable organoleptic difficulty ( smell, significant colour and
turbidity ), consumers health problems, numerous complaints, information in local
media , financial loss up to 105 EUR - C=2,
 large – the endangered people require hospitalisation, professional rescue teams
involved, serious toxic effects in test organisms , information in nationwide media,
financial loss over 105 EUR - C=3,
 category for protection – O.

The questionnaire suggested for the preliminary assessment of SZW protection degree is
given in [5,6]. If the total number of points equals :
 7 ÷ 10 – high protection level - O = 3,
 12 ÷34 – medium protection level - O = 2,
 over 34 – low protection level - O = 1.
In table 5 the risk categories and corresponding point scales are shown.
Table 5. Risk categories
Risk category
Point scale
Tolerable
0,33  r  3,0
Controlled
4,0  r  8,0
Unacceptable
9 r  27
The exemplary application of the method is following:
The probability that the given undesirable event occurs is P = M = 2

Predicted losses are estimated as C = M = 2

The protection level defined on the base of the supplementary questionnaire O = H = 3

The number of the endangered inhabitants using the water pipe N = L = 1

The numerical risk value read from table 4 is: r = 1.33 which, according to table 5,
means the tolerable risk.
74
Rak J., Tchórzewska-Cieślak B.
________________________________________________________________________
6. The five parametric matrix for risk assessment
For very expanded SZW in big city agglomeration it is suggested to use the five
parametric matrix to risk assessment according to the formula ;
PC NE
(5)
r
O
where:
P – point weight connected with the probability that given representative undesirable
event occurs,
C - point weight connected with the size of losses,
N - point weight connected with a number of the endangered inhabitants,
O - point weight connected with SZW protection against the extraordinary threats,
E - point weight connected with the exposure to potential threat.
For the parameters P, C, N, O and E the size level is assigned in the same way as in the
four parametric method. Analogically the risk matrix contains the scale of measures in the
numerical form within the range [0,33  81].. The description of risk components N, P, C
and O is the same as in the four parametric method and it is suggested to assume the
exposure to threat as following:
 Category for exposure – E
- little , a dozen times a year – E = 1,
- medium, several times a week – E = 2,
- high, every day ( constant ) – E = 3.
The point risk scale was presented in tab 6.
Table 6. Risk levels
Risk levels
Point scale
Tolerable
0,33  r  6,0
Controlled
8,0  r  18,0
Unacceptable
24  r  81
7. Conclusions
Water supply system can be counted among the so called critical infrastructure of state,
regions and cities. The ability to estimate risk and to categorise it enables to make right
decisions on its possible reduction. Nowadays the water pipe plants will be forced to
introduce the risk management procedures. The matrix methods for risk assessment are
relatively simple for application so that they are widely used in the risk analysis in SZW.
Review of matrix methods for risk assessment...
75
________________________________________________________________________
They can be modified for different systems depending on the current needs what means
that they can take into account their individual idiosyncrasy . The two and three
parametric matrix can be used at the preliminary risk analyses or for small SZW. The
expanded four and five parametric matrixes should be used by the experts to analyse the
risk connected with water supply systems in the big city agglomerations.
References
1.
Hipel K.W, Kilgour D.M, Zhao N.Z.: Risk analysis of the walkerton drinking water
crisis. Canadian water resources journal.vol.28, no3, p.395-397, 2003.
2. Kempa E.S.: Analiza ryzyka w systemach oczyszczania wód. Ochrona Środowiska,
3(50). Wydaw. PZITS O/Dolnośląski, s. 5-10, Wrocław, 1993.
3. Lubowiecka T., Wieczysty A.: Ryzyko w systemach zaopatrzenia w wodę.
Monografia Komitetu Gospodarki Wodnej PAN „Ryzyko w gospodarce wodnej”, z.
17, s. 113-141, Oficyna Wydawnicza PW, Warszawa, 2000.
4. Łozowska-Stupnicka T.: Ocena ryzyka i zagrożeń w złożonych systemach człowiek –
obiekt techniczny – środowisko. Seria Inżynieria Sanitarna i Wodna. Monografia 270.
Wydawnictwo PK, Kraków, 2000.
5. Mays L.W.: The role of risk analysis in water resources engineering. Department of
civil
and
environmental
engineering.
Arizona
State
University.
www.public.asu.edu/lwmays, s 8-12, 2005.
6. Rak J.: A study of the qualitative methods for risk assessment in water supply systems.
Oficyna Wydawnicza PW, Environment Protection Engineering, z.3-4, s.123-134,
2003.
7. Rak J.: Istota ryzyka w funkcjonowaniu systemu zaopatrzenia w wodę. Oficyna
Wydawnicza PRz, s.1-113, 2004.
8. Rak J., Tchórzewska-Cieślak B.: Czteroparametryczna matryca szacowania ryzyka w
funkcjonowaniu systemu zaopatrzenia w wodę. Wydawnictwo czasopism i książek
technicznych SIGMA NOT, GWiTS, z.2, s.6-9, 2005.
9. PN-EN-1050. Zasady oceny ryzyka, 1999.
10. PN-IEC 60300-3-9. Analiza ryzyka w systemach technicznych, 1999.