INTRODUCTION
All power generating machine such as internal combustion engine, gas turbines
and nuclear reactors need efficient cooling system for safe and smooth operation
of the system. The disaster of the explosion of Fukushima Nuclear Reactors in
Japan was caused by breakdown of cooling system in earthquake and tsunami.
For long term running and safe operation of on board diesel engine, the cooling
system is the most important supporting system of the operation of engine. Cooling
systems and temperature control systems are important to maintain the temperature
of engine at most favorable level of energy efficiency and to ensure the long
life of the parts using in diesel engine. On board there are two main types
of cooling system, one is sea water cooling system and the other is fresh water
cooling system also known as central cooling water system. Fresh water central
cooling system can be further divided into low temperature fresh water cooling
system and high temperature fresh water cooling system. Jacket water cooling
system is a high temperature fresh water cooling system and the safe and smooth
operation of this system is based on the operation conditions of both sea water
and fresh water cooling system (Zhou and Xu, 2011; McGeorge,
2002).
To obtain high reliable water cooling system, failure analysis of the cooling
systems is carried out by using different reliability analysis methods from
design stage to operation stage. In doing these kinds of analysis, the most
challenging problem for engineers is to get the exact value of failure probability
of the basic components (basic events). The failure probability of basic components
(basic events) and the reliability of the cooling system depends on factors
such as ship mobility, states of loading and weather conditions. In this study,
Fault Tree analysis is used to make qualitative analysis of the system and Fuzzy
probability is used for assigning the failure probability of basic events of
FTA to overcome the over and under estimation and to get more reliable and reasonable
results in quantitative analysis of the system. Fuzzy probability is used to
determine the fuzzy failure rate of the basic events based on the statistical
data, influencing factors and expert judgments. Fuzzy Fault Tree Analysis (FFTA)
is used to find out the fuzzy probability of the occurrence of the most undesired
events of the system and fuzzy important index is used to discover the critical
components of the system which can lead to major undesired events of the system
(Tanaka et al., 1983). This study presents in
6 parts (1) Fault tree analysis (2) Fuzzy numbers and its operation (3) Fuzzy
operations for gates of FTA (4) Calculation of Fuzzy important measure (5) Reliability
analysis of the jacket water cooling system (6) Result and discussion and (7)
Conclusion.
Overview of the method of study: The study can be divided into two major
parts, the first one describes about the theory and method of study (from part
15) and the second one describes about the results and discussions of the study
(from part 67). The first part presents about the theory of fault tree analysis
and applying of fuzzy probability in fault tree analysis, detail of the type
of fuzzy number used in this study and their operations are shown in part two.
Part three describes about how to calculate the outcomes of logic gates in FTA
by using fuzzy failure probability. Fuzzy probability ranking method and Graded
Mean Integration Representation (GMIR) distance method which are proposed to
use in calculation of the fuzzy importance measures of the components of the
system are explained in part four. The reliability study of marine diesel engine
jacket water cooling system is widely discussed with three separate sections
in part five. The first section describes about the operation of jacket water
cooling system, the second section discusses about the obtaining of fuzzy failure
probability of components used in the system and the third section explains
about the procedure of fuzzy fault tree calculation for the top undesired event
(failure to maintain desired temperature of jacket cooling water system). And
the results, fuzzy failure probability of jacket water cooling system and fuzzy
important measures of the components of the system which are calculated by two
different methods mentioned in part four, are described in the section of results
and discussions (part 6). Finally the study is concluded by recommending how
this study can be dealt with two major aspects in reliability analysis of any
onboard system and reveals the most critical and least critical components of
the diesel engine jacket water cooling system.
FAULT TREE ANALYSIS
Fault Tree Analysis (FTA) is a powerful method to calculate the probability
of the undesired outcomes of a system at the same time it can calculate how
importance of the basic events causing the major undesired events (Vesely
et al., 1981). Based on the combination of different failure causes
such as hardware failure, common cause failure, environmental impact and human
error, the probability of the undesired outcome is calculated (Yuhua
and Datao, 2005). Logic gates are used as nodes in combination of basic
events which can give middle events and then up to reach the target undesired
top event. OR gate AND gate and NOT gate are the most common static gates used
by FTA as well as in Fuzzy FTA (FFTA). Although, FTA users can easily understand
and clearly find out the causes of top undesired events from FTA systematic
diagram, FTA has weak points in solving problems with uncertain failure rates
of basic events. To overcome these problem researchers combined Fuzzy concept
to FTA and invented Fuzzy Fault Tree Analysis (FFTA) method to solve the problem
of uncertain failure rates of basic events (Mentes and
Helvacioglu, 2011; Ma et al., 2011; Wang
et al., 2011). This study used the integrated concept of FFTA to
calculate the fuzzy probability of undesired major event.
FUZZY NUMBER
Fuzzy probability is a fuzzy number which is expressed by a fuzzy set (Huang
et al., 2001) and characterized by its membership function μ.
Let Ãε(∞, +∞). Tanaka et
al. (1983) described about the LR fuzzy number, suppose L and R as
referring functions of the fuzzy numbers. If:
then Ã is called LR fuzzy number. The parameter m is the mean of the
fuzzy number Ã and α, β is the left and right expansion of
Ã which can be represented as Ã = (m, α, β)_{LR}.
The membership function of Ã is equal to zero when x≤mα or x≥m+β
that means if x is out of the range of (mα, m+β), it does not belong
to Ã.
Fuzzy probability can be represented by a triangular or trapezoidal shape or
bell shaped membership function. Generally triangular and trapezoidal membership
functions are widely used to represent the failure nature of equipments. Triangular
membership functions are used to represent more or less probability estimation
of failure occurrence (e.g., the probability of failure occurrence is about
0.0001) and a trapezoidal membership function is better to used in describing
failure probability interval of equipments (e.g., the probability of failure
occurrence lies between 0.00001 and 0.000025). Due to the failure nature of
cooling system equipments, triangular fuzzy number will be used in further calculations
of this study. Details properties and calculation of triangular fuzzy number
will discuss in following sections.
Triangular fuzzy number: A fuzzy number Ã is termed as triangular
fuzzy number if the membership function of fuzzy number Ã is defined
by the following Eq. 2. A triplet (a, m, b) or Ã =
(ma, m, bm) can be represented the triangular fuzzy number. The left and right
expansions of triangular fuzzy number and the confidence level of probability
of uncertain events can be obtained from statistical data and expert judgment
of the system:
Algebraic operation of fuzzy number: Let
be two fuzzy number .
The basic arithmetic operations of two fuzzy numbers are shown in Eq.
34 and operation between crisp number and fuzzy number
is shown by Eq. 5:
In multiplication of two triangular fuzzy numbers, the following Eq.
6 is used instead of the real product of two triangular fuzzy numbers which
is not a triangular fuzzy number (Mao et al., 2010):
FUZZY OPERATORS
Conventional fault tree methods used Boolean operators AND, OR and NOT gates
with precise probability of basic events to calculate the probability of top
event. If a fuzzy event ‘i’ is represented by a possibility function
fuzzy operators are used to calculate the probability of top event (Ferdous
et al., 2009). Generalized Fuzzy Boolean NOT operator can be denoted
as
and defined as:
In FTA, the result of the AND gate and OR gate operation of the précised
probabilities of basic events G_{i} (where i = 1, 2, 3,..., n) can
be represented as:
and their fuzzy operations can be represented as:
FUZZY IMPORTANCE MEASURE
In conventional FTA, there are three kinds of importance measures, Birnbaum
importance measure, Criticality importance measure and FussellVesely importance
measure, are carried out to know the structural importance, integrated structural
and the reliability importance of basic component or event (criticality Importance)
and importance of the contribution a basic event in different cut sets. In Fuzzy
FTA, fuzzy importance measure of a basic event or component is calculated by
measuring the difference between two fuzzy probabilities of the top event of
a fault tree with and without existence of that basic event (Tyagi
et al., 2010).
In this study, two new fuzzy importance measure methods will be introduced
and compare the results with each other to verify the results.
in Eq. 12 denotes the probability of absolute occurrence
of top event and in
Eq. 13 is the probability of occurrence of top event in absence
of basic event i. It can be shown as follow:
The first proposed method of fuzzy important measure can be executed by ranking
of fuzzy probability of the top events
by applying the main idea of the ranking of fuzzy number method. Detail of this
method is explained in the reference (Thorani et al.,
2012). First the incentre of the triangular fuzzy number
can be found by Eq. 14. The rank of the fuzzy probability
R()
can be calculated by using Eq. 15:
Where:
The higher the rank of the fuzzy number the less fuzzy importance measure of
the basic event for the system. The second proposed fuzzy important measure
of basic events can be evaluated by using fuzzy distance method. First calculate
the fuzzy number (
 )
where i = 1, 2, 3,... for all basic event and find the maximum fuzzy number
of (
 )
Then the fuzzy distance between each fuzzy number (
 )
and the maximum fuzzy number of (
 )
are calculated by Graded Mean Integration Representation (GMIR) distance method
(Chen and Wang, 2006). GMIR of a triangular fuzzy number
(
 )
can be found as follow:
And the distance between two fuzzy numbers can be defined as:
Then the fuzzy important measure can be found by using Eq. 18:
RELIABILITY ANALYSIS OF JACKET WATER COOLING SYSTEM
System analysis of the jacket water cooling system: Jacket water cooling
system cools the most important working parts of a diesel engine such as cylinder
jacket, piston heads, pistons, exhaust valves. Jacket water cooling system locates
in the centre of heat exchanging system. The heat from engine is removed by
High Temperature fresh water cooler (H.T. F.W), the heat from H.T.F.W cooler
is removed by fresh water central cooler and the heat from central cooler is
removed by sea water. The detail analysis of sea water cooling system is not
included in the scope of reliability analysis of the jacket water cooling system.
Components include in this analysis is as shown in Fig. 1.
Failure in temperature control of jacket water cooling system is caused by two
main factors (1) Primary water cooling system (High Temperature Fresh Water
cooling system) and (2) Secondary water cooling system (Low Temperature Fresh
Water Cooling system).
Main Engine Jacket water piping, H.T F.W cooler, circulation pumps, automatic
control three way valves, temperature sensor and controller, automatic pump
controller, H.T F.W high pressure tank, level sensors and water pump are basic
components of the primary water cooling system. Central cooler, lube oil cooler,
circulations pumps, automatic control three way valves, temperature controller,
automatic pump controller, control air supply and temperature sensors are major
components of primary fresh water cooling system. Fault tree diagram of the
failure of the temperature control of jacket water cooling system is as shown
in Fig. 23.
Method of evaluating the fuzzy probability of basic events: Industrial
databases are not productspecific or applicationspecific; that is, they do
not distinguish between harsh or mild environments, process conditions, or levels
of maintenance and inspection.

Fig. 1: 
Components include in the analysis of jacket water cooling
system. This figure describe the overview of the central water cooling system
or fresh water cooling system using onboard and it also shows the clear
boundary between low temperature fresh water cooling system and high temperature
fresh water cooling system. Heat from jacket is removed by H.T.F.W cooling
system and heat from H.T.F.W is removed by L.T.F.W cooling system and heat
from L.T.F.W is removed by sea water cooling system which is not include
in this analysis 

Fig. 2: 
Fault tree diagram of the jacket water cooling system. This
figure describes about the overall fault tree diagram of jacket water cooling
system. It includes both basic events and transfer gates. Detail of transfer
gates are described in Fig. 3. And the descriptions of
transfer gates are described in Table 2 
If the systems are well maintained like on board systems, equipments are rarely
to reach their failure states. But failures are still occurred in both main
engine and auxiliary systems of ships. In the reliability analysis of jacket
water cooling system, failure probability of basic events are obtained from
the manufacturer and the reliability data handbook of Offshore Reliability Engineering
Data handbook (OREDA) which can give the failure data with the most similar
operational environment with onboard operation (SINTEF Industrial
Management, 2002) and also from other reliability data handbook such as
NonElectrical Parts Reliability Data (Denson et al.,
1991). Based on these data expert judgments are added to get the left and
right fuzzy numbers of basic events. In this case, expert judgments are done
by chief engineer, second engineer, third engineer and electrical engineer who
had been worked on board. In addition to their expert knowledge, strong records
of ship maintenance system (PMS, Prevented Maintenance System) are fed to experts
to obtain the expert decision of the system.
Different kind of methods such as mean, median, maximum and mixed operators,
etc., are in use to evaluate the fuzzy opinions of experts.

Fig. 3(ae): 
Transfer gates of the jacket water cooling system (a) Transfer
gate (A), (b) Transfer gate (B), (c) Transfer gate (C), (d) Transfer gate
(D) and (e) Transfer gate (E). This figure describes about detail of each
transfer gate. Each transfer gate is composed with logic gates and basic
events which includes in that system. Name of each basic events and their
fuzzy probability of failure of each basic event is described in Table
1. Boolan algebra and fuzzy probability of failures are used to calculate
the result of each transfer gate 
In this study, the weighted a verage is used because of judgments come from
different level of engineers. Let the triplet G_{ij} = (a_{ij},
m_{ij}, b_{ij}) represent the triangular fuzzy numbers, where
i = 1, 2, 3,..., n and j = 1, 2, 3, 4 are the fuzzy probability given to
even i by expert j. The expression for aggregating the judgments of experts
in a fuzzy number is:
where, G_{i} represents the weighted average fuzzy probability of the
event i and W_{j} denotes the weight of the judgment of engineers. Fuzzy
probabilities of the basic events are given in Table 1.
Fuzzy fault tree calculation of the top event: Detail descriptions of
the transfer gates and middle gates of the fault tree diagram are explained
in Table 2 and 3. By using fault tree analysis
method fuzzy probability of the top gate can be found as follow:
Table 1: 
Fuzzy probability of failure of the basic events of jacket
water cooling system 

This table describes about the triangular fuzzy No. of probability
of failure of each basic events (failure h^{1}). These data are
got from the combination of failure rate data from hand book (OREDA offshore
engineering reliability data handbook, 2002 and non electrical product reliability
data handbook, 1991) and experts’ judgments from engineers worked on
board. Weighted average method as shown in Eq. 19 of
main text is used to calculate the average of fuzzy probability of failure 
Table 2: 
List of the transfer gates with respect to their function
of failure calculation 

This table describes the list of transfer gates which are
used to calculate the subsystem of the fault tree system as shown in Fig.
2 
By applying the Eq. 11 for calculating the fuzzy probability
of transfer gate A:
Table 3: 
List of the gates with respect to their function of failure
calculation 

This table describes the list of the logic gates of the fault
tree system. Lower level of logic gates are composed of basic events and
upper level logic gates are composed of both basic events and logic gates
as shown in Fig. 3 
Applying Boolean algebra the fuzzy probability of the top event can be expressed
as follow:
RESULTS AND DISCUSSION
Fuzzy failure probability of the top event: Fuzzy probability of all
transfer gates can be found by using fuzzy gates equations and the fuzzy probability
of the top undesired event can be calculated by using the results of these transfer
gates. Eq. 20 is used to calculate the top event (failure
to maintain the desired temperature of the jacket water cooling system) and
the result can be shown as a triangular fuzzy number (0.001252095, 0.001589956,
0.001927729) and the fuzzy probability of the final event in failure probability
per hour can be shown as follow:
Fuzzy importance measure of basic events: For calculating the fuzzy
importance measure of the basic events, the probability of occurrence of top
event in absence of basic event x_{i}, i = (1, 2, 3,..., 25, 26)
is calculated as shown in Table 4. From these results, it
can be known the level of importance of the component (basic event) to cause
the undesired top event. In other words, this table can show the importance
of the contribution a basic event in different cut sets of the system. But these
results are in triangular fuzzy numbers, so they cannot be compared as real
number.
Table 4: 
Fuzzy failure probability of the top event in the absence
of the basic events x_{i} 

This table describes about the fuzzy probability of failure
of top event (failure to maintain the desired temperature of the jacket
water cooling system) without considering the failure effect of each basic
event includes in the system. From these results, it can be know the level
of importance of the component (basic event) to cause the undesired top
event. In other words, this table can show the importance of the contribution
a basic event in different cut sets of the system. But these results are
in triangular fuzzy numbers, so they cannot be compared as real number.
Fuzzy number ranking methods are used to determine the level of importance
of each basic event and the result is shown in Table 5 
Fuzzy number ranking methods are used to determine the level of importance
of each basic event.
Two fuzzy importance measures are calculated by applying fuzzy probability
ranking method and Graded Mean Integration Representation (GMIR) distance method
and the detail comparison of the results are shown in Table 5.
From the results, it can be seen that both proposed methods give same ranks
of fuzzy importance measure for basic events and the result can also be comparatively
verified the results of the two proposed methods. As a qualitative analysis
it can be clearly observed that the highest failure rate event of control air
supply X_{9} has the highest fuzzy importance measure as well as the
most critically important event of the system. Due to the redundant configuration
of H.T F.W pumps and L.T F.W pumps, they have the lowest fuzzy important measure
even though their failure rates are higher than some of the equipments. The
ranks of fuzzy importance measures of other basic events or equipments which
are connected with OR gates from bottom layer to top undesired event are found
out with their order of failure rates.
Table 5: 
Comparison of fuzzy importance measure of the basic events
of jacket water cooling system 

This table shows the level of importance of the component
(basic event) to cause the undesired top event by comparing fuzzy failure
probability by using two different methods. The second column shows the
result of using ranking fuzzy number method. The lower the rank, the higher
the importance of the component for the system is. The third column shows
the result of using fuzzy distance method. The larger the distance, the
higher the importance level of the component for the system is. From the
results, it can be seen that both proposed methods give same ranks of fuzzy
importance measure for basic events and the result can also be comparatively
verified the results of the two proposed methods 
CONCLUSION
In this study, fuzzy importance measure methods are presented to deal with
two major aspects in reliability analysis of any onboard system. These are as
follow:
• 
Describing the fuzzy importance measure evaluating procedure
for a system in the case of operating different environments, process conditions,
levels of maintenance and inspection and incomplete information of maintenance
data 
• 
To determine the critically importance of basic events by using fuzzy
importance measure to improve the system reliability, availability and planning
of future maintenance and inspection works 
From the reliability analysis of jacket water cooling system has high reliability
and it can be concluded that the repeated basic event X_{9}, failure
of supply air, is the highest rank of fuzzy importance measure for the occurring
the top event of the system. It further more shows the redundant installations
of water circulation pumps are lowest in fuzzy important measure of the system.
To improve the reliability of cooling water system, availability of supply air
system should be well maintained and monitored, because it is not only high
in fuzzy important measure but also the most repeated event of the system.
ACKNOWLEDGMENT
The authors acknowledged the financial support of the National High Technology
Research and Development of China (863 Program) (No. 2013 AA040203).