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 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
UNACCEPTABLE RISK CONTROLLED RISK TOLERABLE RISK R I S K L E V E L CRIMINAL RESPONSIBILITY CIVIL RESPONSIBILITY COMPANY RESPONSBILITY LACK OF RESPONSIBILITY R S E P O N S I B I L I T Y S C A L E
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 Category of probability- frequency Category of consequences A B C D E Often Probable Occasional Little probability Improbable F G H I J Catastrophic Serious Significant Marginal Negligible Each time risk ( r ) is determined according to the formula:
r = X . Y (1)
where :
X – frequency of the undesirable events occurrence, Y – consequences of the undesirable events,
Using the formula (1) we can obtain the following possibilities of the undesirable events combinations shown as the risk matrix in fig. 2
ExJ
;
DxJ
;
CxJ
;
BxJ
;
AxJ
ExI
;
DxI
;
CxI
;
BxI
;
AxI
ExH
;
DxH
;
CxH
;
BxH
;
AxH
ExG
;
DxG
;
CxG
;
BxG
;
AxG
ExF
;
DxF
;
CxF
;
BxF
;
AxF
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: [Ax 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 P 1 2r 3 1 1 2 3 2 2 4 5 3 3 6 9
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 5103 EUR ; with weight 1.0
- medium loss from 5103 to 5104 EUR, with weight 3.0
- large loss 5104 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
- 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 5103. The levels of risk in the five stage scale are shown in table 3 .
Table 3. The levels of risk
Class Description Numerical values Risk level 1 2 3 4 5 very little little medium large very large 0,05 < r 5 .... 5 < r 50 50 < r 200 200 < r 400 400 < r 5000 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:
O N C P r (4) 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,
Table 4. The four parametric risk matrix
N P L = 1 C L = 1 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 LLLM0,5 LLLL1 LMLH0,66 LMLM1 LMLL2 LHLH1 LHLM1,5 LMLL3 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 LLHM1,5 LLHM3 LMHH2 LMHM3 LMHL6 LHHH3 LHHM4,5 LMLH9 N P M = 2 C L = 1 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 MLMM2 MLML4 MMMH2,66 MMMM4 MMML8 MHMH4 MHMM6 MHML12 H=3 MLHH 2 MLHM 3 MLHL 6 MMHH 4 MMHM 6 MMHL 12 MHHH 6 MHHM 9 MHHL 18 N P H = 3 C L = 1 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 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
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:
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.
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 ;
O E N C P r (5) 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
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. 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.
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