Markov model of risk in the life safety

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©V.D. Gogunsky, Yu.S. Chernega, E.S. Rudenko, 2013
УПРАВЛІННЯ ПРОЕКТАМИ
PROJECT MANAGEMENT
UDC 65.012.123:504.75 V.D. Gogunsky, Dr. tech. sciences, prof.,
Yu.S. Chernega, ing.,
Odesa nat. polytechnic univ.,
E.S. Rudenko, master,
Sea trade port “Yuzhny”
MARKOV MODEL OF RISK IN THE LIFE SAFETY
PROJECTS*
В.Д. Гогунський, Ю.С. Чернега, E.С. Руденко. Марківська модель ризику в проектах безпеки
життєдіяльності. Запропоновано модель для оцінки ризику в проектах і програмах у сфері безпеки жит-
тєдіяльності на основі подання оцінки у вигляді однорідного марківського ланцюга з дискретними ста-
нами і часом. Застосування запропонованої моделі дозволяє перейти від одновимірних до багатовимір-
них оцінок рівня безпеки організаційно-технічних систем.
Ключові слова: ризик, оцінка, система, стани, марківський ланцюг, управління проектами.
В.Д. Гогунский, Ю.С. Чернега, E.В. Руденко. Марковская модель риска в проектах безопасности
жизнедеятельности. Предложена модель для оценки риска в проектах и программах в сфере безопасно-
сти жизнедеятельности на основе представления оценки в виде однородной марковской цепи с дискрет-
ными состояниями и временем. Применение предложенной модели позволяет перейти от одномерных к
многомерным оценкам уровня безопасности организационно-технических систем.
Ключевые слова: риск, оценка, система, состояния, марковская цепь, управление проектами.
V.D. Gogunsky, Yu.S. Chernega, E.S. Rudenko. Markov model of risk in the life safety projects. We
propose a model to assess the risk in projects and programs in the field of safety of life through the provision of
assessment in the form of a homogeneous Markov chain with discrete states and time. The application of this
model allows us to go from one-dimensional to multidimensional assessment of the security of organizational
and technical systems.
Keywords: risk, assessment, system, states, Markov chain, project management.
Introduction. Under the project as “a time-limited purposeful change a single system with the re-
quirements for quality results, possible limits flow of funds and resources and a specific organization” [1].
A key part of building projects in the life safety (LS) is a goal-setting - the rationale and objectives of
the project selection [2].
Formulation of the problem. This raises the problem of management of complex multi-criteria
system including facilities, planned process and many of the project participants, each of which is fo-
cused on their own goals [2]. The proved choice of the objective function determines the efficiency of
the projects LS [3].
Analysis of the publications. Fundamentals of Project Management using the Balanced Scorecard
are proposed in [1]. The risk of reduced life under the influence of harmful factors in the working area
* Research performed by the plan the state of research “Theory and practice competence of project management in organiza-
tional-technical and social systems”, registration № 73-32 (2012)

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of enterprises is based on the well-known Weber-Fechner law [2]. Other approaches for risk assess-
ment developed are in [3...5]. The theoretical basis for the formation of the concept of security in the
organizational and technical system is an axiom of the potential hazard, the principle of Farmer, the
Weber-Fechner law, the principle of minimum Liebig, Shelford tolerance law [4]. The level of influ-
ence of environmental factors on the condition of people is based on the data in the form of “dose —
effect” [6]. In general, the result of a project in LS is some product or service, which are characterized
by cost and risk measure.
The purpose of the article. Development of model representations of the projects objective func-
tion in safety.
Features of the risk assessment. Management of projects and programs in LS is - a complex form
of project activity in the “man — machine — environment” system.
These projects (programs) have the similar to any other project features:
— goal — the safety of operating personnel and the public;
— certain deadlines (start and end);
— established financial, material and human resources.
Differences projects LS [1]:
— the objective function of the project — a technological and environmental security — shows
the desired property of a production environment or territory, and is not related to profit from the pro-
duction and sale of a product or service;
— uniqueness — geographical and climatic characteristics, method of formation, mechanisms of
harm and spread of harmful substances in the environment.
There are two types of governance for LS — proactive and reactive. The difference between
them lies in the fact that the external action on the part of the project team in the first instance is car-
ried out before, while the second – after the accident, which may occur as a result of accidental causes.
Currently dominates the second variant, that usually triggers certain decisions to change the system to
reduce the risk of adverse events in the future.
Methods for measuring and assessing risk are manifold. This is due to the fact that in different
domains of life different parameters to evaluate the efficiency of objects and processes are used. Thus,
while treating a patient doctor controls the “medical risk” that characterizes the success of surgery or
therapy. Civil engineer manages the risk of an object under construction structural failure. Profession-
als in the field of occupational health and environmental safety are focused on risk assessment and
management system to reduce it. They act on behalf of enterprises, customers, the community, society,
or life in general. Risk is mainly assessed by the probability curve (a dimensionless value between 0
and 1), but the frequency of implementation also can be used. The frequency of implementation is — a
number of cases of possible manifestations of risk during a particular period of time.
There are two long-standing views on risk: the first is based on scientific and technical assess-
ments (the so-called theoretical risk), the second depends on the human perception of risk (the so-
called effective risk). These two points of view permanently conflict in social, human and politi-
cal sciences.
The theoretical risk is expressed in the form of statistics that often comes down to the probability
of an undesirable event. Usually, the probability of such events, and some estimate of expected harm
is combined into one plausible result, which combines a set of probabilities of risk, regret and reward
in the expected value for this result.
Thus, in the statistical decision theory, risk assessment function δ(x) for the parameter θ, calcu-
lated for some of the observed parameters x, is defined as the mathematical expectation of the loss
function L
() (, ()) ( | )
R
Lxfxdxθ= θδ ⋅ θ
∫, (1)
where (,())Lxθδ — the loss function of the estimate θ parameter and the estimate δ(x) value; ( | )
f
xθ —
the probability of an undesirable event.

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In practice, as a rule, use particular forms of expression (1) are used, consisting in the fact that the
relationship is greatly simplified, given the specific conditions of the implementation of risk assessment.
The probability of an undesired event is determined by the frequency of implementation of hazards
()
(|) ()
N
t
fx Qx
θ= , (2)
where N(t) — the number of adverse events during the period t; Q(x) — the total number of events in
the system.
Thus, the assessment of health risk when exposed to environmental factors are performed under
the assumption that the level of pollution of the air or the water environment is known [6]. This means
that the pollution event has already taken place: ( | )
f
x
θ
=1.
For the loss function ( , ( ))Lxθδ usually take some measure of the cost of risk units is taken, which
characterizes the effects of an event. Thus, the fatal accident entails paying an employee family of
five-years salary. Thus, the cost measure of risk in terms of the employer may be taken to be the size
of a five-year earnings of the employee. Such a valuation can be established for other levels of severity
of consequences of adverse events.
The present study proposes not to use a measure of the cost of risk, since it is an expression of the
level of development of social relations, and not characteristic of the quality of organizational and
technical system. To characterize the effects, you can use the value of the probability of consequences,
taking into account their severity. This is a ratio of the number of adverse events (number of deaths,
number of cases of disease, disability, etc.) caused by the action on the population of a particular ha-
zard to the total possible number
1
(,()) i
ii s
i
i
n
pL x
n
=
=θδ =
∑
, (3)
1
1
s
i
i
p
=
=
∑, (4)
where i — condition index (events) to determine the severity of the population, i=1…s; pi — proba-
bility state, 0<pi<1; ni — the number of events of a certain severity in the population.
Under these provisions risk is the product of two variables having probabilistic nature. Further, in
this study we consider the characterization of the effects of exposure to adverse environmental factors
for ( | )
f
xθ=1.
Negative aspects of the factors of the “man — machine — environment” are implemented
through the events that serve as indicators of their potential manifestation and contribute to the realiza-
tion of potential dangers in the future. On the recommendation of the International Organization of
Health different levels of damage and, therefore, the probability of impact can be determined by the
structure of events: D1 — norm; D2 — minor violations; D3 — indirect evidence; D4 — denial; D5 —
accident
Represented as a graph of transitions from one state Di to another Dj, which denote the probabili-
ties { 1,5; 1,5; }
ij ijijπ= = ≠ of transitions to other states, as well as the probability
{ 1,5; 1,5; }
ij ijijπ= = =of saving the current state (Fig. 1). These transitional probabilities can be de-
termined by expert methods. The resulting homogeneous Markov chain with discrete states and the
time can be solved by well-known methods [3].
Additionally, compared to the pyramids recommended by the Health International society, on a
graph marked switch to less dangerous levels of damage in one, two or even three levels, which may
take place in case of organizational, preventive or therapeutic measures.
We describe a homogeneous Markov chain with discrete states and discrete time steps and
changes calculated using the probabilities of the states. [4] Under the steps we understand some com-
plex-effects of measures implemented by an object that changes the parameters Di.

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Suppose that at any time t (after any k-th step) index D can be in one of the states: D={D1, D2, …
D5}, that is, to make one of the full group of incompatible events: D1(k), D2(k), …, D5(k). The indicator
D may vary at each step k
D={p1(k), p2(k), …, p5(k)}.
Denote the probability of finding an object in the states j: j =1, n in the completion of steps k
k=1; p1(1)=P(D1(1)); p2(1)=P(D2(1)); …., p5(1)=P(D5(1)).
k=2; p1(2)=P(D1(2)); p2(2)=P(D2(2)); …., p5(2)=P(D5(2));
………………………………………………………….
k=l; p1(l)=P(D1(l)); p2(l)=P(D2(l)); …., p5(l)=P(D5(l)).
Probabilities p1(k), p2(k), …, p5(k) are the probability of the state of a homogeneous Markov
chain, in which the transition probabilities are independent of the step. Considering the property of
incompatible actions, forming a complete set for each step k
p1(k) + p2(k) + … + p5(k) =1.
The transition probabilities πik {i= 1…n; k= 1…n; n=5} can be obtained by expert methods. “A
delay” πii, complement to one the sum of the transition probabilities of transition to other states.
Based on the matrix of transition probabilities, provided that the initial state of the system is
known, it is possible to find the state probabilities p1(k), p2(k), …p5(k) after each k-th step of manage-
ment actions on the system. The equations for calculating the probabilities of the form:
1 1 1.1 2 2.1 3 3.1 4 4.1 5 5.1
2 1 1.2 2 2.2 3 3.2 4 4.2 5 5.2
3 1 1.3 2 2.3 3 3.3 4 4.3 5 5.3
41
(1) () () () () () ,
(1)() () () () (),
(1) () () () () () ,
(1) (
pk pk p k pk pk p k
pk pk pk pk pk pk
pk pk pk pk pk pk
pk p
+ = ⋅π + ⋅π + ⋅π + ⋅π + ⋅π
+=⋅π+⋅π+⋅π+⋅π+⋅π
+ = ⋅π + ⋅π + ⋅π + ⋅π + ⋅π
+= 1.4 2 2.4 3 3.4 4 4.4 5 5.4
5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
)()()()(),
(1) () () () () () .
kpkpkpkpk
pk pk pk pk pk pk
⋅π+⋅π+⋅π+⋅π+⋅π
+= ⋅π+ ⋅π+ ⋅π+ ⋅π+ ⋅π
(5)
The system (5) has 10 variables — transition probabilities πij are known. Five equations include
10 unknown variables. Solving this system requires the number of equations to be equal to the number
of unknown variables, ie five ties must be added, based on the initial conditions. Usually the state
probabilities pi(k), i=1…5 are set as known variables.
Behavior of the system is determined by the initial state {p1(0), p2(0), … p5(0)} and the transition
matrix, which, for each new project has different values of its elements:
1,1 1,2
2,1 2,2 2,3
3,1 3,2 3,3 3,4
4,4 4,5
4,1 4,2 4,3
5,4 5,5
000 0,75 0, 25 0 0 0
00 0,15 0, 75 0,10 0 0
00,05 0,15 0,65 0,15 0
0,02 0,05 0,15 0,63 0,15
0000,250,75
000
ij
ππ
πππ
ππππ
π= =
ππ
πππ
ππ
.
Simulation results example using a Markov chain shows the possibility of a multi-dimensional
assessment of the probability of occurrence of certain events (Fig. 2).
The system can be changed by the influence of perfomers. This is possible by using, for example,
health and safety law, socio-economic, organizational, technical, sanitary and preventive measures
aimed at protecting the health and efficiency of employees [4]. It is necessary to ensure the proper
management of projects in LS. After all the project can go into the category of “bad” and not complete
at all if any errors or project team improper actions take place [5].

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D
1
D
2
D
3
D
4
D
5
π
11
π
22
π
33
π
23
π
34
π
43
π
44
π
55
π
54
π
45
π
41
π
31
π
21
π
42
π
32
π
12
p
i
(k)
0,8
0,6
0,4
0
0,2
10 20 30
k
1
2
345
Fig. 1. A labeled graph of the system: Di — the dis-
crete states; πij — transition probabilities
Fig. 2. Changing the state probabilities of the system
during the work projects: pi(k) — the probability of
state, i=1…5, k — step number — the discrete time
Conclusions. The proposed model reflects the Markov processes in LS. System structure and topol-
ogy of relationships allow to draw preliminary conclusions about the concept of proactive systems man-
agement LS — do not expect serious consequences and accidents. You must use the indicators of levels
of damage occurrence and form events, and introduce new methods for preparation and healthy life.
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©Т.М. Олех, К.В. Колесникова, С.В. Руденко, 2013
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Reviewer Dr. tech. sciences, Prof. of Odesa nat. polytechnic univ. Stanovsky A.L.
Received April 1, 2013
УДК 504.75 Т.М. Олех, математик,
Е.В. Колесникова, канд. техн. наук, доц.,
Одес. нац. политехн. ун-т,
С.В. Руденко, д-р техн. наук, проф.,
Одес. нац. морской ун-т
ЭКОЛОГИЧЕСКАЯ ОЦЕНКА ПРОЕКТОВ
Т.М. Олех, К.В. Колесникова, С.В. Руденко. Екологічна оцінка проектів. Запропоновано підхід до
виявлення екологічних проблем на ранніх стадіях розгляду проектів для включення в них заходів, спря-
мованих на поліпшення якості навколишнього середовища та запобігання, зменшення та компенсацію
екологічного збитку. Розглянуто оцінку якості екологічної складової проекту.
Ключевые слова: проекти, екологічні характеристики, вплив на навколишнє середовище, якість еко-
логічної оцінки.
Т.М. Олех, Е.В. Колесникова, С.В. Руденко. Экологическая оценка проектов. Предложен подход к
выявлению экологических проблем на ранних стадиях рассмотрения проектов для включения в них меро-
приятий, направленных на улучшение качества окружающей среды и предотвращение, уменьшение и ком-
пенсацию экологического ущерба. Рассмотрена оценка качества экологической составляющей проекта.
Ключові слова: проекты, экологические характеристики, влияние на окружающую среду, качество
экологической оценки.
T.M. Olekh, K.V. Kolesnikova, S.V. Rudenko. Environmental assessment of projects. An approach to the
identification of environmental problems at the early stages of the projects to be included in the draft of meas-
ures aimed at improving the quality of the environment and the prevention, reduction and compensation for envi-
ronmental damage. The evaluation of the quality of environmental component of the project is considered.
Keywords: projects, environmental performance, the impact on the environment, the quality of environ-
mental assessment.
Введение. Цель экологической оценки (ЭО) состоит в обеспечении качества проектирования,
чтобы проекты, планы развития, программы и др. были приемлемы с точки зрения окружающей сре-
ды и устойчивости. ЭО используется для прогноза, анализа и интерпретации значимых воздействий