This relationship is developed by probabilistic techniques that assess the possibility of accident events Duijm, The central idea is to develop FTA that begins from a pre-event side, and takes into account the post-event for the purpose of predicting GPS risks. Consequently, Shahriar et al. Bow-tie analysis thus utilizes failure probabilities and associated probabilities instead of crisp probabilities. As a consequence of the low occurrence of BE, the probability of the TE can be approximated by the first half of Equation 1.
This calculation enables experts to provide a more accurate opinion about the failure risk of gas connectors. Expert elicitation is the method of synthesizing and integrating expert judgement, as a critical piece of risk assessment modeling, especially in areas with insufficient or implicit information, van der Sluijs et al. These opinions can be synthesized from different formats, including; qualitative expressions description with words to explain causal relationships or potential origin of data uncertainty , quantitative numeration relative or absolute numbers, such as point estimates or data distribution and graphical data display e.
Therefore certain factors must be considered, if the study design involves the elicitation of experts, such as; the type and objective of the information acquired and effects of these on the quality of information collected, choice and availability of experts, level of expert interactions, study costs and other viable constraints. This study primarily evaluated the probability of BE relative to gas connectors from elicited experts.
This method provides valuable information for assessing the potential risks involved with gas connector failures and for making good decisions on the best alternative. The Delphi method was chosen for this study, since it allows for indirect interaction among different experts and enables experts to give their opinion on the reliability of the gas connectors. Delphi method is significant to this study because it allows a diverse group of experts with different experience and levels of education to give their opinion on risk assessment of gas connectors.
The investigator in this study designed an IRB approved No. Experts were categorized based on three criteria; level of education, job title, and service time. Experts in different fields made judgements about probability of events based on their expertise in the field and relative familiarity with the gas connectors. Since expert opinion about the probability of events tend to vary and may sometimes be vague, natural linguistic expressions were applied to characterize the risk associated with events as follows: Very Low VL , Low L , Medium M , High H , and Very High VH. However, since natural linguistic expressions cannot be handled by common mathematics due to their ambiguity, FST was introduced as a possible solution.
Tf N are used as values to represent the linguistic expressions. Experts were selected from a broad spectrum of fields such as higher education institutions, manufacturing, construction, regulatory agencies, maintenance, operations, and management to provide risk assessment of gas connectors. Considering the diverse expert opinions on the reliability of the different gas connectors, a weighting score is introduced to characterize the relative quality of the different experts as shown in Table 2.
Table 2 was incorporated in the survey sent to all the experts. The weighting factors of 10 categorized experts were determined as shown in Table 3. The weighting factor is calculated by dividing the weighting score of each expert by the sum of the weighting scores. The sum of the weighting factors is 1.
The linguistic expressions of the 10 expert categories on the probability of failure of gas connectors are: very low VL , low L , medium M , high H , and very high VH. The experts were grouped into five distinct groups for simplification. Experts with same characteristics were grouped together.
The output of the expert elicitation process involves linguistic terms used by the experts to judge the failure probability of the gas connectors. The conversion scale shown in Figure 4 was used to represent the expert assessment and the membership functions f x of various linguistic terms, i.
When probability theory is inadequate to represent all types of uncertainties, the FST introduced by Zadeh, has recently been applied by researchers. The main difference between FST and traditional set theory is that in the traditional set theory, x is either a member of the set A or not. This study utilizes Tf N to determine the likelihood of occurrence of a risk and to provide quantitative values to uncertainty in expert opinion. The method of converting expert opinion and respective membership functions is as shown in Equation 2. The number assigned for basic risk events are based are based on the 5-granular system and represents the likelihood as shown in Table 4.
Each expert group gives a numerical value on the failure risk of an event, that is, from 1 to 5. An aggregate of these risks is calculated, and a corresponding f N , from the granularity table is assigned. Knowledge acquisition helps in understanding the relationship between risk and failure. Distinct activities involved include; background analysis, literature review, surveys, and expert elicitation.
The most important part of knowledge acquisition is converting expert opinion into failure probability through bow-tie analysis. Since expert knowledge often conflict, opinion aggregation is often utilized to harness opinion from different experts. The most common method used for opinion aggregation is the weighted average method Table 5 shows aggregated for FGC. For example, in Table 4 , five experts presented the failure risk of misuse by the numeric values 4, 4, 5, 3, and 4 respectively. The average numeric risk value is 4. From the 5-granular system in Table 4 , 4 represents a high likelihood and is represented by the f N of 0.
This value is presented in the column for aggregated Tf N. Based on differences of opinion of the probability of basic events, all the opinions are aggregated into a single opinion Equation 3. The outcome of incorporating fuzzy ratings in FTA yields f N and for the relationship between them to be clearly defined, the process of defuzzification is used to convert f N into a crisp real score Q , which is the Fuzzy Possibility Score F ps.
The F ps characterizes the highest potential confidence an expert can express about a basic event occurring. The fuzzy ranking system proposed by Chen and Hwang shown in Equation 6 was used in this study, with the definition of the minimum f min x and maximum f min x sets. Therefore, the two- sided utility score of fuzzy number Q can be calculated from Equations 7 — 8. Using the thermal conductivity property of a material to conduct heat , density property of metals due to the tightly packed crystal lattice of the metallic structure , thermal expansion tendency of matter to change in volume in response to a change in temperature, through heat transfer , temperature, wall thickness, and malleability factors, the probabilities real numbers of the basic events were determined in the FTA.
Other probabilities for the gas connector failures were obtained through the expert elicitation method. The failure probability of other fuzzy events were determined in the same way. When all the probabilities of the basic events were included, the calculated Pr TE from Equation 1 is 5. An important analysis of BE determines I m for the overall analysis and it is a critical process in the quantitative analysis of gas connector failures. In this study, a combination of failure probabilities and expert opinions were integrated to determine the failure risks of the TE of gas connectors.
The influence of BEs on the probability of TE could be assessed quantitatively, since the failure probabilities of BEs are known.
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The important analysis of BE s determined for the gas connectors and presented in Table 6. The importance measure for lightning hazard X 20 was determined as 1. It is important for risk management systems to include risk analysis of the system components. In quantitative risk analysis, risk to a component R i is measured as the product of likelihood of the occurrence of any undesired event L i and the consequence of the corresponding undesirable event C i , as shown in Equation This method is also incorporated in calculating the failure risks of the three types of gas connectors studied, based on the quantitative probabilities and the elicited expert opinions.
The likelihood of the occurrence of an event is given by experts. The Fuzzy-based bow-tie analysis is used to obtain the likelihood of each output event. Fuzzy synthetic evaluation that utilizes a linearized weighting scheme for evaluation is applied to determine the fuzzy risk. A new fuzzy likelihood scale consisting of five linguistic constants as shown in Table 4 is a result of the transformation of the likelihood functions obtained from the bow-tie analysis. Gas connector damages can result in a number of consequences such as: environmental damage, property damage, social damages, and economic damages.
The magnitude of these consequences is influenced by factors such as perforation, rupture, mechanical forces, and operational characteristics. Figure 5 shows the hierarchical structure for consequences of each output event. The leading path is significant for determining the likelihood of each output event if the likelihood of the critical event is known.
Upon failure of the connector, the gas released in a confined space will result to detonation output event—OE1 , if an ignition source is present. The summary of the total risk risk index from OE were calculated based on the extant study by Tesfamaraim and Saatcioglu and presented in Table 7. The fuzzy consequence of OE for human, societal and the economic facts were determined using assigned weights from the FTA.
Risk index above unity indicates quantifiable risk and enhanced precautions are needed for the safety of people, property and the environment. The outcomes of this study are important and would provide further insight to; risk evaluators, investigators and professionals in effectively managing the risks associated with gas connectors, especially in deciding on the choice of materials for utilities and determining appropriate preventive, protective and corrective measures for the overall risk reduction of gas connector failures. Fuzziness or data uncertainties are characteristically present in diverse factors, affecting the outcome of an analysis.
Bow-tie analysis, integrating the elicitation of experts to provide a quantitative estimation of the possibility of the TE to occur as OE, may be used without categorizing most substantial input events. By introducing a sensitivity analysis SA , this study systematically evaluated quantitative information for the purpose of identifying sources of uncertainty, variability and areas of weakness in the risk analysis process for the gas connectors. As shown, this study identified 33 BEs as presented in Figure 2 with the importance measures, presented in Table 6. Iteration trials were modeled for several scenarios, to approximate the likelihood of events that may lead to outcome events and the associated uncertainties were accounted for in the models by assigning fuzzy probability distribution to the BEs.
Assumptions were made for each factor as a quarter of the mean, based on the standard deviation, observed and estimated values. It was also observed the contributions of each of these input events varied based on the dependency relationship, as compared to other factors. With assumption of independent factors relation, the mean absolute percentage error for the different groups of BEs modeled, varied from The multi-dimensional risk model from complex risk scenarios and outcomes were synthesized with the integration of fuzzy synthetic evaluation FSE and fuzzy rule base FRB techniques in this study.
This is effective for the assessment and quantification of both individual risk and the overall risk of failure events on gas connectors, with the goal of understanding the failure activities and planning for the minimization of the potential risks. The analysis of feedback from the expert elicitation method, incorporated with fuzzy set theories FST minimized the uncertainty and fuzziness of the potential failure events. The calculated Pr TE was determined as 5. Based on experts suggestions and analysis, the FGC x is safer for use and have the lowest failure risk compared to BIP. The outcome of this study will be improved on in future research by collecting actual data from gas connectors by type, usage, age, risk factors and condition, to show the applicability of the method used in this study.
The future study will also involve comprehensive sensitivity analysis that will consider the operator aggregates and weights effects as model predictors and integrate the effect of the individual risk events on the overall gas connector system.
The fuzzy risk assessment method is effective for the purpose of this study, however, it is financially expensive and time consuming. It is recommended that its use should be limited to the major accident scenarios requiring high level of details and precision. You are free to: Share — copy and redistribute the material in any medium or format.
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Download PDF. Cite this article as:. Article Figures and tables References. Abstract Abstract Gas connectors are installed in almost every home in the United States and around the world.
Risk Assessment of Factors Influencing Non-Revenue Water Using Bayesian Networks and Fuzzy Logic
Public Interest Statement Most residential or commercial buildings utilize fuel gas connectors FGC to supply gas to compatible appliances. Introduction The technology of gas pipes and connectors, channeled into residential, industrial and commercial buildings, has evolved tremendously since the early s. Characteristics of the investigated FGC The latest generation of FGC come with a protective coating on the outside of the tube to prevent the material.
Stainless steel FGC The leading advantages of the stainless steel FGC over the coated brass connectors include: ability to be used outdoors or in places where appliances are subject to vibration during normal use; ability to withstand the torsion and bend tests; and ability to offer more resistance to corrosion from air, petroleum products, ammonia, fatty oils, grease, household chemicals, and liquid petroleum and natural gases with high sulfur content Oster, Corrugated stainless steel tube CSST is a flexible, polyethylene-covered tubing product which performs the same function as BIP and copper tubing.
Black iron pipe Traditionally, BIP was the most used distributing flammable gases. Table 1. Table 2. Table 3. Weighting factors of 10 expert elicited No. Table 5. Table 6. Consequence analysis of failures to gas connectors in residential buildings. Table 7.
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Conclusion The multi-dimensional risk model from complex risk scenarios and outcomes were synthesized with the integration of fuzzy synthetic evaluation FSE and fuzzy rule base FRB techniques in this study. Funding The author received no direct funding for this research. References Ahmadi-Nedushan, B. Instream flow determination using a multiple input fuzzy-based rule system: A case study. River Research and Applications , 24 , — Multi-attribute risk assessment for risk ranking of natural gas pipelines.
A reliability approach to transmission expansion planning using fuzzy fault tree model. Electric Power Systems Research , 45 , — Fuzzy multiple attribute decision making methods and applications pp. Berlin: Springer. Fuzzy system reliability analysis by interval of confidence. Fuzzy Sets and Systems , 56 , 29— ARAMIS project: A more explicit demonstration of risk control through the use of bow—tie diagrams and the evaluation of safety barrier performance. Journal of Hazardous Materials , , — Safety-barrier diagrams as a safety management tool.
Using fuzzy set theory to address the uncertainty of susceptibility to drought. A basic application might characterize various sub-ranges of a continuous variable. For instance, a temperature measurement for anti-lock brakes might have several separate membership functions defining particular temperature ranges needed to control the brakes properly.
Each function maps the same temperature value to a truth value in the 0 to 1 range. These truth values can then be used to determine how the brakes should be controlled. While variables in mathematics usually take numerical values, in fuzzy logic applications, non-numeric values are often used to facilitate the expression of rules and facts. A linguistic variable such as age may accept values such as young and its antonym old. Because natural languages do not always contain enough value terms to express a fuzzy value scale, it is common practice to modify linguistic values with adjectives or adverbs.
For example, we can use the hedges rather and somewhat to construct the additional values rather old or somewhat young. Fuzzification operations can map mathematical input values into fuzzy membership functions.
And the opposite de-fuzzifying operations can be used to map a fuzzy output membership function into a "crisp" output value that can be then used for decision or control purposes. Fuzzification is the process of assigning the numerical input of a system to fuzzy sets with some degree of membership. This degree of membership may be anywhere within the interval [0,1].
If it is 0 then the value does not belong to the given fuzzy set, and if it is 1 then the value completely belongs within the fuzzy set. Any value between 0 and 1 represents the degree of uncertainty that the value belongs in the set. These fuzzy sets are typically described by words, and so by assigning the system input to fuzzy sets, we can reason with it in a linguistically natural manner. For example, in the image below the meanings of the expressions cold , warm , and hot are represented by functions mapping a temperature scale.
A point on that scale has three "truth values"—one for each of the three functions. The vertical line in the image represents a particular temperature that the three arrows truth values gauge. Since the red arrow points to zero, this temperature may be interpreted as "not hot"; i. The orange arrow pointing at 0. Therefore, this temperature has 0. The degree of membership assigned for each fuzzy set is the result of fuzzification. Fuzzy sets are often defined as triangle or trapezoid-shaped curves, as each value will have a slope where the value is increasing, a peak where the value is equal to 1 which can have a length of 0 or greater and a slope where the value is decreasing.
Fuzzy logic works with membership values in a way that mimics Boolean logic. There are several ways to this. A common replacement is called the Zadeh operators :. There are also other operators, more linguistic in nature, called hedges that can be applied. These are generally adverbs such as very , or somewhat , which modify the meaning of a set using a mathematical formula. However, an arbitrary choice table does not always define a fuzzy logic function. In the paper,  a criterion has been formulated to recognize whether a given choice table defines a fuzzy logic function and a simple algorithm of fuzzy logic function synthesis has been proposed based on introduced concepts of constituents of minimum and maximum.
A fuzzy logic function represents a disjunction of constituents of minimum, where a constituent of minimum is a conjunction of variables of the current area greater than or equal to the function value in this area to the right of the function value in the inequality, including the function value. Given a certain temperature, the fuzzy variable hot has a certain truth value, which is copied to the high variable. The goal is to get a continuous variable from fuzzy truth values.
This would be easy if the output truth values were exactly those obtained from fuzzification of a given number. Since, however, all output truth values are computed independently, in most cases they do not represent such a set of numbers. Since the fuzzy system output is a consensus of all of the inputs and all of the rules, fuzzy logic systems can be well behaved when input values are not available or are not trustworthy. Weightings can be optionally added to each rule in the rulebase and weightings can be used to regulate the degree to which a rule affects the output values.
These rule weightings can be based upon the priority, reliability or consistency of each rule. These rule weightings may be static or can be changed dynamically, even based upon the output from other rules. Many of the early successful applications of fuzzy logic were implemented in Japan. The first notable application was on the subway train in Sendai , in which fuzzy logic was able to improve the economy, comfort, and precision of the ride. Fuzzy logic is an important concept when it comes to medical decision making.
Since medical and healthcare data can be subjective or fuzzy, applications in this domain have a great potential to benefit a lot by using fuzzy logic based approaches. One of the common application areas that use fuzzy logic is computer-aided diagnosis CAD in medicine. Fuzzy logic can be highly appropriate to describe key characteristics of this lesion. Fuzzy logic can be used in many different aspects within the CAD framework. The biggest question in this application area is how much useful information can be derived when using fuzzy logic.
A major challenge is how to derive the required fuzzy data. This is even more challenging when one has to elicit such data from humans usually, patients. How to elicit fuzzy data, and how to validate the accuracy of the data is still an ongoing effort strongly related to the application of fuzzy logic. The problem of assessing the quality of fuzzy data is a difficult one. This is why fuzzy logic is a highly promising possibility within the CAD application area but still requires more research to achieve its full potential.
In mathematical logic , there are several formal systems of "fuzzy logic", most of which are in the family of t-norm fuzzy logics. These extend the above-mentioned fuzzy logics by adding universal and existential quantifiers in a manner similar to the way that predicate logic is created from propositional logic. The semantics of the universal resp.
The notions of a "decidable subset" and " recursively enumerable subset" are basic ones for classical mathematics and classical logic. Thus the question of a suitable extension of them to fuzzy set theory is a crucial one. A first proposal in such a direction was made by E.
Santos by the notions of fuzzy Turing machine , Markov normal fuzzy algorithm and fuzzy program see Santos Successively, L. Biacino and G.
Gerla argued that the proposed definitions are rather questionable. For example, in  one shows that the fuzzy Turing machines are not adequate for fuzzy language theory since there are natural fuzzy languages intuitively computable that cannot be recognized by a fuzzy Turing Machine. Then, they proposed the following definitions. We say that s is decidable if both s and its complement — s are recursively enumerable.
An extension of such a theory to the general case of the L-subsets is possible see Gerla The proposed definitions are well related with fuzzy logic. Indeed, the following theorem holds true provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property. Any "axiomatizable" fuzzy theory is recursively enumerable. In particular, the fuzzy set of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general.
Moreover, any axiomatizable and complete theory is decidable. It is an open question to give supports for a "Church thesis" for fuzzy mathematics , the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. In order to solve this, an extension of the notions of fuzzy grammar and fuzzy Turing machine are necessary. Once fuzzy relations are defined, it is possible to develop fuzzy relational databases. Medina, M. Vila et al. Fuzzy querying languages have been defined, such as the SQLf by P. Bosc et al. Galindo et al. These languages define some structures in order to include fuzzy aspects in the SQL statements, like fuzzy conditions, fuzzy comparators, fuzzy constants, fuzzy constraints, fuzzy thresholds, linguistic labels etc.
The knowledge graph Weaviate uses fuzzy logic to index data based on a machine learning model called the Contextionary. Fuzzy logic and probability address different forms of uncertainty.
While both fuzzy logic and probability theory can represent degrees of certain kinds of subjective belief, fuzzy set theory uses the concept of fuzzy set membership, i. The concept of fuzzy sets was developed in the mid-twentieth century at Berkeley  as a response to the lacking of probability theory for jointly modelling uncertainty and vagueness. Bart Kosko claims in Fuzziness vs. Probability  that probability theory is a subtheory of fuzzy logic, as questions of degrees of belief in mutually-exclusive set membership in probability theory can be represented as certain cases of non-mutually-exclusive graded membership in fuzzy theory.
In that context, he also derives Bayes' theorem from the concept of fuzzy subsethood.